Matrix Factorizations of the Classical Discriminant by Bradford Hovinen A

Matrix Factorizations Of The Classical Discriminant

Bradford Hovinen

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto

Matrix Factorizations Of The Classical Discriminant

Bradford Hovinen Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2009

The classical discriminant Dn of degree n polynomials detects whether a given univariate polynomial f has a repeated root. It is itself a polynomial in the coefficients of f which, according to several classical results by Bézout,Sylvester, Cayley, and others, may be expressed as the determinant of a matrix whose entries are much simpler polynomials in the coefficients of f. This thesis is concerned with the construction and classification of such determinantal formulae for Dn.

In particular, all of the formulae for Dn appearing in the classical literature are equivalent in the sense that the cokernels of their associated matrices are isomorphic as modules over the associated polynomial ring. This begs the question of whether there exist formulae which are not equivalent to the classical formulae and not trivial in the sense of having the same cokernel as the 1 × 1 matrix (Dn). The results of this thesis lie in two directions. First, we construct an explicit non-classical formula: the presentation matrix of the open swallowtail ﬁrst studied by Arnol’d and Givental. We study the properties of this formula, contrasting them with the properties of the classical formulae.

4 2 Second, for the discriminant of the polynomial x + a2x + a3x + a4 we embark on a classiﬁcation of determinantal formulae which are homogeneous in the sense that the cokernels of their associated matrices are graded modules with respect to the grading deg ai = i. To this end, we use deformation theory: a moduli space of such modules arises from the base spaces of versal deformations of certain

4 3 modules over the E6 singularity {x − y }. The method developed here can in principle be used to classify determinantal formulae for all discriminants, and, indeed, for all singularities.

ii Acknowledgements

The author greatly thanks his thesis advisor, Ragnar-Olaf Buchweitz, for all of his generous support over the past four years. He would like to thank Hubert Flenner, Nagat Karroum, Michael Kunte, Paul Cadman, David Mond, David Eisenbud, Greg Smith, Steve Kudla, Jan Christophersen, and Klaus Altmann for their valuable discussions. Thanks also go to Mike Stillman and Dan Grayson for their assistance with the use of Macaulay 2.

iii Contents

1 The Classical Discriminant 1

1.1 Introduction ...... 1

1.2 A local perspective: Unfoldings ...... 4

1.3 A global perspective: Hilbert schemes ...... 7

1.4 The geometry of ∆n ...... 8 1.5 Structure of this thesis ...... 11

2 Matrix Factorizations 13

2.1 Classical determinantal formulae ...... 14

2.2 Eisenbud’s Theorem ...... 20

2.3 The Cayley Method ...... 24

2.4 On the ranks of the presentation matrices of Σ¯ n,k ...... 35

3 The Open Swallowtail 38

3.1 An algebraic deﬁnition ...... 38

3.2 Construction of the presentation matrix ...... 42

3.3 The construction of Arnol’d ...... 51

3.4 The conductor of the open swallowtail ...... 52

3.5 Application to the root structure of a univariate polynomial ...... 56

4 Deformations of Modules 57

4.1 Deformation theory of modules ...... 58

4.2 Computing versal deformations ...... 60

4.2.1 Liftings and obstructions ...... 62

4.2.2 The ﬁrst-order deformation ...... 65

4.2.3 Further deformations ...... 68

iv 4.3 Rank one MCM modules over xn − yn−1 ...... 71

4.4 Example: Graded modules over ∆4 ...... 76

4.4.1 A classiﬁcation of rank one graded MCM modules over ∆˜ 4 ...... 86 4.5 Structure of Maximal Cohen-Macaulay Modules ...... 87

5 Conclusion 91

A Macaulay 2 package for Module Deformations 93

Bibliography 100

v Chapter 1

The Classical Discriminant

1.1 Introduction

Fix a ﬁeld K of characteristic not equal to two and let

n n−1 n−2 f(x) := x + a1x + a2x + ··· + an−1x + an

be a monic polynomial of degree n ≥ 1 over K with roots α1, . . . , αn in some splitting ﬁeld of f. The quantity

Y 2 D(f) = (αi − αj) (1.1) 1≤i

IV]). The formula (1.1) is a symmetric polynomial in the roots α1, . . . , αn and is hence a polynomial in the symmetric functions of those roots, which are the coeﬃcients of f. We call the universal version of this polynomial the classical discriminant of degree n polynomials, denoted by Dn. The utility of D(f) suggests that one might wish to evaluate it on either a single polynomial or a family of polynomials — for example, a polynomial f(x, t) ∈ K[x, t], which may be written

n n−1 f(x, t) = x + f1(t)x + ··· + fn(t).

Evaluation of D(f) (with respect to x) in this case is just a matter of specializing Dn to the coeﬃcients a1 = f1(t), . . . , an = fn(t). One may then identify, say, for which values t0 of t the specialized polynomial f(x, t0) has a repeated root by computing the roots of the resulting polynomial in t. However, the direct

1 Chapter 1. The Classical Discriminant 2

Figure 1.1: The “swallowtail”

approach of writing the polynomial Dn and substituting values of the coeﬃcients is intractable, since the number of terms in Dn grows very quickly with the degree. This motivates the provision of eﬃcient formulae for Dn. Formulae which allow for evaluation of D(f) on a family of polynomials as above are particularly desirable.

We begin with an essential property of Dn.

Proposition 1.1.1. The universal discriminant Dn is irreducible as an element of K[a1, . . . , an].

Proof. Suppose that Dn factors nontrivially into D1D2. Then the factorization must also hold in the

n n−1 splitting ﬁeld E of the universal polynomial f(x) = x + a1x + ··· + an. Namely, it must be that some nontrivial factor of

Y 2 D(f) = (αi − αj) 1≤i

2 they are not symmetric. In the second case, there is some factor (αi − αj) which divides D1 and not D2

2 and some factor (αk − αl) which divides D2 and not D1. Let σ be the permutation which reverses αi and αk, reverses αj and αl, and leaves the remaining roots ﬁxed. Then σ swaps the two aforementioned factors, again contradicting the premise that D1 and D2 be symmetric. The only possibility is that one of D1 or D2 be trivial, and the claim follows.

Proposition 1.1.1 implies that the ideal (Dn) ⊆ K[a1, . . . , an] is prime. In particular, the algebraic properties of Dn may be understood directly by studying the geometry of the vanishing locus Z(Dn), which is a hypersurface in the aﬃne space deﬁned by the coordinates a1, . . . , an. Abusing language slightly, we call this hypersurface again the discriminant and denote it ∆n, or ∆ when n is understood.

4 2 Figure 1.1 shows the discriminant ∆˜ 4 of the polynomial x +a2x +a3x+a4, also known as the swallowtail. Chapter 1. The Classical Discriminant 3

The “true” degree four discriminant ∆4 is the product of ∆˜ 4 and an aﬃne line, as will be explained in Section 1.2. In analogy, discriminants of higher degree are sometimes called generalized swallowtails.

This thesis is concerned primarily with the study of the geometry of ∆n. In particular, we develop the philosophy that the geometry of ∆n is intimately related to the existence of determinantal formulae for D(f), formulae which represent Dn as the determinant of a matrix.

The universal polynomial

n n−1 n−2 f(x) = x + a1x + a2x + ··· + an

is quasihomogeneous of degree n with weights deg x = 1 and deg ai = i for i = 1, . . . , n. The formula

n Y f(x) = (x − αi) i=1 suggests that the roots (treated for the moment as indeterminates) should also have degree 1. In that case, (1.1) indicates that the D(f) is also quasihomogeneous of degree

n · 2 = n(n − 1). 2

Indeed, when D(f) is written in terms of the coeﬃcients a1, . . . , an according to the above degrees, that is the case (c.f. [Bou03, Proposition A.IV.6.10]). This implies further that O∆n = K[a1, . . . , an]/(Dn) is a graded ring with respect to the grading speciﬁed above.

Quasihomogeneity allows the properties of ∆n to be connected with its formal germ ∆ˆ n at 0. Namely,

∗ there is a group action K on ∆n given by

2 n λ · (a1, . . . , an) 7→ (λa1, λ a2, . . . , λ an).

Therefore, any property which is true in a neighbourhood of the origin is true on all of ∆n. In the next section, we shall show how the formal germ of ∆n is constructed using the theory of unfoldings of maps.

This allows us to bring the powerful tools of deformation theory to bear on the study of ∆n.

For the remainder of this thesis, we assume that the ﬁeld K has characteristic zero. Chapter 1. The Classical Discriminant 4

1.2 A local perspective: Unfoldings

In this section, we construct the formal germ of ∆n at 0 using the theory of unfoldings of maps. We roughly follow the development in [Tei77]. To avoid unnecessary technical complications, we develop this theory in a very restricted context, namely, unfoldings of analytic maps C → C. In particular, all future results which use the theory of unfoldings will require that K = C.1 The reader may consult, e.g., [BF96] for an extensive treatment of unfoldings from a deformation-theoretic standpoint.

Roughly speaking, an unfolding of a map C → C is a family of maps which varies continuously in the parameters of some base space S. More precisely:

Deﬁnition 1.2.1. Let ψ :(C, 0) → (C, 0) be the germ of a holomorphic map. A (smooth) unfolding of ψ is a pair ((S, 0), Ψ) where (S, 0) is the germ of a smooth analytic space and Ψ : (S × C, 0) → (C, 0) is the germ of a map such that Ψ |{0}×C= ψ. We shall normally refer to the deformation ((S, 0), Ψ) merely by Ψ, suppressing the base space (S, 0).

Given a holomorphic map ψ : C → C, an unfolding of ψ is a pair (S, Ψ) where S =∼ Ck for some k ≥ 0 and Ψ : S × C → C is a holomorphic map such that the germ of Ψ at 0 is an unfolding, in the above sense, of the germ of ψ at 0.

Unfolding a map reveals what happens under small perturbations of the map. A map is called stable if no small perturbation fundamentally changes the properties of the map. The precise meaning of stability is dependent on the sense in which two germs are considered to be equivalent: the properties of interest are precisely those which are preserved in an equivalence class of germs. There are several senses of equivalence relations of germs which appear in the literature, c.f. [BF96]. We shall be concerned with the geometry of the fibre of the map over zero, so, for us, a stable map is one whose fibre over zero, up to isomorphism, does not change under small perturbations. The natural relation for such study is extended contact equivalence, and we frame our definitions thus.

A condition closely related to stability is versality: an unfolding is versal if it represents all nontrivial ways of perturbing the map. If an unfolding be versal, then no further unfolding thereof can change the properties of the map in a nontrivial way, so versal unfoldings are stable maps. We deﬁne versality precisely as follows:

Deﬁnition 1.2.2. Let ((S, 0), Ψ) and ((T, 0), Ψ0) be unfoldings of a germ ψ :(C, 0) → (C, 0). We say that ((T, 0), Ψ0) is induced from ((S, 0), Ψ) if there exists a holomorphic germ Φ : (T × C, 0) → (S × C, 0) 0 and an invertible element u ∈ O(T ×C,0) such that Ψ and (t, z) 7→ u(t, z)(Ψ ◦ Φ)(t, z) deﬁne the same

1This is not a severe restriction in practise, thanks to the Lefschetz principle. Chapter 1. The Classical Discriminant 5 germ. We say that ((S, 0), Ψ) is versal if every unfolding of ψ is induced from ((S, 0), Ψ). We say that ((S, 0), Ψ) is semiuniversal or miniversal if ((S, 0), Ψ) is versal and the dimension of S is minimal. An unfolding Ψ0 of a holomorphic map ψ is induced from an unfolding Ψ of ψ if the germ at zero of Ψ0 is induced from the germ at zero of Ψ. The unfolding Ψ is versal or miniversal if its germ at zero is.

Our basic example is the map x 7→ xn, from which arises our object of study. We ﬁrst construct a miniversal unfolding thereof.

n n−2 Proposition 1.2.3. The map Ξ:(˜ x, a2, . . . , an) 7→ x + a2x + ··· + an is a miniversal unfolding of the map x 7→ xn.

0 n Proof. Suppose (T, Ξ ) is another unfolding of the map x 7→ x . Write T = Spec C b1, . . . , bk . We may J K expand Ξ0 as a Taylor series about the origin in T × C:

0 2 Ξ (x, b1, . . . , bk) = g0(b1, . . . , bk) + g1(b1, . . . , bk)x + g2(b1, . . . , bk)x + ··· .

0 n That Ξ is an unfolding of x 7→ x implies that gn(0,..., 0) = 1 and gi(0,..., 0) = 0 for all i 6= n.

0 0 0 The Weierstraß Preparation Theorem implies that Ξ = uξ , where u is a unit in OT ×C and ξ = n−1 n hn + hn−1x + ··· + h1x + x , hi ∈ OT for each i.

0 We now define the map Φ : T × X → S × X in stages. First, let S = Spec C a1, . . . , an and define J K 0 0 0 00 0 Φ : T × X → S × X via Φ (b1, . . . , bk, x) = (h0, . . . , hn−1, x). Now define Φ : S × X → S × X via

00 1 1 n−i Φ (a1, . . . , an, x) = (k2, . . . , kn, x + n a1), where ki ∈ OS0×X is the coefficient of z + n a1 in the n n−1 1 Taylor expansion of z + a1z + ··· + an about the point z = − n a1. Note that the coefficient of 1 n−1 1 n z + n a1 in this Taylor expansion is zero and that the coefficient of z + n a1 is one. Finally, let Φ := Φ00 ◦ Φ0. It is an easy verification (left to the reader) that Ξ0 and (t, z) 7→ u(t, z)(Ξ˜ ◦ Φ)(t, z) define the same germ, and thus (T, Ξ0) is induced from Ξ.˜ Since (T, Ξ0) was arbitrary, Ξ˜ is versal.

Our initial focus of study is the ﬁbre over zero of the unfolding of x 7→ xn. We may think of it as the germ of an incidence variety: points on this ﬁbre are pairs (s0, x0) such that x0 is a zero of the

perturbed map ψs0 := (x 7→ Ξ(s0, x)). We deﬁne two germs which are fundamental.

Deﬁnition 1.2.4. Let (S, Ψ) be an unfolding of ψ : C → C. Let p :Ψ−1(0) → S is the restriction of −1 1 the projection S × X → S to Ψ (0). The critical locus C(Ψ) is the support of the module ΩΨ−1(0)/S of 1 relative diﬀerentials of the map p. The discriminant ∆(Ψ) is the support of the direct image p∗ΩΨ−1(0)/S.

Suppose the unfolding Ψ is given by a polynomial vanishing at zero, which is certainly the case for the unfolding Ψ of x 7→ xn in Proposition 1.2.3. Then p is a ﬁnite map: the preimages of s ∈ S correspond Chapter 1. The Classical Discriminant 6

−1 to the roots of the perturbed polynomial ψs. Furthermore, the ﬁbre Ψ (0) is the vanishing locus of

n n−1 x + a1x + ··· + an, which is a smooth manifold, as can be seen from the Jacobian criterion. In this case, the critical locus C(Ψ) is just the branch locus: the locus of points on Ψ−1(0) in a neighbourhood of which p is not bijective.

The discriminant of a miniversal unfolding contains all of the essential geometric content of the discriminant of any versal unfolding. This is important since we are studying primarily the discrimi-

n n−1 nant of the versal unfolding (x, a1, . . . , an) 7→ x + a1x + ··· + an, not of the miniversal unfolding

n n−2 (x, a2, . . . , an) 7→ x + a2x + ··· + an. The following proposition makes this precise.

Proposition 1.2.5. Let ((S, 0), Ψ) be a semiuniversal unfolding of ψ :(C, 0) → (C, 0) and let ((S0, 0), Ψ0) be a versal unfolding of ψ. Then ∆(Ψ0) is analytically isomorphic to the product of ∆(Ψ) and a smooth factor.

Proof. Versality of Ψ0 implies that there exists a map Φ : (S0 × C, 0) → (S × C, 0) and a unit u ∈ 0 0 OS ×C,0 such that Ψ and (s, z) 7→ u(s, z)(Ψ ◦ Φ)(s, z) deﬁne the same germ. In particular, Φ restricts 0 ∼ to a map ∆(Ψ ) → ∆(Ψ) and thus O∆(Ψ0) is a ﬁnitely-generated algebra over O∆(Ψ), say O∆(Ψ0) =

O∆(Ψ) t1, . . . , tk /I, whose presentation is chosen so that k is minimal. J K 00 Let us form a new unfolding ((T, 0), Ψ ) of ψ. The germ (T, 0) is S × Spec C t1, . . . , tk . Since J K k is minimal, the Zariski tangent spaces at zero of T and of S0 are isomorphic as C-vector spaces. Geometrically, T is just the product of S and the germ of k-dimensional aﬃne space. The map Ψ00 :

T × C → C is just (s, t, z) 7→ Ψ(s, z). Clearly Ψ may be induced from Ψ00, so ((T, 0), Ψ00) is versal. 00 ∼ 00 ∼ 0 Clearly also ∆(Ψ ) = ∆(Ψ) × Spec C t1, . . . , tk . It therefore suﬃces to show that ∆(Ψ ) = ∆(Ψ ). J K 00 0 0 By versality of Ψ , there exists a map Φ :(T × C, 0) → (S × C, 0) and a unit v ∈ OT ×C,0 such that Ψ00 and (t, z) 7→ v(t, z)(Ψ0 ◦ Φ0)(t, z) deﬁne the same germ. Thus Φ0 maps ∆(Ψ00) to ∆(Ψ0).

Now let φ : O∆(Ψ0) → O∆(Ψ00) be the associated map of algebras. Let π be the natural projection ∼ 0 O∆(Ψ00) = O∆(Ψ) t1, . . . , tk → O∆(Ψ0). Then π ◦ φ is an automorphism of O∆ 0 , so ∆(Ψ ) is a retract J K Ψ of ∆(Ψ00). Since ∆(Ψ0) and ∆(Ψ00) have isomorphic tangent spaces at zero, they are isomorphic, as claimed.

Now we are ready to connect the classical discriminant ∆n with the discriminant of a versal unfolding of x 7→ xn.

n n−1 Proposition 1.2.6. The map Ξ:(x, a1, . . . , an) 7→ x + a1x + ··· + an is a versal unfolding of the

n map x 7→ x . Its discriminant is the classical discriminant ∆n. In particular, ∆n is isomorphic to the ˜ ˜ product of the the discriminant of Ξ, which we call the reduced discriminant ∆n, and Spec C[a1]. Chapter 1. The Classical Discriminant 7

Proof. That the above map is versal follows immediately from Proposition 1.2.3, since Ξ,˜ and hence every unfolding of x 7→ xn, can be induced from it. Observe that Ξ−1(0) is just the incidence variety

n n−1 {(a1, . . . , an, x) ∈ S ×C | x +a1x +···+an = 0}. The preimages of (a1, . . . , an) under the projection −1 n n−1 Ξ (0) → S are the points (a1, . . . , an, α) where α is a root of f(x) = x + a1x + ··· + an. Thus a generic point over (a1, . . . , an) has n distinct preimages. The critical locus of the projection is the branch locus, and its image ∆(Ξ) is just the locus of points whose associated polynomial has less than n distinct roots, namely, the classical discriminant ∆n.

1.3 A global perspective: Hilbert schemes

We now give a projective version of the discriminant. Consider the space X = (P1)×n. There is a natural action of the symmetric group Sn on X given by permutation of the factors. The quotient of X by

1 this action is called the Hilbert scheme of n points on P , denoted Hn, or just H when n is understood.

An orbit in X under the action by Sn may be viewed as the set of roots of some nonzero homogeneous polynomial

n n−1 n−2 2 n−1 n F (x, y) = a0x + a1x y + a2x y + ··· + an−1xy + any .

This sets a natural one-to-one correspondence up between those orbits and polynomials F (x, y), up to nonzero scalar multiples of the latter. Thus the Hilbert scheme is isomorphic to Proj [a , . . . , a ] = n . K 0 n PK

Now consider the critical locus C of the quotient map q : X → Hn. This locus consists of points on

X where the action of Sn is not faithful. Its image under q is the discriminant locus ∆n in Hn, which we identify with the set of homogeneous polynomials F (x, y) with a repeated linear factor, modulo nonzero scalar action. Clearly this homogeneous realization of ∆n agrees, on the affine piece {y 6= 0 and a0 6= 0}, with the affine version defined above.

Consider now the product H × P1. Let

n n−1 n−1 n F := a0x + a1x y + ··· + an−1xy + any ∈ K[a0, . . . , an, x, y]

∂F be the universal homogeneous polynomial of degree n in x and y and view its partial derivatives Fx := ∂x ∂F 2 1 ¯ and Fy := ∂y as sections of OH×P (1, n − 1). Consider the incidence variety ∆ deﬁned by the sections 1 Fx and Fy. It is a smooth codimension-two subvariety of H × P . The Euler identity

nF = xFx + yFy

2 ∂g ∂g In general, for a section g of O 1 (j, k), j, k ≥ 0, we write gx for and gy for . H×P ∂x ∂y Chapter 1. The Classical Discriminant 8

shows that on the aﬃne pieces Uy := {y 6= 0} and Ux := {x 6= 0}, respectively, ∆¯ coincides with the

n n 1 1 varieties deﬁned by the sections { y F,Fx} ⊆ Γ(Uy, OH×P (1, n − 1)) and { x F,Fy} ⊆ Γ(Ux, OH×P (1, n − 1)). Thus points on ∆¯ are pairs (f, t) ∈ H × P1 such that t is a repeated root of f, the latter viewed as a 1 ¯ homogeneous polynomial. In particular, the projection map pH : H × P → H, restricted to ∆, deﬁnes a map π : ∆¯ → ∆.

We now deﬁne a family of subvarieties of ∆n which is of fundamental importance: the caustics Σn,k for 2 ≤ k ≤ n. They are a particular subset of the Thom-Boardman stratiﬁcation, c.f. [Tei77]. Let

¯ 1 Σn,k be the subvariety of H × P deﬁned by the kth-order partial derivatives of F , viewed as sections

1 ¯ of OH×P (1, n − k). It is again an easy computation to see that Σn,k is smooth. Furthermore, since the deﬁning ideal of Σ¯ n,k on a given aﬃne set is quasihomogeneous with respect to the weights given in

Section 1.1, Σ¯ n,k is connected.

Deﬁnition 1.3.1. The kth caustic of ∆n, denoted Σn,k, is the image of Σ¯ n,k under the projection pH .

For n ≥ k ≥ 2, the caustic Σn,k is the locus of polynomials of degree n with a root of multiplicity at least k. The caustics ∆n = Σn,2 ⊇ Σn,3 ⊇ · · · ⊇ Σn,n form a stratiﬁcation of ∆n. For each k > 2, Σn,k is a subvariety of Σn,k−1 of codimension one.

Proposition 1.3.2. The restriction of pH to Σ¯ n,k is the normalization of Σn,k. In particular, π : ∆¯ → ∆ is the normalization of ∆, and the normalizations of each of the caustics Σn,k are smooth.

Proof. Fix n ≥ k ≥ 2. The map πk is ﬁnite since the number of preimages of a point on Σn,k is the number of roots of multiplicity k of the associated degree n polynomial, which is clearly a ﬁnite number. In addition, a generic polynomial of degree n with a root of multiplicity k has exactly one root of multiplicity exactly k, so πk is generically one-to-one. Finally, Σ¯ n,k is smooth, hence normal. Thus the map πk is the normalization of Σn,k, as claimed.

1.4 The geometry of ∆n

Here we collect some basic results about the geometry of the discriminant ∆n and its caustics Σn,k. The

ﬁrst theorem, whose proof we defer to Chapter 2, characterizes the singular loci of the caustics Σn,k as the locus of points corresponding to polynomials with greater degeneracies than are required to be on

Σn,k.

Theorem 1.4.1. Let n ≥ k ≥ 2. The singular locus of the k-th caustic Σn,k is the locus of polynomials with either a root of multiplicity strictly greater than k or more than one root of multiplicity at least Chapter 1. The Classical Discriminant 9 k. The former locus has codimension one and the latter has codimension k − 1 if 2k ≤ n and is empty otherwise.

Deﬁnition 1.4.2. The self-intersection locus of Σn,k (sometimes called the Maxwell set in the case k = 2) is the locus of polynomials with more than one root of multiplicity at least k.

The next corollary will be useful in Chapter 3.

Corollary 1.4.3. The singular locus of the caustic Σn,3 is contained in its intersection with the self- intersection locus in ∆n.

Proof. By Theorem 1.4.1, the singular locus of Σn,3 consists of Σn,4 and the locus of polynomials of degree n with more than one triple root. The former locus is contained in the self-intersection locus since a single root of multiplicity four is two pairs of double roots, while the latter locus is clearly contained in the self-intersection locus.

We now use the theory of unfoldings to describe the geometry of ∆n near certain points other than the origin.

Theorem 1.4.4. Let p ∈ Σn,3 be a point which is not on the self-intersection locus of ∆n. Then the germ of ∆n at p is formally isomorphic to the product of ∆˜ 3 and a smooth factor.

Proof. Let p ∈ Σn,3 be as given and write p = (a1, . . . , an). We denote by p(x) the polynomial corresponding to p. We claim that the germ of Ψ at p is a versal unfolding of x 7→ p(x). This is a consequence of the following general result in deformation theory, for whose proof the reader may consult, e.g., [Fle81].

Theorem 1.4.5 (Openness of Versality). Let Ψ:(S × C, 0) → C be a versal unfolding of ψ : C → C.

Then there is an open neighbourhood U of 0 in S ×C such that Ψ is a versal unfolding of ψs : z 7→ Ψ(s, z) for every s ∈ U.

∗ The C action on ∆n described in Section 1.1 implies that, in fact, U may be taken to be the entire discriminant. Applying Proposition 1.2.5, we ﬁnd that the germ of ∆n at p is formally isomorphic to the product of a versal deformation of p(x) and a smooth factor. The choice of p implies that its corresponding polynomial is of the form p(x) = g(x)(x − α)3, where g(x) has distinct roots which do not include α. But then g(x) is invertible in a neighbourhood of α, so a versal unfolding of p(x) is just

3 2 (x, a1, a2, a3) 7→ g(x)((x − α) + a1(x − α) + a2(x − α) + a3), which is equivalent to a versal unfolding of x 7→ x3. The claim now follows from Proposition 1.2.6. Chapter 1. The Classical Discriminant 10

We now restrict to the affine subsets {y 6= 0 and a0 6= 0} of the caustics Σn,k. It will be convenient to work in different coordinates, namely, those associated to the divided powers of x. For 1 ≤ i ≤ n, set si := (n!/(n − i)!)ai. Then s1, . . . , sn are identified with the coefficients of the polynomial

(n) (n−1) (n−2) x + s1x + s2x + ··· + sn, where x(k) := xk/k! is the kth divided power of x. For n ≥ k > 0, differentiation of a polynomial in x with respect to x defines a finite map Σn,k → Σn−1,k−1, which in turn defines a tower of varieties terminating at the discriminant ∆n = Σn,2. With respect to the coordinates s1, . . . , sn, this map is just projection (s1, . . . , sn) 7→ (s1, . . . , sn−1). This tower is central to the study of the affine discriminant. We give here a few general geometric results which will be useful in Chapter 3.

Proposition 1.4.6. For a ﬁxed n, the varieties Σ¯ n,k, k ≥ 2, share the same normalization in the sense that the normalization map Σ¯ n−1,k−1 → Σn−1,k−1 factors through Σn,k.

Proof. We construct a parametrization of Σn,k in the following manner: For a polynomial f with a root t of multiplicity k, its Taylor series around t has the form

(n) (n−1) (k) f(x) = (x − t) + u1(x − t) + ··· + un−k(x − t) , (1.2)

Expansion of (1.2) deﬁnes a map Spec K[t, u1, . . . , un−k] → Σn,k which is clearly a normalization. Dif- ferentiation of (1.2) yields

0 (n−1) (n−2) (k−1) f (x) = (x − t) + u1(x − t) + ··· + un−k(x − t) ,

which is of the same form as that corresponding to the parametrization of Σn−1,k−1. The claim follows.

The following result of Givental shows how the functions sn−2+i on Σn+k,k+2 for k ≥ i > 0 embed ∼ in the normalization Σ¯ n+k,k+2 = ∆¯ n. This will be useful for computations later on.

R x 00 (i−1) Lemma 1.4.7. For i > 0, sn−2+i = ± 0 f (t)t dt.

Proof. See [Giv82], Lemma 2.

The following application of Lemma 1.4.7 will be useful in Chapter 3.

Proposition 1.4.8. For n ≥ 2 and k ≥ 0, Ω1 =∼ Ω1 . Σ¯ n+k,k+2/Σn+k,k+2 Σ¯ n,2/Σn,2 Chapter 1. The Classical Discriminant 11

Proof. The modules Ω1 and Ω1 are the cokernels of the Jacobian matrices of Σ¯ n+k,k+2/Σn+k,k+2 Σ¯ n,2/Σn,2 0 the normalization maps π : Σ¯ n+k,k+2 → Σn+k,k+2 and π : Σ¯ n,2 → Σn,2. We use the local coordi- ∼ nates x, s1, . . . , sn−2 for Σ¯ n+k,k+2 = Σ¯ n,2 and local coordinates s1, . . . , sn+k, respectively s1, . . . , sn, for

Σn+k,k+2, respectively Σn,2. It suﬃces to show that, for i > 1, the form dsn+i−2 is a Σ¯ n,2-linear combination of ds1, . . . , dsn−1. For i > 0, diﬀerentiating the formula for sn+i−2 given in Lemma 1.4.7 with respect to the local coordinates on Σ¯ n,2, we obtain the form

n + i − 4 i − 1 ds = ± f 00(x)x(i−1) dx + x(n+i−3) ds + ··· + x(i) ds . n+i−2 i − 1 1 i − 1 n−2

Setting i = 1, we obtain the form

00 (n−2) dsn−1 = ± f (x) dx + x ds1 + ··· + x dsn−2 .

(i−1) Therefore, for i > 1, dsn+i−2 is the sum of ±x dsn−1 and a suitable Σ¯ n,2-linear combination of ds1, . . . , dsn−2. The claim follows.

1.5 Structure of this thesis

The remainder of this thesis is organized as follows. Chapter 2 introduces determinantal formulae and describes three such formulae for Dn from the classical literature. It then describes the formalism of matrix factorizations, with which we study such formulae. There is then a detailed study of the Cayley method described in [GKZ94]. The main original contribution in Chapter 2 is a generalization of the

Cayley method to construct presentation matrices of OΣ¯ n,k over OH. In the ﬁnal section of this chapter, this construction is used to prove Theorem 1.4.1.

Chapter 3 contains the main original results of this thesis, namely, the construction of a new determinantal formula which is not equivalent to the classical ones. The principal contribution is Theorem 3.1.2,

1 which characterizes Ω∆¯ /∆. Based on that, the open swallowtail is deﬁned as the kernel of the universal 1 derivation O∆¯ → Ω∆¯ /∆ and, in view of Theorem 3.1.2, a presentation matrix thereof is constructed in Section 3.2. In Section 3.3, the algebraic deﬁnition of Section 3.1 is proved to be equivalent to the geometric version introduced by Arnol’d in [Arn81]. The remainder of Chapter 3 is devoted to further study of the properties of the open swallowtail and its presentation matrix, making ample use of the tools provided by the algebraic construction.

Chapter 4 discusses the theory of deformations of modules and describes how to use its tools to construct moduli spaces of determinantal formulae for Dn. The ﬁrst part is devoted to a detailed Chapter 1. The Classical Discriminant 12 description of an algorithm for computing versal deformations of matrix factorizations. This algorithm slightly generalizes the Massey Product Algorithm described in detail in [Siq01] in that the algorithm described in this thesis works in the relative case where the singularity is deformed along with the module over it. This generalization is necessary to use the algorithm to construct a moduli space of liftings of a module over a curve singularity to ∆n. The middle part of Chapter 4 is an elementary treatise on the structure of graded rank one maximal Cohen-Macaulay modules over the curve singularity

n n−1 Γn := {x − y = 0}. As proved at the beginning of Chapter 4, all graded rank one MCM modules over ∆n restrict to graded rank one MCM modules over Γn, so the latter modules are starting points for the classiﬁcation of the former. The next section contains the main contribution of this chapter: an exposition of the use of this algorithm with the computer algebra system Macaulay 2 [GS] to classify graded rank one MCM modules over ∆˜ 4. The last section is devoted to some theoretical results which describe the theory of these deformations and may, in principle, be used to construct deformations without resorting to computer calculation. Chapter 5 includes some concluding remarks and some musings about possible directions for future research along the lines of this thesis. Chapter 2

Matrix Factorizations

In this chapter, we introduce our source of formulae for the classical discriminant, namely, representations of the discriminant as the determinant of a matrix whose entries are in the ring of coeﬃcients

K[a1, . . . , an]. These so-called determinantal formulae allow us to realize our goal of eﬃciently computing the discriminant both of individual polynomials and of families thereof, since the problem is reduced to computing the determinant of a matrix deﬁned over a suitable polynomial ring. Furthermore, we shall see that the matrices involved in these formulae provide far richer information about the root structure of the polynomial than the discriminant alone.

We begin with a treatment of the formulae discovered in the 18th and 19th centuries. We then state the well-known result that all of these formulae are, in a sense which we shall make precise, equivalent. This motivates the main question to which this thesis is devoted:

Are there any nontrivial determinantal formulae for Dn which are inequivalent to the classical ones?

We then introduce Eisenbud’s [Eis80] construction, which connects determinantal formulae with the

homological properties of the coordinate ring O∆n of the discriminant hypersurface. This indispensable tool permits us to view determinantal formulae from a geometric viewpoint and provides a detailed understanding of the properties of the matrices of the formulae.

In the last section of this chapter, we introduce the Cayley method, which can be used to derive each of the classical determinantal formulae. We generalize this method slightly to construct resolutions of the normalizations of the caustics introduced in Chapter 1. This is of interest in its own right, since the presentation matrices reveal information about the root structure of the polynomial, but it will also be important in Chapter 3 when constructing the presentation matrix of the so-called open swallowtail.

13 Chapter 2. Matrix Factorizations 14

2.1 Classical determinantal formulae

There are three speciﬁc classical formula for Dn, two of which come in both aﬃne and projective versions. We describe them in turn.

The Vandermonde formula. Consider the Vandermonde matrix in the roots α1, . . . , αn:

   1 1 ··· 1 1       α1 α2 ··· αn−1 αn      V :=  α2 α2 ··· α2 α2  .  1 2 n−1 n   . . . .   ......   . . . . .     n−1 n−2 n−1 n−1 α1 α2 ··· αn−1 αn 

Q T It is a classical result in linear algebra that det(V ) = 1≤i

T Y 2 det(VV ) = (αi − αj) = D(f). 1≤i

Furthermore, the (i, j) entry of VV T is

n n X i j X i+j αkαk = αk . k=1 k=1

This is a power sum symmetric function, which is in particular symmetric in the roots α1, . . . , αn. Thus

T the entries of VV may be written in terms of the elementary symmetric functions a1, . . . , an. The

T formula Dn = det VV is called the Vandermonde formula for Dn.

The Sylvester formula. For k > 0, let K[x]≤k be the (k + 1-dimensional) vector space of polynomials of degree at most k. Consider the map

φ : K[x]≤n−2 ⊕ K[x]≤n−1 → K[x]≤2n−2

(g, h) 7→ gf + hf 0.

The polynomial f lies on the discriminant if and only if f and f 0 share a common factor, which is the case if and only if φ is not surjective. Since φ is a map of ﬁnite-dimensional vector spaces of the same dimension, this is the case if and only if φ is not an isomorphism. Hence the determinant of φ equals the discriminant of f up to some nonzero constant factor. The resulting matrix has the form shown in Figure 2.1. Chapter 2. Matrix Factorizations 15

 1 n    a 1 (n − 1)a n   1 1   . .  a a .. (n − 2)a (n − 1)a ..   2 1 2 1   ......   ......   . . 1 . . n    an an−1 a1 an−1 2an−2 (n − 1)a1   . . . .   ......   an . an−1 .     .. ..   . an−1 . 2an−2    an an−1

Figure 2.1: Aﬃne Sylvester Matrix

 na a   0 1  (n − 1)a na 2a a   1 0 2 1   . .  (n − 2)a (n − 1)a .. 2a 2a ..   2 1 3 2   ......   ......   . . na0 . . a1     an−1 2an−2 (n − 1)a1 nan (n − 1)an−1 2a2   . . . .   ......   an−1 . nan .     .. ..   . 2an−2 . (n − 1)an−1    an−1 nan

Figure 2.2: Projective Sylvester Matrix

There is also a projective version of the Sylvester formula. Namely, let

n n−1 n f(x, y) = a0x + a1x y + ··· + any be a homogeneous polynomial in two variables. An argument analogous to that above shows that the map

Φ: K[x, y]n−2 ⊕ K[x, y]n−2 → K[x, y]2n−3

(g1, g2) 7→ fxg1 + fyg2,

where K[x, y]l is the degree l piece of K[x, y], is not an isomorphism of vector spaces if and only if fx and fy have a factor in common, which is the case if and only if f has a repeated linear factor. Thus a matrix representing Φ constitutes a determinantal formula for the homogeneous discriminant

∆n = ∆n(a0, . . . , an). The matrix is shown in Figure 2.2.

The Sylvester formula can more generally be used to compute the resultant of two univariate polynomials; see, e.g., [vzGG03, Chapter 6] for a readable treatment. Chapter 2. Matrix Factorizations 16

The B´ezoutformula. Consider the rational function

f 0(x)f(y) − f 0(y)f(x) B(x, y) = . (2.1) y − x

Since the numerator vanishes when x = y, B(x, y) is in fact a polynomial which my be written

n−1 X i j B(x, y) = bijx y . i,j=0

The coeﬃcients bij are polynomials in the variables a1, . . . , an. Let B be the matrix whose (i, j) entry

n is bi+1,j+1. We may view B as a bilinear form on the vector space K . The following proposition shows that B is a determinantal formula for D(f).

Proposition 2.1.1. We have det B = D(f).

Proof. (See also [BEvB06, Theorem 5.6].) We show ﬁrstly that the bilinear form B is nondegenerate if and only if the roots of f are distinct. Let α be a root of f. We claim ﬁrst that α is a repeated root of f if and only if (x − α) divides B(x, y). One direction is obvious from (2.1). Suppose now that α is a root but not a repeated root of f. Specializing B, we obtain

f 0(α)f(y) − f 0(y)f(α) B(α, y) = y − α f(y) = f 0(α) . y − α

Since α is not a repeated root of f, f 0(α) 6= 0, so B(α, y) is some nonzero polynomial in y, whence the claim.

For α ∈ K, denote by v(α) the vector (1, α, α2, . . . , αn−1). Suppose now that α is a repeated root. Then (x−α) divides B(x, y), so, for all β ∈ K, v(β)T Bv(α) = 0. But vectors of the form v(β) span Kn, so T n w Bv(α) = 0 for all vectors w ∈ K , and B is degenerate. Suppose now that the roots (α1, . . . , αn) of f are distinct. Then (x−αi) does not divide B(x, y) for any i = 1, . . . , n. In particular, for i = 1, . . . , n, the

T map β 7→ v(β) Bv(αi) is not identically zero. Since α1, . . . , αn are distinct, the vectors v(α1), . . . , v(αn) span Kn. Thus B is nondegenerate. We have thus shown that det B = uD(f) for some nonzero constant u. We now show that u = 1 by testing on the polynomial f(x) = xn − 1. Let ξ be a primitive nth root of unity in some extension

ﬁeld of K. Then the roots of f are 1, ξ, . . . , ξn−1. We ﬁrst compute D(f) via the Vandermonde formula. This requires computing the power sum symmetric functions of the roots of f, for which we appeal to Chapter 2. Matrix Factorizations 17 the following well-known formula (see, e.g., [vzGG03, Lemma 8.7]):

 n−1  X n, if n | l, (ξj)l =  j=0 0, otherwise.

Thus, letting V be the Vandermonde matrix in the roots 1, ξ, . . . , ξn−1, the matrix VV T has the form

  n       n     T  ..  VV =  .  .      n       n 

Thus D(f) = det(VV T ) = (−1)b(n−1)/2cnn.

Now we compute the determinant of the B´ezoutmatrix. We have

f 0(x)f(y) − f 0(y)f(x) (yn − 1)(nxn−1) − (xn − 1)(nyn−1) = y − x y − x xn−1yn−1(y − x) + yn−1 − xn−1 = n y − x (y − x) xn−1yn−1 + xn−2 + xn−3y + ··· + yn−2 = n y − x

= n xn−1yn−1 + xn−2 + xn−3y + ··· + yn−2 .

Thus the matrix B has the form    n     .   ..        B =  n  .     n       n

Its determinant is therefore (−1)b(n−1)/2cnn = D(f), as claimed.

The formula described above is an aﬃne version of the B´ezoutformula, since the polynomial f is

n n−1 n not homogeneous. There is also a projective version. Let f(x, y) = a0x + a1x y + ··· + any be a Chapter 2. Matrix Factorizations 18 homogeneous polynomial in the indeterminates x and y. We set

fx(x0, y0)fy(x1, y1) − fy(x0, y0)fx(x1, y1) BP (x0, y0, x1, y1) := . x0y1 − x1y0

Then BP (x0, y0, x1, y1) is a bihomogeneous polynomial which we write

n−2 X i n−i−2 j n−j−2 BP (x0, y0, x1, y1) = bijx0y0 x1y1 . i,j=0

The matrix BP with entries bij then has determinant equal to the homogeneous discriminant of f. This matrix is (n − 1) × (n − 1), as opposed to the aﬃne B´ezoutmatrix B described above which is n × n.

As with the Sylvester formula, the B´ezoutformula may be easily generalized to compute the resultant of two univariate polynomials.

The classical formulae introduced above are related to one another. To explain how, we introduce a deﬁnition which generalizes the ordinary notion of equivalence of matrices.

Deﬁnition 2.1.2. Let A and B be two matrices over a commutative ring R. We say that A and B are weakly equivalent if they have isomorphic cokernels when viewed as maps of free modules over R.

Remark 2.1.3. If R is a local ring, then the condition that A and B are weakly equivalent is equivalent to the existence of invertible matrices U1,V1,U2,V2 over R such that

    A0 0  A0 0      U1AV1 =   and U2BV2 =   ,  0 U 0  0 V 0 with U 0 and V 0 invertible. An analogous condition holds when R is graded and A and B are presentation matrices of graded R-modules.

Theorem 2.1.4. The matrices of the Vandermonde, affine Sylvester, and affine Bézoutformulae are pairwise weakly equivalent. The projective versions of the Sylvester and Bézoutformulae restricted to the affine piece {a0 6= 0} are equivalent to the affine versions.

Proof. See [Bou03, A.IV.78] for a proof of the equivalence of the affine Sylvester and Vandermonde formulae. The equivalence of the projective Sylvester and Bézoutformulae on an affine set follows from Examples 2.3.2 and 2.3.3 in Section 2.3 below. It remains to show that the affine Sylvester formula is equivalent to the projective Sylvester formula and that the affine Bézout formula is equivalent to the projective Bézoutformula. We address each in turn. Chapter 2. Matrix Factorizations 19

First, we consider the Sylvester matrix. It is necessary to specify row and column operations on the affine Sylvester matrix of Figure 2.1 so that the resulting matrix is of a block diagonal form in which one block is invertible and the other agrees with the projective Sylvester matrix of Figure 2.2 with a0 = 1. To do this, first scale the left-hand block of the affine matrix by a factor of n and then subtract the first n − 1 columns of the right-hand block from the left-hand block. Then switch the order of the two blocks and apply row operations to eliminate the entries below the (1, 1) entry. The resulting matrix is of the desired form.

Now we consider the B´ezoutmatrix. Let B¯P (u, v) := BP (u, 1, v, 1). In view of the Euler identity

nf = xfx + yfy, we have

f (u, 1)f(v, 1) − f(u, 1)f (v, 1) nB(u, v) = n x x v − u f (u, 1)(vf (v, 1) + f (v, 1)) − (uf (u, 1) + f (u, 1))f (v, 1) = x x y x y x v − u

= B¯P (u, v) + fx(u, 1)fx(v, 1). (2.2)

Pn−1 i j Write fx(u, 1)fx(v, 1) = i,j=0 ciju v and let C be the matrix whose (i, j) entry is ci+1,j+1. Then, for i 1 ≤ i < n, the i-th row, respectively i-th column, of C equals the product of n an−i and the last row, respectively last column, of C. In particular, there exists an invertible matrix U such that U T CU is a matrix with a unit in the (n, n) entry and zero elsewhere. Furthermore, denoting by B˜P the n × n

matrix whose upper-left (n − 1) × (n − 1) submatrix is the specialization BP |a0=1 and whose last row

T and column are both zero, we have U B˜P U = B˜P . In view of (2.2),

nB = B˜P + C.

Thus

T T nU BU = B˜P + U CU.

The matrix on the left-hand side is clearly equivalent to B, while the matrix on the right-hand side is a block-diagonal form where one block is BP and the other is invertible. The claim follows.

Within this equivalence class, the projective B´ezoutformula is a minimal presentation in the sense that no matrix of size strictly less than (n−1)×(n−1) can be weakly equivalent to the classical formulae. Chapter 2. Matrix Factorizations 20

One can see this already by examining the Vandermonde formula: the matrix is of size n × n, and the only entry with a unit is the (1, 1) entry.

Theorem 2.1.4 begs the question of whether there exist determinantal formulae for the classical discriminant which are inequivalent to the classical ones. Of course, there exists a formula in which Dn is placed in a 1 × 1 matrix. This is inequivalent to the classical formulae for the reason stated above. This formula, and any other formula weakly equivalent to it, is trivial. We now restate our main question given at the beginning of this chapter.

Are there any nontrivial determinantal formulae for Dn which are not weakly equivalent to the classical ones?

2.2 Eisenbud’s Theorem

We now introduce matrix factorizations of a polynomial, which generalize determinantal formulae. In this formalism Eisenbud frames his celebrated results which connect determinantal formulae with homological algebra.

Deﬁnition 2.2.1. Let R be a commutative ring and f ∈ R \{0}.A matrix factorization of f is a pair of square matrices (A, B) such that AB = f · id = BA, where id is the identity matrix of suitable size. If f is a non-zerodivisor of R, then one of the two equations suﬃces.

Example 2.2.2. Let f ∈ R and suppose that A is a k × k matrix such that det A = f. Let B = adj A, the adjugate (or classical adjoint) of A. That is,

  k+1  det A11 − det A21 ··· (−1) det Ak1    k+2   − det A12 det A22 ··· (−1) det Ak2   B :=  . . . .  ,  . . .. .   . . .     k+1 k+2 2k  (−1) det A1k (−1) det A2k ··· (−1) det Akk 

where Aij is the matrix resulting from removing the ith row and jth column from A. Then (A, B) is a matrix factorization of f. This shows how matrix factorizations generalize determinantal formulae.

Let f ∈ R be a non-zerodivisor. We deﬁne a morphism of matrix factorizations (A, B) → (A0,B0) of f in the following way. View A, respectively A0, as a map A : G → F , respectively a map A0 : G0 → F 0, Chapter 2. Matrix Factorizations 21 of free R-modules and B, respectively B0, as a map B : F → G, respectively B0 : F 0 → G0. A morphism (A, B) → (A0,B0) consists of a pair (α, β) of maps α : F → F 0 and β : G → G0 making the diagram

G - F - G B A

β α β (2.3) ? ? ? G0 - F 0 - G0 B0 A0 commute.

Matrix factorizations of a non-zerodivisor f ∈ R form a category MF(f) in the following way. The objects of MF(f) are matrix factorizations of f. The morphisms of MF(f) are morphisms of matrix factorizations modulo those morphisms which factor through the trivial matrix factorization (1, f). In this category, two matrix factorizations (A, B) and (A0,B0) of f such that A and A0 are weakly equivalent are isomorphic.

If (A, B) is a matrix factorization of f, then it follows from Cramer’s rule that f annihilates coker A, so coker A is naturally a module over R/(f). Furthermore, provided that f is a non-zerodivisor, A : G → F is injective as a map of R-modules. Hence the complex

A 0 - G - F - coker A - 0 is a resolution of coker A as an R-module. A morphism (α, β):(A, B) → (A0,B0) of matrix factorizations by definition defines a morphism from the associated resolution of coker A to that of coker A0, and therefore a module homomorphism coker A → coker A0. Thus (A, B) 7→ coker A defines a functor, also denoted coker, from MF(f) to the category mod(R/(f)) of finitely-generated R/(f)-modules. Further- more, if R is Cohen-Macaulay and f ∈ R is a regular element, it follows from the Auslander-Buchsbaum formula that depth coker A = dim R − 1 = dim R/(f), so coker A is a maximal Cohen-Macaulay module over R/(f). Denote by MCM(R/(f)) the full subcategory of mod(R/(f)) consisting of maximal Cohen-Macaulay modules over R/(f). The following result due to Eisenbud is fundamental.

Theorem 2.2.3 ([Eis80]). Suppose that R is a regular local ring and that f ∈ R is a regular element. Then the functor coker is an equivalence of the categories MF(f) and MCM(R/(f)).

Proof sketch. We show here only how to define a pseudoinverse Υ for coker. For a proof that Υ is well-defined and that it is indeed a pseudoinverse of coker, see, e.g. [Yos90, Theorem 7.4]. Let M be a maximal Cohen-Macaulay module over R/(f). We first resolve M over R. Since depth M = dim R/(f) = Chapter 2. Matrix Factorizations 22

depth R − 1, pdR M = 1, so the resolution has the following form:

0 - G - F - M - 0. (2.4) A

We pull (2.4) back through the map given by multiplication by f, which is zero on M since M is a module over R/(f):

0 - G - F - M - 0 6 A 6 6 f f f = 0

0 - G - F - M - 0 A

The map f : F → F therefore takes values in the kernel of the projection F → M. Thus it lifts uniquely to a map B : F → G as follows:

- - - - 0 G F M 0 6 A 6 6

f B f f = 0 . (2.5)

0 - G - F - M - 0 A

Commutativity of (2.5) implies that B ◦ A = f = A ◦ B. Given two maximal Cohen-Macaulay modules M and M 0 with a morphism φ : M → M 0, we may deﬁne a morphism of associated matrix factorizations (α, β) by lifting φ to a map of minimal resolutions of M and M 0, as in the following diagram:

0 - G - F - M - 0 A

β α φ . ? ? ? 0 - G0 - F 0 - M 0 - 0 A

The morphism (α, β) is not uniquely determined by φ. However, since morphisms in MF(f) are taken modulo those which factor through the matrix factorization (1, f), the class of (α, β) as a morphism in MF(f) is uniquely determined. These constructions deﬁne a functor Υ from the category MCM(R/(f)) of maximal Cohen-Macaulay modules over R/(f) to the category of matrix factorizations of f.

The data in a matrix factorization (A, B) of f also completely describe a resolution of coker A over R/(f), as the following theorem shows. Chapter 2. Matrix Factorizations 23

Theorem 2.2.4 ([Eis80]). Let R be a regular local ring, f ∈ R a regular element, and (A, B) a matrix factorization of f. Denote by A¯ : G¯ → F¯ and B¯ : F¯ → G¯ the maps resulting from reducing the entries of A and B modulo f. Then coker A has a two-periodic resolution

A¯ B¯ A¯ ··· - G¯ - F¯ - G¯ - F¯ - coker A - 0. (2.6)

Proof. See, e.g., [Yos90, Proposition 7.2].

The next proposition shows precisely which maximal Cohen-Macaulay modules correspond to determinantal formulae, at least when f is irreducible.

Proposition 2.2.5 ([Eis80]). With the same hypotheses as in Theorem 2.2.3, assume now that f ∈ R is irreducible and let (A, B) be a matrix factorization of f by k × k matrices. Then det A = uf i and det B = u−1f k−i, where i is the rank of coker A as an R/(f)-module and u ∈ R is a unit.

Proof. Since f is irreducible in the regular local ring R, the ideal generated by f is prime. Consider the localization R(f). Over this localized ring, A is equivalent to a diagonal matrix with diagonal entries

α1 αk P i f , . . . , f , where i := j αj ≤ k. Clearly then det A is associate to f , and this remains true over

R. Furthermore, in R(f), the length of the cokernel of A i, and, reducing modulo f, we see that this length is precisely the dimension of the cokernel of A over K(R/(f)), which is the rank of A. The claim follows.

In particular, matrix factorizations associated to determinantal formulae for f correspond to rank one maximal Cohen-Macaulay modules over R/(f). A natural question is to which module the classical determinantal formulae introduced in Section 2.1 correspond. The following theorem, which is an immediate consequence of Examples 2.3.2 and 2.3.3 below combined with Proposition 1.3.2, answers this.

Theorem 2.2.6. The cokernels of the classical determinantal formulae introduced in Section 2.1 are all locally isomorphic to the normalization O∆¯ of O∆.

If R is a graded regular ring and f ∈ R is a homogeneous element, then one may define the category GrMF(f) of graded matrix factorizations in analogy with the above definition. Namely, the objects of GrMF(f) are pairs of matrices (A, B) which define homomorphisms of graded free modules such that the compositions AB and BA are multiplication by f. The morphisms are pairs of homomorphisms α, β of graded free modules making (2.3) commute. In this case, the functors coker and Υ induce inverse Chapter 2. Matrix Factorizations 24 equivalences GrMF(f) and GrMCM(R/(f)), the category of graded maximal Cohen-Macaulay modules over the graded ring R/(f).

2.3 The Cayley Method

In this section, we develop a method for resolving over the ambient space H of ∆ modules which are

locally isomorphic to the normalizations OΣ¯ n,k of the caustics Σn,k. Originally due to Cayley, this method was developed to construct determinantal formulae for the equation of the dual variety X∨ of a given projective variety X. In our case, the variety X is P1, embedded via the nth Veronese embedding n ∨ in P , and X is ∆n. Our treatment omits many of the technical points required for the general case. See [GKZ94, Chapter 2] for a thorough, modern treatment of this method.

¯ 1 The normalization Σn,k of the caustic Σn,k is of codimension k in H × P , while its vanishing ideal Ik

1 is deﬁned by the k − 1-order partial derivatives of F viewed as sections of OH×P (1, n − k + 1), of which there are k. Thus it is a global complete intersection and its structure sheaf is resolved over the ambient

k 1 space via the Koszul complex K• on the aforementioned sections of OH×P (1, n − k + 1) deﬁning Ik. For ∼ example, Figure 2.3 shows resolutions by locally free sheaves of OΣ¯ n,2 = O∆¯ and OΣ¯ n,3 as modules over

1 OH×P .

One might hope to construct a resolution of pH∗OΣ¯ n,k over H by constructing the maps of the spectral p,q q • k sequence associated to the double complex E0 := Ip , where Ip is an injective resolution of Kp for each k 0 ≤ p ≤ k. Indeed, since K• is quasi-isomorphic to OΣ¯ n,k , the resulting spectral sequence converges to l l 0,0 E := R pH∗OΣ¯ n,k , which vanishes for l 6= 0. However, the resulting spectral sequence has nonzero E∞ −1,1 and E∞ . Thus such a computation can produce only a resolution of the associated graded module of

a ﬁltration of pH∗OΣ¯ n,k , which is not exactly what we need.

k To correct this problem, we twist K• by such a line bundle as to ensure that, in the spectral sequence p,q associated to the resulting double complex, E∞ = 0 except for p = q = 0. We ﬁrst select a pivot k k 1 j ∈ {1, . . . , k} and twist K• by OH×P (0, j(n − k + 1) − 1). The resulting complex K• (j(n − k + 1) − 1) is then a resolution of the twisted structure sheaf OΣ¯ n,k (0, j(n − k + 1) − 1), which is isomorphic to OΣ¯ n,k 1 1 on any subset of H ×P over which OH×P (0, j(n−k +1)−1) is trivial — in particular any aﬃne subset. l Since the choice of twist is positive, we still have R pH∗OΣ¯ n,k (0, j(n − k + 1) − 1) = 0 when l 6= 0. As we shall see, our choice of twist not only yields a spectral sequence with the desired properties, but greatly facilitates the computation.

We now describe how a resolution of OΣ¯ n,k (0, j(n − k + 1) − 1) arises from the maps of the spectral Chapter 2. Matrix Factorizations 25

0 0 6 6

OΣ¯ n,2 OΣ¯ n,3 6 6

O O 6 6

8 9 8 9 :Fx Fy ; :Fxx Fxy Fyy ;

O(−1, 1 − n)⊕2 O(−1, 2 − n)⊕3 . 6 6 8 9 8 9 0 Fyy −Fxy Fy > > > > >−Fyy 0 Fxx > :−Fx; > > : Fxy −Fxx 0 ;

O(−2, 2 − 2n) O(−2, 4 − 2n)⊕3 6 6 8 9 Fxx > > >Fxy > > > :Fyy ;

0 O(−3, 6 − 3n) 6

Figure 2.3: Koszul complexes resolving OΣ¯ n,2 and OΣ¯ n,3 Chapter 2. Matrix Factorizations 26

0 1 1 OH ⊗K H P , OP (j(n − k + 1) − 1) 6 −1,0 d1 . . 6 −j+1,0 d1

0 1 ⊕( k ) 1 j−1 OH(−j + 1) ⊗K H P , OP (n − k)

1 1 ⊕( k ) 1 j+1 OH(−j − 1) ⊗K H P , OP (k − n − 2) 6 −j−2,1 d1 . . 6 −k,1 d1

1 1 1 OH(−k) ⊗K H P , OP ((j − k)(n − k + 1) − 1) .

Figure 2.4: First page of the spectral sequence

sequence. It follows from the projection formula that, for l, p, q ∈ Z,

l l 1 1 ∼ 1 R pH∗OH×P (p, q) = OH(p) ⊗K H (P , OP (q)).

0 1 1 1 In particular, R pH∗OH×P (p, q) = 0 when q < 0 and R pH∗OH×P (i, j) when q > −2. The ﬁrst page of the spectral sequence is shown in Figure 2.4. If j = 1, the second page of the spectral sequence is

0 1 O ⊗ H , O 1 (n − k) H K P P

− d 2 2 ,1

−3,1 coker d1 .

−2,1 The cokernel of d2 is pH∗OΣ¯ n,k (n − k), and the spectral sequence degenerates after this step. If j > 1, Chapter 2. Matrix Factorizations 27 then the second page of the spectral sequence is

pH∗OΣ¯ n,k (j(n − k + 1) − 1) . .

−j+1,0 ker d1

d − j− 2 1 ,1

−j−2,1 coker d1 .

k In this case, since the original complex K• is exact except at homological degree zero and since

l R pH∗OΣ¯ n,k (j(n − k + 1) − 1) = 0

−j−1,1 for l < 0, d2 must be an isomorphism. Again, the spectral sequence degenerates after this point.

k k 0 1 1 ⊕( ) 0 1 ⊕( ) 1 j+1 1 j−1 Let d : OH(−j − 1) ⊗K H P , OP (k − n − 2) → OH(−j + 1) ⊗K H P , OP (n − k) −j−1,1 be a lifting of d2 . For 0 < i < k, let

  −i,0 d1 , if i < j,   ∂i := d0, if i = j,    −i−1,1 d1 , if i > j.

The maps ∂i for 0 < i < k are the diﬀerentials of a complex C• which, in view of the above observations, ∼ is exact except at the right. We have coker ∂1 = pH∗OΣ¯ n,k (j(n − k + 1) − 1). In this manner the maps of the spectral sequence give rise to a resolution of pH∗OΣ¯ n,k (j(n − k + 1) − 1) over H by locally free sheaves.

To describe the maps explicitly, it is convenient to compute in the ﬁbre over a ﬁxed point [a0 :

··· : an] ∈ H. The restriction of C• to this ﬁbre is a map of vector spaces whose diﬀerentials vary

• k polynomially in the coordinates a0, . . . , an. In doing so, we replace the injective resolutions Il of Kl ˇ k with Cech complexes which compute the cohomology of Kl , as shown in Figure 2.5. The ﬁrst page of the spectral sequence is shown in Figure 2.6.

Recall Serre’s theorem characterizing Ha( b, O(c)) for a, b, c ∈ . Set b = Proj [x , . . . , x ] and P Z PK K 0 b let V denote the degree one part of K[x0, . . . , xb]. Chapter 2. Matrix Factorizations 28

Γ(Ux, O(j(n − k + 1) − 1)) - L - - 0 Γ(Uxy, O(j(n − k + 1) − 1)) 0 d0,0 Γ(Uy, O(j(n − k + 1) − 1)) H 6 6 −1,0 −1,1 dV dV

. . . . 6 6

−j+1,0 −j+1,1 dV dV

⊕( k ) Γ Ux, O(n − k) j−1 ⊕ k 0 - L - Γ U , O(n − k) (j−1) - 0 −j+1,0 xy k ⊕( ) dH Γ Uy, O(n − k) j−1 6

6−j,0 −j,1 dV dV

⊕(k) Γ Ux, O(−1) j ⊕ k 0 - L - Γ U , O(−1) (j) - 0 −j,0 xy k ⊕( ) dH Γ Uy, O(−1) j 6

6−j−1,0 −j−1,1 dV dV

⊕( k ) Γ Ux, O(k − n − 2) j+1 ⊕ k 0 - L - Γ U , O(k − n − 2) (j+1) - 0 −j−1,0 xy k ⊕( ) dH Γ Uy, O(k − n − 2) j+1 6

6−j−2,0 −j−2,1 dV dV

. . . . 6 6

−k,0 −k,1 dV dV

Γ(Ux, O((j − k)(n − k + 1) − 1)) 0 - L - Γ(U , O((j − k)(n − k + 1) − 1)) - 0 −k,0 xy Γ(Uy, O((j − k)(n − k + 1) − 1)) dH

k Figure 2.5: Double complex for K• over [a0 : ··· : an]. Chapter 2. Matrix Factorizations 29

0 1 1 H P , OP (j(n − k + 1) − 1) 6 −1,0 d1 . . 6 −j+1,0 d1

0 1 ⊕( k ) 1 j−1 H P , OP (n − k)

1 1 ⊕( k ) 1 j+1 H P , OP (k − n − 2) 6 −j−2,1 d1 . . 6 −k,1 d1

1 1 1 H P , OP ((j − k)(n − k + 1) − 1) .

Figure 2.6: First page of the spectral sequence on a ﬁbre over a point Chapter 2. Matrix Factorizations 30

Theorem 2.3.1 (Serre). We have

 Sym (V ), if k ≥ 0 and a = 0,  c  a b  H (P , O(c)) =∼ (Sym (V ))∗, if k < −j and a = b,  −c−b−1   0, otherwise.

∗ Here Symc(−) denotes the cth symmetric power of the argument and (−) is the K-dual.

Proof. See [Har77, Theorem 5.1].

−i,0 −i,1 k In view of these identifications, the differentials d1 and d1 are just the maps of K• restricted to −1,0 −k,1 the given fibre. In particular, the matrices of the first and last maps, d1 and, respectively, d1 are generalized Sylvester matrices whose structure we now describe.

−1,0 Assume that j > 1. Then the map d1 is

∂k−1F ∂k−1F (g , . . . , g ) 7→ g + ··· + g , 1 k ∂xk−1 1 ∂yk−1 k where each gi is a homogeneous polynomial in K[x, y] of degree (j − 1)(n − k + 1) − 1. The associated matrix is therefore divided vertically into k blocks of (j − 1)(n − k + 1) − 1 columns each. Each block is associated with a given k − 1-order partial derivative of F . Each block is of the form

   α0       α α   1 0     ..   α2 α1 .     . . .   . . .. α   0    ,   αn−k+1 αn−k α1     . .   α .. .   n−k+1     ..   . αn−k       αn−k+1

where α0, . . . , αn−k are the coeﬃcients of the associated partial derivative of F . In particular, the entries are linear in the coeﬃcients a0, . . . , an.

−k,1 Assume now that j < k − 1. Then the map d1 is

∂k−1F ∂k−1F g 7→ g, . . . , g , ∂xk−1 ∂yk−1 Chapter 2. Matrix Factorizations 31 where g is a homogeneous element of K[x−1, y−1] of degree (j −k)(n−k +1)−1 < 0. Its matrix then has a form similar to the transpose of the matrix described above: it is divided horizontally into k blocks of −(j −k +1)(n−k +1) rows each. The i-th row of a given block contains the coeﬃcients of the associated k − 1-order partial derivative of F shifted to the right i − 1 places. The dimension of the vector space of homogeneous elements of K[x−1, y−1] of degree (j − k)(n − k + 1) − 1 is (k − j)(n − k + 1), so the matrix is of size k(k − j − 1)(n − k + 1) × (k − j)(n − k + 1).

It remains to characterize the map

1 1 ⊕( k ) 0 1 ⊕( k ) ∂j : H P , O(k − n − 2) j+1 → H P , O(n − k) j−1 .

Since Hi(P1, O(−1)) = 0 for all i, the map

k k k −j,0 ⊕(j) ⊕(j) ⊕(j) dH :Γ Ux, O(−1) ⊕ Γ Uy, O(−1) → Γ Uxy, O(−1) is an isomorphism. Thus the map

k k k −j,0 −1 ⊕(j) ⊕(j) ⊕(j) (dH ) :Γ Uxy, O(−1) → Γ Ux, O(−1) ⊕ Γ Uy, O(−1)

is well-deﬁned and ∂j is the composition

−j−1,1 k d k ⊕( ) V ⊕( ) Γ Uxy, O(k − n − 2) j+1 −→ Γ Uxy, O(−1) j

−j,0 −1 d k k ( H ) ⊕( ) ⊕( ) −→ Γ Ux, O(−1) j ⊕ Γ Uy, O(−1) j

−j,0 d k k V ⊕( ) ⊕( ) −→ Γ Ux, O(n − k) j−1 ⊕ Γ Uy, O(n − k) j−1

k 1 1 ⊕( ) restricted to H P , O(k − n − 2) j+1 .

The matrix of ∂j is B´ezout-type, in the sense that its entries are quadratic binominals in the variables a0, . . . , an, just as in the B´ezoutmatrix.

Example 2.3.2. By applying this method to ∆ = Σn,2 and setting the pivot j = 2, we recover the classical

k projective Sylvester formula for the discriminant. Namely, the complex K• is twisted by 2n − 3. The Chapter 2. Matrix Factorizations 32

ﬁrst page of the spectral sequence is

0 1 1 OH ⊗K H P , OP (2n − 3) 6 −1,0 d1

0 1 ⊕2 1 OH(−1) ⊗K H P , OP (n − 2) .

−1,0 0 1 ⊕2 ⊕2 1 ∼ The map d1 takes g1, g2 ∈ H P , OP (n − 2) = K[x, y]n−2 to

0 1 1 ∼ Fxg1 + Fyg2 ∈ H P , OP (2n − 3) = K[x, y]2n−3.

See [GKZ94, Example 2.11] for more information.

Example 2.3.3. The projective B´ezoutformula for the discriminant arises when when we apply this method to ∆ = Σn,2 with pivot j = 1. The ﬁrst page of the spectral sequence in this case is

0 1 1 OH ⊗K H P , OP (n − 2)

1 1 1 OH(−2) ⊗K H P , OP (−n) .

There are no nonzero diﬀerentials. The second page of the spectral sequence therefore has the same

−2,1 1 1 0 1 1 1 entries as the ﬁrst page, with a map d2 : OH(−2)⊗K H P , OP (−n) → OH ⊗K H P , OP (n − 2) . −2,1 See [GKZ94, Proposition 5.4] for a proof that the B´ezoutformula is a matrix representing d2 .

Example 2.3.4. We construct a resolution of a module M locally isomorphic to OΣ¯ 4,3 over the ambient space of ∆4. We set the pivot j = 1. The resolution will have the form

- ⊕4 - ⊕6 - ⊕2 - - 0 OH(−3) OH(−2) OH M 0. ∂2 ∂1

We compute explicit formulae for the matrices representing ∂1 and ∂2. Removing from the double complex modules which have no eﬀect on the computation, we obtain the diagram shown in Figure 2.7.

For n = 4, we obtain the diagram shown in Figure 2.8. A basis of H1(P1, O(−5)) is

{x−4y−1, x−3y−2, x−2y−3, x−1y−4} and a basis for H1(P1, O(−3)) is {x−2y−1, x−1y−2}. Chapter 2. Matrix Factorizations 33

Γ(Ux, O(n − 3)) ⊕ Γ(Uy, O(n − 3)) 6 −1,0 dV

- ⊕3 ⊕3 - ⊕3 - 0 Γ Ux, O(−1) ⊕ Γ Uy, O(−1) Γ Uxy, O(−1) 0 d−1,0 H 6 −2,1 dV

⊕3 Γ Uxy, O(1 − n) 6 −3,1 dV

Γ(Uxy, O(3 − 2n))

Figure 2.7: Truncated zeroth page of the spectral sequence

Γ(Ux, O(1)) ⊕ Γ(Uy, O(1)) 6 −1,0 dV

- ⊕3 ⊕3 - ⊕3 - 0 Γ Ux, O(−1) ⊕ Γ Uy, O(−1) Γ Uxy, O(−1) 0 d−1,0 H 6 −2,1 dV

⊕3 Γ Uxy, O(−3) 6 −3,1 dV

Γ(Uxy, O(−5))

Figure 2.8: Truncated zeroth page of the spectral sequence for n = 4 Chapter 2. Matrix Factorizations 34

The map ∂2 is

g 7→ (Fxxg, Fxyg, Fyyg).

Thus, with respect to the bases above, the map ∂2 has matrix

  12a 6a 2a 0   0 1 2       0 12a0 6a1 2a2       3a1 4a2 3a3 0    .    0 3a 4a 3a   1 2 3       2a2 6a3 12a4 0       0 2a2 6a3 12a4

We now compute a matrix representing ∂1. To do so, we must compute its action on each element of a

1 1 ⊕3 ⊕3 basis of H P , O(−3) . We write e1, e2, e3 for each of the three summands of O(−3) . First, we −2,1 −2 −1 ⊕3 ﬁnd dV (x y e1). It is given by the triple (h1, h2, h3) ∈ Γ(Uxy, O(−1) ), where

h1 = 0,

−2 −1 h2 = −Fyyx y

2 2 −2 −1 = −(2a2x + 6a3xy + 12a4y )x y

−1 −1 −2 = −2a2y − 6a3x − 12a4x y,

−2 −1 h3 = Fxyx y

2 2 −2 −1 = (3a1x + 4a2xy + 3a3y )x y

−1 −1 −2 = 3a1y + 4a2x + 3a3x y.

−1,0 Mapping these through dH , we obtain the element

−1 −1 −1 −2 −1 −2 γ := ((0, −2a2y , 3a1y ), (0, −6a3x − 12a4x y, 4a2x + 3a3x y))

⊕3 ⊕3 −1,0 of Γ Ux, O(−1) ⊕ Γ Uy, O(−1) . When applying the the map dV , it does not matter on which Chapter 2. Matrix Factorizations 35 element of the pair we calculate, so we will calculate with the ﬁrst one.

−1,0 −1 −1 dV (γ) = Fxx · 0 + Fxy · (−2a2y ) + Fyy · (3a1y )

2 2 −1 2 2 −1 = −2(3a1x + 4a2xy + 3a3y )a2y + 3(2a2x + 6a3xy + 12a4y )a1y

2 = (18a1a3 − 8a2)x + (36a1a4 − 6a2a3)y.

In the same manner we compute the image under the last diﬀerential of each basis element of

1 1 ⊕3 H P , O(−3) .

From this, we obtain the matrix representing ∂1:

 T  18a a − 8a2 36a a − 6a a   1 3 2 1 4 2 3     2   6a2a3 − 36a1a4 18a3 − 48a2a4     2  12a1a2 − 72a0a3 4a2 − 144a0a4    .    144a a − 4a2 72a a − 12a a   0 4 2 1 4 2 3    2   18a1 − 48a0a2 6a1a2 − 36a0a3     2   36a0a3 − 6a1a2 18a1a3 − 8a2 

0 Example 2.3.5. We may also construct a resolution of a module M locally isomorphic to OΣ¯ 4,3 over the ambient space of O∆ by setting the pivot to j = 2. In this case, the resolution has the form

- ⊕2 - ⊕6 - ⊕4 - 0 - 0 OH(−3) 0 OH(−1) 0 OH M 0. ∂2 ∂1

0 0 The reader may verify using the same technique as above that ∂1 and, respectively, ∂2 are given by the transposes of the matrix representing ∂2 and, respectively, ∂1 in Example 2.3.4.

2.4 On the ranks of the presentation matrices of Σ¯ n,k

In this section, we present an interpretation of the ranks of the presentation matrices of Σ¯ n,k over the ambient space of Σn,k and use this result to prove Theorem 1.4.1 characterizing the singular locus of

Σn,k.

Proposition 2.4.1. Fix n ≥ k > 1. Let A be a presentation matrix of a sheaf over the ambient space of Σn,k which is locally isomorphic to Σ¯ n,k. Let f be a polynomial of degree n and let Af be the matrix Chapter 2. Matrix Factorizations 36

A specialized to the coeﬃcients of f. Then the nullity of Af is

X max(µ(α) − (k − 1), 0) α where the sum is over all distinct roots α of f and µ(α) is the multiplicity of α as a root of f.

Proof. The discussion in Section 2.3 shows that A is obtained as the last differential in the pushforward of the Koszul complex defining Σ¯ n,k. Specializing to f corresponds to specializing the Koszul complex to the fibre L over f. It therefore suffices to compute the length l of the cokernel of the last differential d in the restriction of this Koszul complex to L. We have that

X l = length coker dα α∈L where dα is the localization of d at the point α ∈ L. However, d is given by the kth order partial derivatives of f. Thus, for α ∈ L, length coker dα is just the minimum of the multiplicities of α as a root of each of the k − 1-order partial derivatives of f, which is µ(α) − (k − 1) if α is a root of multiplicity k and 0 otherwise. The claim follows.

We can now prove Theorem 1.4.1 as a corollary.

Proof of Theorem 1.4.1. Proposition 2.4.1 implies that the radical of the first Fitting ideal F1(Σ¯ n,k) as a module over the ambient space of Σn,k is the ideal defining the locus of polynomials of degree n with p more than one root of multiplicity k or with a root of multiplicity greater than k. By [Eis95] F1(Σ¯ n,k) is also the ideal defining the locus where Σ¯ n,k cannot be generated by one element as a module over

Σn,k, which is the locus where the normalization map Σ¯ n,k → Σn,k is not an isomorphism. Since Σ¯ n,k is smooth, the normalization map is a desingularization, so the above locus is just the singular locus of

Σn,k. The codimensions are obtained by parameterizing the above loci and counting the number of parameters. It suﬃces to perform this computation in the aﬃne setting. Consider the locus of polynomials of degree n with a root of multiplicity greater than k. A general polynomial of degree n with a root of multiplicity at least k + 1 is of the form

k+1 n−(k+1) n−(k+1)−1 (x − t) (x + s1x + ··· + sn−(k+1)),

so the locus of such polynomials has dimension n − (k + 1) + 1 = n − k and its codimension in Σn,k is (n − k + 1) − (n − k) = 1. Chapter 2. Matrix Factorizations 37

Now consider the locus of polynomials of degree n with more than one root of multiplicity k. If 2k > n, then a polynomial of degree n cannot have more than one root of multiplicity k, so this locus is empty. If 2k ≤ n, then a general polynomial with more than one root of multiplicity k is of the form

k k n−2k n−2k−1 (x − t1) (x − t2) (x + s1x + ··· + sn−2k),

so the locus of such polynomials has dimension n − 2k + 2 and its codimension in Σn,k is (n − k + 1) − (n − 2k + 2) = k − 1.

We also obtain a nice characterization of the rank of a minimal presentation of the normalization ∆¯ n of the discriminant ∆n.

Corollary 2.4.2. Let n > 0 and let A be a minimal presentation of ∆¯ n as a module over ∆n. Let f be a polynomial of degree n and let Af be the specialization of A to f. Then rank Af is the number of distinct roots of f.

Proof. This follows from Proposition 2.4.1. Chapter 3

The Open Swallowtail

This chapter contains the main results of the thesis. We construct a hitherto undiscovered determinantal formula for Dn as the presentation matrix of a canonically deﬁned O∆-module called the open swallowtail. We then connect the algebraic deﬁnition with the original construction of the open swallowtail, due to Arnol’d and extensively studied by Givental. Finally, we use the related geometry of the open swallowtail to describe some properties of the presentation matrix. In particular, when specialized to a particular polynomial, the rank of the presentation matrix yields information on the root structure of that polynomial which the classical formulae cannot discern.

3.1 An algebraic deﬁnition

The following proposition rephrases Proposition 1.3.2.

Proposition 3.1.1. Let n > k ≥ 2. The variety Σ¯ n,k+1 embeds in Σ¯ n,k as a smooth subvariety of

codimension 1. In particular, OΣ¯ n,k is a Cohen-Macaulay module over O∆n of codepth k − 2.

The following theorem is key. It characterizes the sheaf of relative diﬀerentials Ω1 and provides Σ¯ n,k/Σn,k the main motivation for the algebraic deﬁnition of the open swallowtail, as well as the means to construct

1 1 ¯ its presentation explicitly. Denote by p the restriction of the projection H × P → P to Σn,k.

1 ∼ ∗ 1 Theorem 3.1.2. For n > k ≥ 2, we have Ω¯ = OΣ¯ ⊗ p Ω 1 as OΣ¯ -modules. Σn,k/Σn,k n,k+1 P /K n,k

0 Proof. For notational convenience, let Γ := Σn,k and let Γ := Σn,k+1. Let π : Γ¯ → Γ, respectively

π0 : Γ¯0 → Γ0 be the normalization maps as given in Chapter 1. Let i :Γ ,→ H and ¯i : Γ¯ ,→ H × P1 be the

38 Chapter 3. The Open Swallowtail 39 natural embeddings. This gives rise to the diagram

Γ¯ - Γ π

¯i i (3.1) ? ? 1 - H × P H. pH

1 ∼ ∗ 1 ∗ 1 1 1 1 We have Ω 1 = p Ω ⊕ p Ω 1 , where p1 : H × → and pH : H × → H are the H×P /K H H/K 1 P /K P P P natural projections. The Zariski-Jacobi sequence for the maps H×P1 → H → K is just the split sequence associated to this direct sum decomposition:

- ∗ 1 - 1 - 1 ∼ ∗ 1 - 0 p Ω Ω 1 Ω 1 = p Ω 1 0. (3.2) H H/K H×P /K H×P /H 1 P /K

The relations in (3.1), along with the deﬁnition of p, imply that ¯i∗p∗ Ω1 =∼ π∗i∗Ω1 and that H H/K H/K ¯∗ ∗ 1 ∼ ∗ 1 ¯∗ i p Ω 1 = p Ω 1 . Furthermore, because (3.2) is split, it remains exact after the application of i ⊗−. 1 P /K P /K In view of the commutativity of (3.1), we obtain the following commutative diagram:

∗ 1 - 1 - 1 - π Ω Ω¯ Ω¯ 0 Γ/K Γ/K Γ/Γ 6 6

- ∗ ∗ 1 - ∗ 1 - ∗ 1 - 0 π i Ω ¯i Ω 1 p Ω 1 0. H/K H×P /K P /K

The top row is the Zariski-Jacobi sequence associated to the maps Γ¯ → Γ → K and the bottom row is ∗ ∗ 1 ∗ 1 ¯∗ 1 1 (3.2). The vertical maps are the surjections π i Ω → π Ω and i Ω 1 → Ω¯ induced by H/K Γ/K H×P /K Γ/K ¯ ∗ 1 1 the embeddings i and i. There is therefore an induced surjection ρ : p Ω 1 → Ω , as in the following P /K Γ¯/Γ diagram: 0 0 0 6 6 6 ∗ 1 - 1 - 1 - π Ω Ω¯ Ω¯ 0 Γ/K Γ/K Γ/Γ 6 6 6 (3.3) ∃ρ

- ∗ ∗ 1 - ∗ 1 - ∗ 1 - 0 π i Ω ¯i Ω 1 p Ω 1 0. H/K H×P /K P /K

∗ 1 ¯0 ¯ It now suffices to prove that ker ρ = J ⊗ p Ω 1 , where J is the ideal sheaf defining Γ in Γ. We P /K ∗ 1 ∗ 1 ¯ 1 have an injection J ⊗ p Ω 1 ,→ p Ω 1 . Let I be the ideal sheaf defining Γ in H × . The cotangent P /K P /K P sequence d 2 - ¯∗ 1 - 1 - I/I i Ω 1 Ω¯ 0 H×P /K Γ/K Chapter 3. The Open Swallowtail 40 then gives rise to the diagram 0 0 6 6

1 - 1 Ω¯ Ω¯ Γ/K Γ/Γ 6 6 ρ (3.4)

∗ 1 - ∗ 1 ¯i Ω 1 p Ω 1 H×P /K P /K 6 6

I/I2 ker ρ whose middle rows come from (3.3).

2 ¯∗ 1 ∗ 1 ∗ 1 We claim ﬁrst that the composition I/I → i Ω 1 → p Ω 1 factors through J ⊗ p Ω 1 H×P /K P /K P /K 2 ∗ 1 ∂i+j and that the resulting map I/I → J ⊗ p Ω 1 is surjective. For i, j ≥ 0, let Fi,j := i j F . Then P /K ∂ x∂ y

I is deﬁned by the sections {Fi,k−i−1 | 0 ≤ i ≤ k − 1} of O(1, n − k + 1). The kernel of the surjection ¯∗ 1 1 ¯∗ 1 i Ω 1 → Ω¯ is the subsheaf of i Ω 1 generated by the sections H×P /K Γ/K H×P /K

dFi,k−i−1 = α0 da0 + ··· + αn dan + Fi+1,k−i−1 dx + Fi,k−i dy (3.5)

for 0 ≤ i ≤ k −1, where α0, . . . , αn are sections of O(0, n−k +1). The summand α0 da0 +···+αn dan is

∗ ∗ 1 ∗ 1 in the component π i Ω and the summand Fi+1,k−i−1 dx + Fi,k−i dy is in the component p Ω 1 . H/K P /K 2 ¯∗ 1 Since J is generated by Fi,k−i for 0 ≤ i ≤ k, the image of the composition I/I → i Ω 1 → H×P /K ∗ 1 ∗ 1 ∗ 1 p Ω 1 lies in the image of the injection J ⊗ p Ω 1 ,→ p Ω 1 , showing the existence of a map P /K P /K P /K 2 ∗ 1 2 ¯∗ 1 ∗ 1 χ : I/I → J ⊗ p Ω 1 through which the composition I/I → i Ω 1 → p Ω 1 factors. P /K H×P /K P /K We now show that χ is surjective by a local calculation. Recall that, on Γ,¯

0 = (n − k)Fi,k−i−1 = xFi+1,k−i−1 + yFi,k−i.

On the aﬃne piece {y 6= 0}, we therefore have

x F = − F . i,k−i y i+1,k−i−1

By induction, xk−i F = (−1)k−i F . i,k−i y k,0 Chapter 3. The Open Swallowtail 41

1 1 ∗ 1 In particular, since Ω 1 is generated by (y dx − x dy) on the aﬃne piece {y 6= 0}, J ⊗ p Ω 1 is P /K y2 P /K generated by 1 F (y dx − x dy) k,0 y2 on this aﬃne piece. On the other hand,

xk−i−1 xk−i F dx + F dy = (−1)k−i−1 F dx + (−1)k−i F dy i+1,k−i−1 i,k−i y k,0 y k,0 xk−i−1 1 = (−1)k−i−1y F (y dx − x dy) y k,0 y2

1 Thus the image of χ is also generated by Fk,0 y2 (y dx−x dy) on this aﬃne piece, and is therefore surjective on it. The argument that χ is surjective on the other aﬃne piece {x 6= 0} is entirely symmetric.

∗ 1 ∗ 1 ∗ 1 The argument above implies that the injection J ⊗ p Ω 1 → p Ω 1 maps J ⊗ p Ω 1 into P /K P /K P /K ∗ 1 1 ∗ 1 ker ρ : p Ω 1 → Ω . To prove the result, it remains to show that the image of J ⊗ p Ω 1 is in fact P /K Γ¯/Γ P /K ¯ 1 1 all of ker ρ. To do this, we restrict to an aﬃne open subset U of Γ on which ΩΓ¯/Γ is trivial. Since ΩΓ¯/Γ is a cyclic module, applying the local trivialization identiﬁes Γ(U, ker ρ) with an ideal I ⊆ Γ(U, OΓ¯ ). The above arguments show that I contains Γ(U, J ), which, since Γ¯0 is smooth and connected, is reduced and irreducible of codimension one. Hence any ideal properly containing Γ(U, J ) has codimesion at least two.

1 We claim that ΩΓ¯/Γ is supported in codimension one. In view of the observation above, this will complete the proof of the theorem. To prove the claim, it suffices to restrict our attention to the affine subset {y 6= 0} and we shall do so. Let ∆ := ∆n−k+2 be the discriminant of degree n+k−2 polynomials. As discussed in Section 1.4, on the affine set {y 6= 0}, differentiation defines a map Γ → ∆. Furthermore, by Proposition 1.4.6, Γ¯ is isomorphic to the normalization ∆¯ of ∆ in such a way as is compatible with the differentiation map. Now let p ∈ Γ¯0 ⊆ Γ¯ be a sufficiently general point, namely, a point corresponding to a polynomial with exactly one root of multiplicity k + 1. The imagep ¯ of p under the map Γ¯ → ∆ lies on the caustic Σn−k−2,3. By Theorem 1.4.4, in a neighbourhood ofp ¯, ∆ is isomorphic to the product ˜ ∼ 3 2 ∼ ˜ of ∆3 = Spec K[x, y]/(x − y ) and a smooth factor. Let N = Spec K[t] be the normalization of ∆3. 1 As verified by routine calculation, Ω is supported on the preimage of the singular locus of ∆˜ 3. In N/∆˜ 3 1 particular, sincep ¯ lies on the singular locus of ∆, Ω∆¯ /∆ is supported atp ¯. Proposition 1.4.8 indicates 1 ∼ 1 ¯ 1 ¯0 that ΩΓ¯/Γ = Ω∆¯ /∆ as Γ-modules. Thus ΩΓ¯/Γ is supported at p. Since the subset of Γ from which p was 1 ¯0 ¯0 chosen is open and dense, ΩΓ¯/Γ is supported generically on Γ . Since Γ has codimension one, the claim is proved.

Once we have characterized Ω1 , the following theorem, which motivates the deﬁnition of the Σ¯ n,k/Σn,k Chapter 3. The Open Swallowtail 42 open swallowtail, is immediate.

1 Theorem 3.1.3. The universal derivation d : O∆¯ → Ω∆¯ /∆ is surjective.

1 Proof. This follows immediately from Theorem 3.1.2: locally, Ω∆¯ /∆ is cyclic and d takes a local section ∂g g(x, a1, . . . , an) ∈ O∆¯ to ∂x dx.

Deﬁnition 3.1.4. The n-th (algebraic) open swallowtail Sn is ker d. It is an O∆-subalgebra of O∆¯ . We refer Sn via S when the degree is understood.

The following proposition shows that S indeed deﬁnes a determinantal formula for Dn.

Proposition 3.1.5. The open swallowtail is a maximal Cohen-Macaulay module of rank 1 over O∆.

Proof. We have that O∆¯ is maximal Cohen-Macaulay over O∆ and, by Proposition 3.1.1 and Theorem 1 3.1.2, Ω∆¯ /∆ has codepth 1. The exact sequence

- - - 1 - 0 S O∆¯ Ω∆¯ /∆ 0

1 of O∆ modules implies that the depth of S is at least the minimum of depth O∆¯ and depth Ω∆¯ /∆ + 1.

Thus S is maximal Cohen-Macaulay. That S has rank 1 follows from its being embedded in O∆¯ .

Moreover, the formula deﬁned by S is nontrivial and not equivalent to the classical formulae, as we shall see in the next section.

3.2 Construction of the presentation matrix

We now describe how to construct a presentation matrix of S. We do this with the mapping cone construction applied to the short exact exact sequence

- - - 1 - 0 S O∆¯ Ω∆¯ /∆ 0.

Applying Theorem 3.1.2 and the constructions in Section 2.3 allows us to construct OH-resolutions of

1 O∆¯ and Ω∆¯ /∆.

We use the Cayley method with pivot j = 1 to construct resolutions of pH∗ (O∆¯ (n − 2)) and

1 ∼ ∗ 1 p Ω (n − 3) = p O¯ ⊗ p Ω 1 (n − 3) H∗ ∆¯ /∆ H∗ Σn,3 P /K Chapter 3. The Open Swallowtail 43

0 0 6 6 1 pH∗ (O∆¯ (n − 2)) pH∗ Ω∆¯ /∆(n − 3) 6 6

0 1 0 1 1 O ⊗ H , O 1 (n − 2) O ⊗ H , Ω 1 (n − 3) H K P P H K P P /K 6 6 A ∂1

1 1 1 1 1 ⊕3 O (−2) ⊗ H , O 1 (−n) O (−2) ⊗ H , Ω 1 (1 − n) H K P P H K P P /K 6 6 ∂2

1 1 1 0 O (−3) ⊗ H , Ω 1 (3 − 2n) H K P P /K 6 0

1 Figure 3.1: Resolutions of pH∗ (O∆¯ (n − 2)) and pH∗ Ω∆¯ /∆(n − 3) over OH. over H. The resolutions have the form shown in Figure 3.1, in which A is the (projective) B´ezoutmatrix and ∂1 and ∂2 are the maps described in Section 2.3.

There is, however, one diﬃculty: while the universal derivation

1 d : pH∗O∆¯ → pH∗Ω∆¯ /∆

is a well-deﬁned map of OH-modules, it cannot be twisted to form a map

1 pH∗ (O∆¯ (n − 2)) → pH∗ Ω∆¯ /∆(n − 3) .

We shall therefore construct maps D0 and D1 making the diagram shown in Figure 3.2 commute and

1 such that D0 is a lifting of d on the affine subset Uy = {y 6= 0} of H×P . The mapping cone construction 0 will then yield a module S which is isomorphic to S on Uy, but may not agree with S globally. The affine piece Uy corresponds to univariate polynomials of degree n, so this restriction suffices for our purposes.

1 As in Section 2.3, we shall fix a point f ∈ H and compute in the fibre of pH over f. On Uy,Ω 1 is P /K 1 1 trivial, being generated freely by d(x/y) = y2 (y dx − x dy). In addition OP (i) is trivial for every i; the i 1 1 map g 7→ y g is an isomorphism Γ(Uy, OP ) → Γ(Uy, OP (i)). We define

1 D˜ :Γ(U , O 1 (n − 2)) → Γ U , Ω 1 (n − 3) 0 y P y P /K Chapter 3. The Open Swallowtail 44

0 0 6 6 0 1 - 0 1 1 O ⊗ H , O 1 (n − 2) O ⊗ H , Ω 1 (n − 3) H K P P H K P P /K D0 6 6 A ∂1

1 1 - 1 1 1 ⊕3 O (−2) ⊗ H , O 1 (−n) O (−2) ⊗ H , Ω 1 (1 − n) H K P P H K P P /K D1 6 6 ∂2

1 1 1 0 O (−3) ⊗ H , Ω 1 (3 − 2n) H K P P /K 6 0

Figure 3.2: Maps D0 and D1

n−3 2−n 1 ˜ by composing with the local trivializations: for h ∈ Γ(Uy, OP (n − 2)), D0(h) = y d y h = hx d(x/y). We then deﬁne

0 1 0 1 1 D : H , O 1 (n − 2) → H , Ω 1 (n − 3) 0 P P P P /K

0 1 ˜ 1 as the restriction of D0 to H P , OP (n − 2) .

Now we construct D1. To do so, we lift D˜ 0 through each map in the spectral sequence through which

1 1 ⊕3 0 1 1 1 0 1 1 1 1 1 the diﬀerentials H P , OP (1 − n) → H (P , OP (n − 3)) and H (P , OP (−n)) → H (P , OP (n − 2)) are found.

We begin with the lifting

1 ⊕3 - 1 Γ U , Ω 1 (−1) Γ U , Ω 1 (n − 3) y P /K y P /K 6 6 ˜ (1) ˜ D1 D0 .

⊕2 1 - 1 Γ Uy, OP (−1) Γ(Uy, OP (n − 2))

˜ (1) ⊕2 Finding D1 amounts to ﬁnding, for given g1, g2 ∈ Γ Uy, O(−1) , some sections

⊕3 g˜1, g˜2, g˜3 ∈ Γ Uy, O(−1) such that x (˜g f +g ˜ f +g ˜ f ) d = D (g f + g f ). 1 xx 2 xy 3 yy y 0 1 x 2 y Chapter 3. The Open Swallowtail 45

Using the identities

1 f = (xf + yf ) x n − 1 xx xy 1 f = (xf + yf ), y n − 1 xy yy we compute directly

D˜ 0(g1fx + g2fy) x = (g f + g f + g f + g f ) d 1x x 1 xx 2x y 2 xy y 1 1 x = g (xf + yf ) + g f + g (xf + yf ) + g f d n − 1 1x xx xy 1 xx n − 1 2x xy yy 2 xy y 1 1 1 1 x = g + xg f + g + yg + xg f + yg f d . 1 n − 1 1x xx 2 n − 1 1x n − 1 2x xy n − 1 2x yy y

Thus

1 g˜ = g + xg 1 1 n − 1 1x 1 1 g˜ = g + yg + xg 2 2 n − 1 1x n − 1 2x 1 g˜ = yg . 3 n − 1 2x

˜ (1) This provides the lifting D1 .

The next lifting 1 ⊕3 - 1 ⊕3 Γ U , Ω 1 (−1) Γ U , Ω 1 (−1) xy P /K y P /K 6 6 ˜ (2) ˜ (1) D1 D1

⊕2 ⊕2 1 - 1 Γ Uxy, OP (−1) Γ Uy, OP (−1) ˜ (1) is easy — we just use the same deﬁnition as for D1 . We proceed to the ﬁnal lifting

1 ⊕3 - 1 ⊕3 Γ U , Ω 1 (1 − n) Γ U , Ω 1 (−1) xy P /K xy P /K 6 6 ˜ ˜ (2) D1 D1 .

⊕2 1 - 1 Γ(Uxy, OP (−n)) Γ Uxy, OP (−1) Chapter 3. The Open Swallowtail 46

1 This is tantamount to ﬁnding, for a given section g ∈ Γ(Uxy, OP (−n)), sections

1 ⊕3 h , h , h ∈ Γ U , Ω 1 (1 − n) 1 2 3 xy P /K such that

1 h f − h f = g + xg 2 yy 3 xy 1 n − 1 1x 1 1 −h f + h f = g + yg + xg 1 yy 3 xx 2 n − 1 1x n − 1 2x 1 h f − h f = yg , 1 xy 2 xx n − 1 2x

where g1 := fyg and g2 := −fxg. We compute each in turn.

1 h f − h f = g + xg 2 yy 3 xy 1 n − 1 1x 1 = f g + x(f g + f g) y n − 1 y x xy 1 1 1 = (xf + yf )g + x (xf + yf )g + f g n − 1 xy yy n − 1 n − 1 xy yy x xy 1 1 1 1 = y g + xg f + x 2g + xg f . n − 1 n − 1 x yy n − 1 n − 1 x xy

1 1 −h f + h f = g + yg + xg 1 yy 3 xx 2 n − 1 1x n − 1 2x 1 1 = −f g + y(f g + f g ) − x(f g + f g ) x n − 1 xy y x n − 1 xx x x 1 1 1 = − (xf + yf )g + y f g + (xf + yf )g + n − 1 xx xy n − 1 xy n − 1 xy yy x 1 1 − x f g + (xf + yf )g n − 1 xx n − 1 xx xy x 1 1 1 = y2g f − x(2g + xg )f + (n − 1)2 x yy n − 1 n − 1 x xx 1 1 1 −yg + yg + xyg − xyg f n − 1 n − 1 x n − 1 x xy 1 1 1 = y2g f − x 2g + xg f . (n − 1)2 x yy n − 1 n − 1 x xx Chapter 3. The Open Swallowtail 47

1 h f − h f = yg 1 xy 2 xx n − 1 2x 1 = − y(f g + f g ) n − 1 xx x x 1 1 = − y(f g + (yf + xf )g ) n − 1 xx n − 1 xy xx x 1 1 1 = − y2g f − y g + xg f . (n − 1)2 x xy n − 1 n − 1 x xx

Thus we conclude

1 h = − y2g 1 (n − 1)2 x 1 1 h = y g + xg (3.6) 2 n − 1 n − 1 x 1 1 h = − x 2g + xg . 3 n − 1 n − 1 x

1 Thus, for a section g ∈ Γ(Uxy, OP (−n)), we set

1 1 1 1 1 D˜ (g) := − y2g , y g + xg , − x 2g + xg . 1 (n − 1)2 x n − 1 n − 1 x n − 1 n − 1 x

1 1 ˜ 1 We then deﬁne D1 as the restriction of D1 to H P , OP (−n) .

The map of resolutions which we have just constructed gives rise to the mapping cone

0 0 G2

∂2 ? ? ? - - G2 ⊕ F1 F1 G1 D1

B A ∂1 . ? ? ? - - G1 ⊕ F0 F0 G0 D0

E ? ? ? G0 0 0

The matrix A is just the projective B´ezoutmatrix. The matrix representing B has the form

  ∂2 D1     .  0 A  Chapter 3. The Open Swallowtail 48

Recall from Section 2.3 that ∂2 has a Sylvester-like structure, namely, it divides horizontally into three blocks, each one corresponding to one second-order partial derivative of F . Each block then has n − 2 rows and the ith row of a given block contains the coeﬃcients of the associated partial derivative of F shifted to the right by i − 1 positions.

The map D0 is essentially diﬀerentiation with respect to x. Since D0 is surjective, the complex

0 - G ⊕ F - G ⊕ F - G - 0 2 1 B 1 0 0

0 has homology S at G1 ⊕ F0 and is exact elsewhere. In particular, there is a subcomplex

- - - 0 F¯0 G0 0,

n−2 n−3 n−3 where F¯0 is the sub-vector space of F0 generated by x , x y, . . . , xy . Removing this subcomplex,

0 we obtain a presentation of S over OH. We have proved:

Theorem 3.2.1. The matrix   ∂2 D1    , (3.7)    0 A(yn−2) where ∂2 and D1 are as given above and A(yn−2) is the row of the B´ezoutmatrix A corresponding to the

n−2 0 basis element y of H (X, O(n − 2)), presents a module isomorphic on the open aﬃne set Uy to the open swallowtail Sn.

We may also use this construction to count the minimal number of generators of Sn on the aﬃne piece {a0 6= 0}

Proposition 3.2.2. On the aﬃne piece {a0 6= 0}, the open swallowtail Sn is minimally generated by n − 2 elements. In particular, it is not free and not isomorphic to the normalization of ∆n.

Proof. After restricting to the aﬃne piece {a0 6= 0}, the now invertible variable a0 appears in exactly n − 2 of the 2(n − 2) + (n − 1) = 3n − 5 columns of (3.7). Furthermore, there are invertible elements in each of the n − 1 columns of D1. None of the other columns has an invertible element. Thus a minimal presentation matrix of Sn is of size (n − 2) × (n − 2).

Example 3.2.3. We construct a presentation matrix of the open swallowtail for the degree 4 discriminant. In accordance with the above discussion, we construct a presentation of the open swallowtail over the Chapter 3. The Open Swallowtail 49

aﬃne piece {y 6= 0}. Recall from Example 2.3.4 that the matrix representing ∂2 is

  12a 6a 2a 0   0 1 2       0 12a0 6a1 2a2       3a1 4a2 3a3 0  ∂ =   . 2    0 3a 4a 3a   1 2 3       2a2 6a3 12a4 0       0 2a2 6a3 12a4

The map D1 is given by the formulae (3.6) above. We compute

1 1 D (x−3y−1) = − yx−4, 0, − x−2y−1 1 3 3 2 1 4 D (x−2y−2) = x−3, x−2y−1, − x−1y−2 1 9 9 9 1 2 5 D (x−1y−3) = x−2y−1, x−1y−2, − y−3 . 1 9 9 9

Thus D1 has the form    0 0 1   9       0 0 0    1   0 0 D =  9  . 1    0 0 2   9    − 1 0 0  3     4   0 − 9 0

The matrix A is the B´ezoutmatrix

   a a − 16a a 2a a − 12a a 3a2 − 8a a   1 3 0 4 2 3 1 4 3 2 4    A =  2  . 2a1a2 − 12a0a3 4a2 − 8a1a3 − 16a0a4 2a2a3 − 12a1a4    2   3a1 − 8a0a2 2a1a2 − 12a0a3 a1a3 − 16a0a4  Chapter 3. The Open Swallowtail 50

Putting these together, S0 is presented by

 1  12a0 6a1 2a2 0 0 0   9     0 12a 6a 2a 0 0 0   0 1 2     1   3a1 4a2 3a3 0 0 9 0    0  2  S = coker  0 3a1 4a2 3a3 0 0  .  9     2a 6a 12a 0 − 1 0 0   2 3 4 3     4   0 2a2 6a3 12a4 0 − 9 0     2   0 0 0 0 a1a3 − 16a0a4 2a2a3 − 12a1a4 3a3 − 8a2a4

Let η1, η2, η3 be respectively the (7, 5), (7, 6), and (7, 7) entries – that is, the B´ezoutentries – of the presentation matrix above. Then, after applying necessary row operations, we obtain the following minimal presentation matrix:

   0 12a0 6a1 2a2      −24a −9a 0 3a  0  0 1 3  S = coker  ,    12a1 18a2 18a3 12a4      γ1 γ2 γ3 0  where

γ1 = −9 · 12a0 · η3 − 9 · 3a1 · η2 + 3 · 2a2 · η1,

γ2 = −9 · 6a1 · η3 − 9 · 4a2 · η2 + 3 · 6a3 · η1,

γ3 = −9 · 2a2 · η3 − 9 · 3a3 · η2 + 3 · 12a4 · η1.

Restricting to monic polynomials by setting a0 = 1 and making a few simpliﬁcations, we get a minimal presentation

  2  3(2a3 − a1a2) 4a4 − a   2  Γ(Uy, S) = coker  .  1 3 2 1 1 1  γ3 − 2 a1γ2 + 16 a1γ1 8 a3γ1 − 6 a2γ2 + 16 a1a2γ1

If we further restrict to the reduced discriminant by setting a1 = 0 and again simplify, we get the presentation matrix   2  a3 4 4a4 − a2   .  2 2 2 3  3a2a3 + 4a4 4a4 − a2 a3 27a3 − 8a2 − 96a2a4  Chapter 3. The Open Swallowtail 51

Figure 3.3: A schematic diagram of the open swallowtail Σ4

3.3 The construction of Arnol’d

The original deﬁnition of the open swallowtail is due to Arnol’d [Arn81]. In this section, we describe his construction and show that it is equivalent to Deﬁnition 3.1.4. Recall the tower of caustics

· · · → Σn+i+1,i+3 → Σn+i,i+2 → Σn+i−1,i+1 → · · · → Σn+1,3 → Σn,2 = ∆n (3.8) deﬁned in Section 1.4. The following proposition appears in [Giv82].

Proposition 3.3.1. For n ≥ 2 and for i > n − 3, the diﬀerentiation map Σn+i,i+2 → Σn+i−1,i+1 is an algebraic isomorphism. Thus the tower (3.8) stabilizes at i = n − 3.

Proof. See [Giv82, Theorem 1].

Deﬁnition 3.3.2. The (geometric) open swallowtail associated to ∆n, denoted Σn (or Σ when n is understood), is the variety obtained at the point where the tower (3.8) stabilizes, namely, Σ2n−3,n−1 in the notation above.

The term open swallowtail is used because Σ is a partial normalization of ∆ in which the self- intersection locus bifurcates, but the caustic remains. This is intuitively clear: a polynomial of degree 2n − 3 cannot have two roots of multiplicity n − 1. However, it certainly can have a root of multiplicity strictly greater than n − 1. Chapter 3. The Open Swallowtail 52

Since the differentiation map is finite, the coordinate ring OΣ of Σ is a finite module over O∆ which, in view of Proposition 1.4.6, embeds in O∆¯ . As such, it agrees with the algebraic open swallowtail introduced above. More precisely, the module Γ(Uy, S), to which we shall refer again as S, is a module over the affine discriminant O∆ which also by definition embeds in O∆¯ . We have the following result.

Theorem 3.3.3. The image of OΣ in O∆¯ equals S.

1 ∼ 1 Proof. Proposition 1.4.8 shows that ΩΣ¯/Σ = Ω∆¯ /∆, which implies that OΣ is contained in S. We claim that the embedding OΣ ,→ S is surjective. From Theorem 3.1.2, we see that the kernel of d consists

00 of those elements g ∈ O∆¯ which, after being diﬀerentiated with respect to x, are divisible by f . Such elements are of the form

Z x 00 g(x, s1, . . . , sn−2) = h1(t, s1, . . . , sn−2)f dt + h2(s1, . . . , sn−2). 0

Writing h1 as a polynomial in t with coeﬃcients in K[s1, . . . , sn−2], we see that g − h2 is an O∆-linear combination of elements of the form

Z x f 00(t)t(i−1) dt, i ≥ 1 0

and therefore, by Lemma 1.4.7, in the image of OΣ. On the other hand, h2 is in the image of O∆ and a fortiori of OΣ. This proves the claim.

In [SvS04], using Givental’s results in [Giv88], Sevenheck and van Straten show that Σ is a Cohen-

Macaulay variety, and hence that OΣ is maximal Cohen-Macaulay over O∆. Our results yield a new, algebraic proof that the open swallowtail of Arnol’d is Cohen-Macaulay.

Corollary 3.3.4. The open swallowtail Σ is Cohen-Macaulay.

Proof. Combine Theorem 3.3.3 with Proposition 3.1.5.

3.4 The conductor of the open swallowtail

We now study the conductor c of the inclusion O∆ → OΣ. Our main result along these lines is

Theorem 3.4.1. The closed subset of ∆ deﬁned by c is the self-intersection locus of ∆.

We start with the following immediate corollary of Theorem 1.4.1. Chapter 3. The Open Swallowtail 53

Proposition 3.4.2. The closed subset of ∆ deﬁned by the conductor d of the normalization map O∆ ,→

O∆¯ is the singular locus of O∆. It has two irreducible components: the caustic and the self-intersection locus.

Lemma 3.4.3. The conductor d is reduced at points of the caustic which are not in the self-intersection locus.

Proof. Such points correspond to polynomials of degree n with exactly one root of multiplicity exactly three and with all other roots distinct. Theorem 1.4.4 implies that, locally at such points, ∆ is the product of a smooth factor and the reduced degree three discriminant ∆˜ 3. The claim now follows from the equivalent claim for ∆˜ 3, which is an easy calculation.

1 The following lemma indicates that Ω∆¯ /∆ is a Gorenstein module over O∆.

Lemma 3.4.4. We have Ext1 (Ω1 , O ) ∼ Ω1 . O∆ ∆¯ /∆ ∆ = ∆¯ /∆

1 Proof. Proposition 3.1.1 and Theorem 3.1.2 imply that Ω∆¯ /∆ is presented as a quotient of O∆¯ by a principal ideal thereof. That is, we have an exact sequence

- - - 1 - 0 O∆¯ O∆¯ Ω∆¯ /∆ 0

of O∆¯ -modules. Treating it as an exact sequence of O∆ modules and applying HomO∆ (−, O∆), we obtain the sequence

- - - 1 1 - 1 0 HomO (O ¯ , O∆) HomO (O ¯ , O∆) Ext Ω ¯ , O∆ Ext (O ¯ , O∆) = 0, ∆ ∆ η ∆ ∆ O∆ ∆/∆ O∆ ∆

where the equality on the right follows from O∆¯ being maximal Cohen-Macaulay over the Gorenstein ring O∆. Since O∆¯ is a maximal Cohen-Macaulay module on a hypersurface and is presented by a symmetric matrix, it is self-dual. The map η is just multiplication by the same non-zerodivisor which presents Ω1 . Thus both Ω1 and Ext1 (Ω1 , O ) have the same presentations as O -modules, ∆¯ /∆ ∆¯ /∆ O∆ ∆¯ /∆ ∆ ∆ and the claim follows.

For lack of a suitable reference, we include the following standard lemma with proof.

Lemma 3.4.5. Suppose that X → Y is a ﬁnite, birational map of irreducible aﬃne varieties with conduc-

tor c. Then the map HomOY (OX , OY ) → OY sending α to α(1) is an isomorphism of HomOY (OX , OY ) onto c. Chapter 3. The Open Swallowtail 54

Proof. Denote the map above by ϕ. Since c is an ideal both of OX and of OY , given an element x ∈ c, there exists an OX -homomorphism y 7→ xy ∈ OX . This maps 1 ∈ OX to x ∈ c, so the image of ϕ contains c.

We now show that c contains the image of ϕ. This entails showing that, if α ∈ HomOY (OX , OY ), then α(1) annihilates OX /OY , or, equivalently, α(1) · x ∈ OY for all x ∈ OX . Let x ∈ OX and c ∈ c.

Then, since cx ∈ OY ,

c · (α(1) · x − α(x) · 1) = cα(1) · x − cα(x) · 1

= α(1) · cx − cα(x) · 1

= cxα(1) · 1 − cα(x) · 1

= (α(cx) − α(cx)) · 1

= 0.

Since c ∈ c was arbitrary, c annihilates α(1) · x − α(x). Since OX is torsion-free as an OY -module,

α(1) · x = α(x) ∈ OY , as required.

Now suppose that α ∈ ker ϕ. Then α(1) = 0, so OY ⊆ ker α. Thus α factors through a map

OX /OY → c. But X → Y is birational, so OX /OY is torsion, and thus any map OX /OY → c is zero. Hence ϕ is injective, and thus an isomorphism.

Proof of Theorem 3.4.1. We begin with the short exact sequence

- - d- 1 - 0 OΣ O∆¯ Ω∆¯ /∆ 0 which results from combining the deﬁnition of the open swallowtail S with Theorem 3.3.3. Applying

HomO∆ (−, O∆), we obtain

j 1 1 0 - Hom (O ¯ , O ) - Hom (O , O ) - Ext Ω , O - 0, (3.9) O∆ ∆ ∆ O∆ Σ ∆ O∆ ∆¯ /∆ ∆

where exactness on the right follows from O∆¯ being maximal Cohen-Macaulay and O∆ being Gorenstein.

Now let c be the conductor of O∆ ,→ OΣ and d be the conductor of O∆ ,→ O∆¯ . Lemma 3.4.5 implies Chapter 3. The Open Swallowtail 55

∼ ∼ that c = HomO∆ (OΣ, O∆) and d = HomO∆ (O∆¯ , O∆) and that the following diagram commutes:

i d ⊂ - c w w w w w∼ w∼ w w w w -j HomO∆ (O∆¯ , O∆) HomO∆ (OΣ, O∆), where i : d → c is the natural inclusion and j is the same map as given in (3.9). Applying these identiﬁcations, Lemma 3.4.4, and Theorem 3.1.2 to (3.9), we obtain a commutative diagram

0 0 ? ? - i - - - 0 d c OΓ¯ 0

? ? O ======O ∆ ∆ (3.10)

? ? - O∆/d O∆/c

? ? 0 0.

The map i in (3.10) is an isomorphism outside of the caustic Γ. Thus, outside of Γ, the support of c equals that of d. Proposition 3.4.2 implies that this support is precisely the self-intersection locus.

It remains to show that, among points on Γ, c is supported only at the intersection of Γ and the self-intersection locus. The snake lemma applied to (3.10) implies the existence of an exact sequence

- - - - 0 OΓ¯ O∆/d O∆/c 0.

Let p be a point of Γ not in the self-intersection locus. Then, by Corollary 1.4.3, p is a smooth point of ∼ Γ, so the map Γ¯ → Γ is an isomorphism at p. Also, at p, by Lemma 3.4.3, O∆/d = OΓ. Thus O∆/c is not supported at p. This proves the claim. Chapter 3. The Open Swallowtail 56

3.5 Application to the root structure of a univariate polynomial

We now give a compelling application of the matrix of the open swallowtail. Recall Corollary 2.4.2, which states that the rank of a minimal presentation of the normalization of ∆n, specialized to a particular polynomial f, is the number of distinct roots of f. The rank of the matrix of the normalization is not able to distinguish between degenerate repeated roots and multiple repeated roots. For example, it cannot detect whether a polynomial has n − 2 distinct roots because it has two distinct pairs of double roots or because it has one root of multiplicity 3. We show here that the matrix of the open swallowtail can make this distinction.

Theorem 3.5.1. Let B be a minimal presentation of OΣ over the ambient ring of O∆. Suppose B is

n n−1 specialized to some polynomial f(x) = x + a1x + ··· + an. The nullity of the resulting matrix is at least 2 if and only if f(x) has more than one distinct pair of double roots.

Proof. Fix i ≥ 0. The ideal deﬁning the locus of polynomials f(x) for which the specialized matrix B has nullity at least 2 is the radical of the Fitting ideal F1(Σ). By [Eis95], Proposition 20.6, the points deﬁned by F1(Σ) are those where OΣ cannot be generated by one element, that is, where the map Σ → ∆ is not an isomorphism. The locus of such points is precisely the zero locus of the conductor of the map Σ → ∆, which, by Theorem 3.4.1, is precisely the self-intersection locus, whence the claim.

It is natural to ask what would be the meaning of the nullity of the matrix of the open swallowtail being strictly greater than two. It is likely that, for i ≥ 0, the nullity of this matrix is at least i+2 if and only if the polynomial f is the ith derivative of a polynomial with more than one root of multiplicity i+2. A proof analogous to that of Theorem 3.5.1 has some diﬃculty here because the intermediate varieties

Σn+i,i+2 for 0 < i < n − 3 are not in general Cohen-Macaulay. Thus this belief is still conjectural. Chapter 4

Deformations of Modules

In this chapter, we develop tools which one may use to construct moduli spaces of graded rank one maximal Cohen-Macaulay modules on the aﬃne discriminant. We begin with an observation.

Proposition 4.0.2. Suppose that n > 2. Then the sequence (a1, . . . , an−1) is a regular sequence on ∼ n−1 n b(n−1)/2c+n−1 n O∆n . Moreover, O∆n /(a1, . . . , an−2) = K[an−1, an]/(c1an −c2an−1), where c1 = (−1) n b(n−2)/2c+n−1 n−1 and c2 = (−1) (n − 1) .

∗ n−1 n Proof. Since, for c1, c2 ∈ K , the equation c1an − c2an−1 = 0 deﬁnes a curve singularity and the dimension of ∆˜ n is n − 2, the former claim follows from the latter. That the resulting curve singularity has the given form follows from quasihomogeneity of Dn: the only terms of degree n(n − 1) involving

n−1 n just an−1 and an are an and an−1. The calculation of c1 is done in Proposition 2.1.1 by computing n the discriminant of the polynomial x − 1. The calculation is c2 is similar and done by computing the discriminant of the polynomial xn − x.

˜ Remark 4.0.3. Since the reduced discriminant ∆n is Spec O∆n /(a1), the above proposition also implies that (a , . . . , a ) is a regular sequence on ∆˜ and that O /(a , . . . , a ) =∼ [a , a ]/(c an−1 − 2 n−1 n ∆˜ n 2 n−2 K n−1 n 1 n n ˜ c2an−1). In this chapter, we shall work exclusively with ∆n to simplify computations.

n−1 n After a change of coordinates, the curve Spec K[an−1, an]/(c1an − c2an−1) may be written as

n n−1 Spec K[x, y]/(x − y ).

Let Γn denote this curve. Suppose we have a maximal Cohen-Macaulay module M on ∆n. Then

M/(a2, . . . , an−2)M is maximal Cohen-Macaulay of the same rank over Γn. Thus we may reduce the classiﬁcation problem to the following two steps.

57 Chapter 4. Deformations of Modules 58

1. Find all graded rank one maximal Cohen-Macaulay modules over Γn.

2. Given a graded rank one maximal Cohen-Macaulay module over Γn, ﬁnd all ways of lifting it to

∆˜ n.

We address the second problem in the next two sections and leave the ﬁrst problem to Section 4.3. We then show in Section 4.4 the classiﬁcation of graded rank one MCM modules over ∆˜ 4.

4.1 Deformation theory of modules

In this section we introduce deformation theory of modules and cast the lifting problem therein. Through- out this section and the next, all spaces are formal germs over C with distinguished point 0, except where stated otherwise. A deformation of a module is analogous to an unfolding of a curve: it represents what can happen under perturbations of the presentation matrix, subject to the condition that the resulting matrix still naturally defines a module over the given curve. In order to construct liftings of a module defined on Γn to ∆˜ n, we add another twist: not only is the module deformed, but the curve singularity Γn is deformed as well in the direction of ∆˜ n. In this setting, a deformation consists of perturbed modules over the resulting perturbed singularities. More precisely, we have the following definition.

Definition 4.1.1. Let q : Y → Σ be a flat map with fibre X over 0. Let M0 be a module over X.A deformation of M0 is a triple (S, M, ϕ) such that S is a space equipped with a map to Σ, M is a module over Y ×Σ S which is flat over S, and ϕ is an isomorphism of the restriction of M to the fibre over 0 ∈ S with M0. The space S is called the base space of the deformation. We shall henceforth suppress the isomorphism ϕ and refer to the deformation (S, M, ϕ) by just the pair (S, M). A morphism of deformations (S, M) → (T,N) consists of a Σ-morphism f : S → T — that is, a

∼ ∗ map which commutes with the maps S → Σ and T → Σ — and an isomorphism M = (f × idY ) N of

OY ×ΣS-modules such that the isomorphism is compatible with the given ones over 0.

With objects and morphisms deﬁned as above, deformations of the OX -module M0 form a category denoted Def(Y,M0). ˜ Example 4.1.2. In the situation at hand, let Y be the formal germ of ∆n at 0, Σ = Spec C a2, . . . , an−2 , J K and the map Y → Spec C a2, . . . , an−2 given by (a2, . . . , an) 7→ (a2, . . . , an−2). The ﬁbre X over J K (0,..., 0) of this map is Γn.

Suppose we are given a deformation (S, M) of M0, and suppose we have a section γ :Σ → S of the map S → Σ. Then γ induces an embeddingγ ˜ : Y → Y ×Σ S such that pS ◦ γ˜ = γ ◦ q, where Chapter 4. Deformations of Modules 59

pS : Y ×Σ S → S is the projection to S, as in the following diagram:

γ˜ q - Y ×Σ S Σ

γ p S - S.

The pullback Mγ of M viaγ ˜ is then a lifting of M0 to Y . Furthermore, if M0 is maximal Cohen-Macaulay on X, it follows from ﬂatness and semicontinuity that Mγ is also maximal Cohen-Macaulay on Y . Thus deformations give rise to liftings of maximal Cohen-Macaulay modules on X to Y . The idea of a versal deformation captures the notion of a given deformation encompassing all possible ways of deforming a module. In particular, in a versal deformation of M0 over X, all liftings of M0 to Y appear in the manner outlined above. It is analogous to the notion of versality in the unfolding of maps.

Deﬁnition 4.1.3. A deformation (S, M) is versal if it satisﬁes the following property. Given any morphism (T,N) → (S, M) and any morphism (T,N) → (T 0,N 0) such that the map T → T 0 is a closed embedding, there is a morphism (T 0,N 0) → (S, M) making the following diagram commute.

(T,N)

- (S, M) -

∃ ? (T 0,N 0)

A deformation (S, M) is semiuniversal if it is versal and S is of minimal dimension.

In particular, setting (T,N) to ({0},M0), we see that a versal deformation induces all families of modules whose ﬁbre over the special point is M0. Further, a versal deformation encodes all liftings of

0 M0 to Y . Namely, suppose we have a module M over Y which lifts M0 and let (S, M) be a versal

0 deformation of M0. Then the natural morphism ({0},M0) ,→ (Σ,M ) satisﬁes the hypothesis above, so there is a morphism (Σ,M 0) → (S, M) whose associated map Σ → S is a section of the projection S → Σ. Thus any such module M 0 may be recovered as the ﬁbre over the image of this section in Chapter 4. Deformations of Modules 60

Y ×Σ S. We have thus reduced the lifting problem to the problems of ﬁnding a versal deformation and of classifying sections Σ → S.

The existence of a versal deformation of a given module M0 is a highly nontrivial fact, proved in a more general setting by Schlessinger in [Sch68]. We state a version speciﬁc to deformations of modules.

Theorem 4.1.4 (Schlessinger, 1968). If Ext1 (M ,M ) has ﬁnite dimension as a -vector space, then OX 0 0 C there exists a semiuniversal deformation of M0 in the deformation theory described above. The relative tangent space at 0 of the map S → Σ in a miniversal deformation (S, M) of M0 is isomorphic to Ext1 (M ,M ). OX 0 0

Given that, in our situation, X is always a curve with an isolated singularity, we know a priori that Ext1 (M ,M ), being a ﬁnitely-generated module supported on a single point, is ﬁnite-dimensional. OX 0 0 Thus Schlessinger’s theorem always applies in our case.

When OY and OΣ are graded, the map OΣ → OY is a graded ring homomorphism, and M0 is a graded OY -module, there exists a category GrDef(Y,M0) of graded module deformations. Objects are again pairs (S, M) such that S is a variety equipped with a map S → Σ, OS is a graded ring, and

OΣ → OS is a homomorphism of graded rings. The module M is a graded module over the ﬁbre product

Y ×Σ S. Morphisms (S, M) → (T,N) have the same data as above with the added requirement that

# f : S → T be induced by a graded ring homomorphism f : OT → OS. Schlessinger’s theorem applies after substituting for Ext1 (M ,M ) the degree-zero part thereof in the above statement. OX 0 0

In particular, a versal deformation of M0 always exists in the graded category provided that M0 is

ﬁnitely-generated over X, even when X does not have isolated singularities. Since ∆˜ n has a graded coordinate ring, this implies that there exists a ﬁnite-dimensional moduli space of rank one MCM modules which lift M0.

4.2 Computing versal deformations

In this section, we assume that Y is a hypersurface and X is a linear section thereof. We describe an algorithm — the Massey Product Algorithm — to construct the versal deformation of a maximal Cohen-

Macaulay module M0 deﬁned on X. The algorithm builds the deformation module M and its base space

S one inﬁnitesimal neighbourhood at a time: each iteration i constructs a deformation (Si,Mi) which is versal to order i, meaning that it satisﬁes the versality criterion for morphisms (T,N) → (T 0,N 0) whose

i+1 i+1 respective maximal ideals mT and mT 0 satisfy mT = 0 and mT 0 = 0. Taking the limits of {Si} and

{Mi} yields a versal deformation (S, M) of M0. Chapter 4. Deformations of Modules 61

We first describe the algorithm in a general context. To do so, we introduce some definitions. Let S be an analytic space with a map to Σ and let I be an OS-module. The following definition captures the notion of extending a given partially-constructed deformation into the next infinitesimal neighbourhood of zero.

Deﬁnition 4.2.1. An extension of S by M is a Σ-monomorphism of spaces S,→ S0 equipped with a short exact sequence of OS-modules

0 - I - O 0 - O - 0 u S S

2 0 such that u(I) is an ideal in OS0 and (u(I)) = 0. Given two extensions (S , u, I) and (T, v, J ), a morphism of extensions is a set of maps (α, β), making the diagram

- - - - 0 I OS0 OS 0 u w w w α β w w ? ? w 0 - J - O - O - 0 v T S commute, such that β is a homomorphism of OΣ-algebras and α is linear over β.

With these deﬁnitions, extensions of S by I form a category denoted ExΣ(S, I). The isomorphism classes in this category from a set denoted ExΣ(S, I).

Given a module I over OS, we may form the trivial extension of S by I, denoted S[I] by taking

OS[I] := S ⊕I with multiplication defined via (s, x)(t, y) := (st, sy+tx). Geometrically, this corresponds to taking the first infinitesimal neighbourhood of S in the space defined by I.

The set ExΣ(S, I) has a natural OS-module structure, under which the trivial extension S[I] is the zero element. See [BF, Section 2.4] for more information.

Finally, we deﬁne extensions of deformations. Let a = (S, M) be a deformation and I an OS-module.

Deﬁnition 4.2.2. An extension of a by I is a deformation a0 = (S0,M 0) such that S0 is an extension of

0 ∼ S by I and M ⊗OS0 OS = M as OS-modules. Morphisms of extensions are morphisms of deformations as deﬁned above whose associated maps of ringed spaces commute with the inclusion maps from S.

Thus extensions form a category denoted ExΣ(a, I). Isomorphism classes of extensions of a by I form an OS-module denoted ExΣ(a, I). 0 0 0 0 ∼ Extensions a = (S ,M ) of a by I such that S = S[I] form a subcategory ExΣ(a/S, I) of ExΣ(a, I).

The set of isomorphism classes in this category is denoted ExΣ(a/S, I). It is again an OS-module. Chapter 4. Deformations of Modules 62

0 0 0 There is a natural functor ExΣ(a, I) → ExΣ(S, I) given by (S ,M ) 7→ S . This functor induces an

OS-module homomorphism

p : ExΣ(a, I) → ExΣ(S, I).

Furthermore, the inclusion functor ExΣ(a/S, I) → ExΣ(a, I) induces an OS-module homomorphism

ExΣ(a/S, I) → ExΣ(a, I).

The two morphisms above clearly compose to zero and thus form an exact sequence which is part of the Kodaira-Spencer sequence, c.f. [BF, Section 3.2]:

- -p ExΣ(a/S, I) ExΣ(a, I) ExΣ(S, I). (4.1)

4.2.1 Liftings and obstructions

We begin with a general treatment of lifting M0. Let (S, MS) be a deformation of M0 with S an Artinian

# germ. Let π : S,→ R be a closed embedding over Σ such that ker π : OR → OS is annihilated by the maximal ideal mR of OR. Letting mS be the maximal ideal of OS, assume furthermore that the map

2 2 n+1 n+2 # mR/mR → mS/mS induced by π is an isomorphism. Then, if mS = 0, mR = 0. Let I := ker π . Then the sequence - - - - 0 I OR OS 0

0 deﬁnes an element in ExΣ(S, I) which we also denote by R.A lifting of MS to R is a module M over

∗ 0 R ×Σ Y such that π MR = MS. Thus MS corresponds to an element a = ExΣ(a, I), where a is the pair

0 (S, MS). With the notation of the last section, the image of the class of a under p is the class of R.

In general, however, the class of R may not be in the image of p. We will characterize a module of obstructions, denoted ObΣ(a/S, I), along with a homomorphism η : ExΣ(S, I) → ObΣ(a/S, I) making the following sequence exact:

- -p -η ExΣ(a/S, I) ExΣ(a, I) ExΣ(S, I) ObΣ(a/S, I). (4.2)

If the image of the class of R under η is zero in ObΣ(a/S, I), then the class of R has preimages in

ExΣ(a, I). The set of such preimages is a coset of the image of ExΣ(a/S, I) in ExΣ(a, I). In the following, we develop explicit descriptions of ExΣ(a/S, I) and ObΣ(a/S, I) and of the map η.

The following exposition is based on [Siq01]. Denote by L• the resolution of M0 over OX given by Chapter 4. Deformations of Modules 63

the matrix factorization (A0,B0):

- B-0 A-0 B-0 A-0 - - L• : ··· F0 G0 F0 G0 F0 M0 0.

∼ ∼ ∼ ∼ Let F and G be free OY -modules such that F0 = F ⊗OY OX (= F ⊗OΣ C) and G0 = G ⊗OY OX (=

G ⊗OΣ C). Let

˜ - A- B- A- - - L• : ··· G ⊗OΣ OS F ⊗OΣ OS G ⊗OΣ OS F ⊗OΣ OS MS 0,

be a free resolution of MS, so that A ≡ A0 mod mS and B ≡ B0 mod mS.

0 0 0 To lift MS to R is to specify the diﬀerentials A and B of a free resolution of MR so that A − A ≡ 0 ≡ B0 − B mod I. Let (A,˜ B˜) be any lifting of (A, B) to R. Then the homomorphism π# induces a commutative diagram

- B- A- B- - ··· F ⊗OΣ OS G ⊗OΣ OS F ⊗OΣ OS G ⊗OΣ OS ··· 6 6 6 6

πF πG πF πG . (4.3)

- - - - - ··· F ⊗OΣ OR G ⊗OΣ OR F ⊗OΣ OR G ⊗OΣ OR ··· B˜ A˜ B˜

The bottom row of (4.3) is not a complex, since A˜B˜ 6= 0 6= B˜A˜ in general. However, since AB = 0 = BA and the above diagram is commutative, the homomorphisms B˜A˜ : F ⊗Σ OR → F ⊗Σ OR and A˜B˜ :

G⊗OΣ OR → G⊗OΣ OR map F and G respectively into ker πF and ker πG. Furthermore, mR annihilates ˜ ˜ ˜ ˜ ∼ I, so mR(F ⊗OΣ OR) ⊆ ker BA and mR(G⊗OΣ OR) ⊆ ker AB. Since F ⊗OΣ OR/mR(F ⊗OΣ OR) = F0 and ∼ ˜ ˜ ˜ ˜ G ⊗OΣ OR/mR(G ⊗OΣ OR) = G0, AB and BA induce OX -module homomorphisms ω0 : F0 → F0 ⊗OΣ I and ω1 : G0 → G0 ⊗OΣ I. These together form a morphism of complexes ω : L• → (L• ⊗OΣ I)[2].

It is easy to verify that the resulting diagram

- B- A- B- - ··· F0 ⊗OΣ I G0 ⊗OΣ I F0 ⊗OΣ I G0 ⊗OΣ I ··· 6 6 6 6

ω0 ω1 ω0 ω1

··· - F - G - F - G - ··· 0 B 0 A 0 B 0 commutes, so ω deﬁnes an element of Ext2 (M ,M ⊗ I) ∼ Ext2 (M ,M ) ⊗ I. Furthermore, OX 0 0 OΣ = OX 0 0 OΣ ˜0 ˜0 ˜ ˜0 ˜0 ˜ ˜ 1 given another lifting (A , B ) of L• to OR,(A , B ) diﬀers from (A, B) by an element of Hom (L•, L•)⊗OΣ Chapter 4. Deformations of Modules 64

I, so ω, as an element of Ext2 (M ,M ) ⊗ I does not depend on the choice of lifting. We have thus OX 0 0 OΣ deﬁned a map η : Ex (S, I) → Ext2 (M ,M ) ⊗ I which is in fact a homomorphism of O -modules. Σ OX 0 0 OΣ S

If MS can be lifted to a module MR over R, then one lifting of the complex L˜ • to OR is given by the diﬀerentials A0 and B0 in the resolution of M . In that case, ω = 0 as an element of Ext2 (M ,M )⊗I. R OX 0 0

Conversely, if ω = 0 as an element of Ext2 (M ,M ) ⊗ I, then there exists a map of chain OX 0 0 OΣ complexes δ : L• → L• ⊗OΣ I such that ω = −dδ. Suppose that δ is given by the matrix pair (AR,BR)

0 0 2 and let A := A˜ + AR and B := B˜ + BR. Then ARBR = 0 = BRAR since (ker π) = 0. Thus we have

0 0 A B = (A˜ + AR)(B˜ + BR)

= A˜B˜ + AB˜ R + ARB˜ + ARBR

= ω + dδ

= 0.

0 0 0 0 Similarly B A = 0. Thus A and B form diﬀerentials of a complex which restricts to L˜ •. The cokernel of A0 is the desired lifting of M . Furthermore, given an element of Ext1 (M ,M ) ⊗ I represented S OX 0 0 OΣ by the matrix pair (E,F ), (A0 + E,B0 + F ) also satisﬁes

(A0 + E)(B0 + F ) = A0B0 + EB0 + A0F + EF

= EB0 + A0F

= 0,

0 0 0 0 0 0 since A B = 0 and mS annihilates I. Similarly, (B +E)(A +F ) = 0. Thus (A +E,B +F ) also deﬁnes a lifting of (A, B) to R. This proves the following result, which appears as Theorem 1.1 in [Siq01].

Lemma 4.2.3 ([Siq01]). Let S be an analytic space over Σ with a Σ-morphism i : S0 ,→ S. Let MS be

∗ # a module over S ×Σ Y such that i (M) = M0. Let π : S,→ R be a Σ-morphism with mR ker π = 0, where mR is the maximal ideal of OR, and let (A,˜ B˜) be a lifting to OR of the matrix factorization

(A, B) associated to MS. Then there exists a lifting MR of MS to R if and only if the homomorphism # ˜ ˜ ˜ ˜ of complexes ω : L• → L• ⊗OΣ ker π given by the matrix pair (AB, BA) is zero as an element of Ext2 (M ,M ) ⊗ ker π#. There is a surjection from Ext1 (M ,M ) ⊗ ker π# to the set of liftings OX 0 0 OΣ OX 0 0 OΣ

MR. Chapter 4. Deformations of Modules 65

Lemma 4.2.3 suggests that

Ex (a/S, I) ∼ Ext1 (M ,M ) ⊗ I and (4.4) Σ = OX 0 0 OΣ

Ob (a/S, I) ∼ Ext2 (M ,M ) ⊗ I. (4.5) Σ = OX 0 0 OΣ

Indeed, the ﬁrst isomorphism follows from [BF, Proposition 5.3.1]. We take the second as a deﬁnition.

The homomorphism η : ExΣ(S, I) → ObΣ(a/S, I) is given by the procedure described above.

Lemma 4.2.3 also gives an outline of the deformation process, which we deﬁne inductively. Let n ≥ 0.

n+1 Let S be a space equipped with a map S → Σ and let Sn := Spec OS/mS , where mS is the maximal ideal of S. Let (Sn,Mn) be a deformation and let (A, B) be the matrix factorization associated to Mn.

n+2 0 Let Rn+1 := Spec OS/mS . To lift M further, we must ﬁnd the smallest ideal I ⊆ ORn+1 such that M 0 can be lifted to Sn+1 := Spec ORn+1 /I . Denote again by (A, B) an arbitrary lifting of (A, B) to Sn+1. We then ﬁnd a pair of matrices (A00,B00) such that (A + A00)(B + B00) = 0 = (B + B00)(A + A00). Then

00 00 0 (A + A ,B + B ) is a matrix factorization corresponding to a lifting M of M to Sn+1.

In this manner, we construct a series

S0 ,→ S1 ,→ S2 ,→ S3 ,→ · · ·

of Artinian bases and correspondingly a series of modules M0,M1,... such that Mi is an OSi×ΣY -module and restricts to Mi−1 over Si−1 for each i > 0. Taking the inductive limit of this directed system, we obtain a versal deformation (S, M) of M0.

In practise, the first-order deformation into S1 differs from the remaining deformations in some key respects. For this reason, we treat the first deformation in one section and the remaining deformations thereafter.

4.2.2 The ﬁrst-order deformation

Now we construct the first-order deformation. Let mΣ be the maximal ideal of OΣ. Let f be the defining ¯ 2 equation of Y in its ambient space and let f be the reduction of f modulo mΣ. The problem at hand is 2 to find a pair (A1,B1) of matrices over mΣ/mΣ ⊗OΣ OY solving the equation

¯ (A0 + A1)(B0 + B1) = f. (4.6) Chapter 4. Deformations of Modules 66

Rewriting (4.6), we have ¯ A0B0 + A0B1 + A1B0 = f (4.7)

2 2 since A1B1 ∈ mΣ/mΣ ⊗OΣ OY = 0. Equivalently,

¯ f − A0B0 = A0B1 + A1B0. (4.8)

The left-hand side of (4.8) deﬁnes an element f1 of

Ext2 (M ,M ) ⊗ m /m2 . OX 0 0 OΣ Σ Σ

Concretely, this map is the composition of:

• the isomorphism L• → L•[2] which expresses the two-periodicity of L•, and

• multiplication by the terms of f which are linear in the variables deﬁning Σ.

If this element vanishes in Ext2 (M ,M ) ⊗ m /m2 , then there exists a solution (Ap,Bp) of (4.8). OX 0 0 OΣ Σ Σ 1 1

Otherwise, it is not possible to extend M0 along all directions of Σ. If this is the case, it is still possible to construct a versal deformation (S, M) of M0, but the morphism S → Σ is not dominant. In particular, there can be no section Σ → S, and thus no lifting of M0 to Y . Now assume that f vanishes in Ext2 (M ,M ) ⊗ m /m2 so that a solution of (4.8) exists, and 1 OX 0 0 OΣ Σ Σ p p pick such a solution (A1,B1 ). Let (E1,F1),..., (Ed,Fd) be morphisms of complexes representing a -basis of Ext1 (M ,M ). Then a general solution of (4.8) is C OX 0 0

p A1 := A1 + ξ1E1 + ··· + ξdEd,

p B1 := B1 + ξ1F1 + ··· + ξdFd, where ξ , . . . , ξ are indeterminates corresponding to a -basis of Ext1 (M ,M )∗. Let a , . . . , a be 1 d C OX 0 0 1 k 2 a minimal set of generators of mΣ and let S1 := Spec C[a1, . . . , ak, ξ1, . . . , ξd]/(a1, . . . , ak, ξ1, . . . , ξd) . Picking a diﬀerent basis of Ext1 (M ,M ) and changing the particular solution (Ap,Bp) corresponds OX 0 0 1 1 to the application of an automorphism of S1, which is immaterial. The method for computing the ﬁrst deformation may be summarized thus:

1. Find a basis of Ext1 (M ,M ) and let (E ,F ),..., (E ,F ) be the corresponding maps of com- OX 0 0 1 1 d d plexes.

p p p p 2. Find a particular solution (A1,B1 ) to the system f1 = A0B1 + A1B0. Chapter 4. Deformations of Modules 67

p p 3. Return the pair (A1 + ξ1E1 + ··· + ξdEd,B1 + ξ1F1 + ··· + ξdFd).

The Macaulay 2 [GS] routine FirstDeformation listed in Appendix A implements the above procedure. As input, it takes the coordinate ring of the ambient space of Y ×Σ S, the matrices A0 and B , the even and odd diﬀerentials d and d of the two-periodic complex Hom• (M ,M ), a basis of 0 0 1 OX 0 0 Ext1 (M ,M ) represented as a list of pairs of matrices (E ,F ), and a basis of Ext2 (M ,M ). It OX 0 0 i i OX 0 0 returns A1 and B1, or gives an error if the extension into Σ is obstructed.

Example 4.2.4. Consider the polynomial f(z, a) = z2 − a2 and the matrix factorization (z, z) of z2. Let

2 M0 be the associated module over C[z]/(z ) = OX , that is, M0 = C[z]/(z). We will construct the ﬁrst deformation of M0 in the associated deformation theory.

First, we have Ext1 (M ,M ) ∼ and Ext2 (M ,M ) ∼ . Thus we require one parameter ξ. OX 0 0 = C OX 0 0 = C The specialization of (4.8) to this case is

0 = zB1 + A1z.

Clearly, the left-hand side vanishes in Ext2 (M ,M )⊗(a)/(a)2, so there is no obstruction. A particular OX 0 0 solution is the matrix pair (0, 0), while the matrix pair (−1, 1) spans Ext1 (M ,M ). Thus the general OX 0 0 solution is

A1 = ξ,

B1 = −ξ.

2 Example 4.2.5. Now consider the polynomial f(z, a) = z − a with X and M0 as in Example 4.2.4. The specialization of equation (4.8) to this case is

2 zB1 + A1z = (z − a) − A0B0

= −a.

This equation clearly has no solution, since the left-hand side is contained in the ideal (z) ⊆ C[a, z]/(z2 − a, a2), while the right-hand side is not. Alternatively, the right-hand side is the product of a and a unit and so cannot vanish in Ext2 (M ,M ) ⊗ (a)/(a)2. In this case, it is not possible to extend the matrix OX 0 0 factorization (z, z) to a matrix factorization of f. This fact is clear since f is irreducible. Chapter 4. Deformations of Modules 68

4.2.3 Further deformations

We proceed now to deformations into larger inﬁnitesimal neighbourhoods of S0. Suppose that we have constructed (Sn,Mn) as described in §4.2.1 and let A and B be the associated matrices. Abusing notation slightly, we again denote the liftings of the matrices A and B to Rn+1 with the same notation.

It is necessary to ﬁnd An+1 and Bn+1 such that

¯ f − AB = A0Bn+1 + An+1B0, (4.9) and ¯ f − BA = B0An+1 + Bn+1A0, (4.10)

¯ n+1 where f is the reduction of f modulo mΣ .

First, we identify the obstruction ideal. Following the exposition in §4.2.1, (f − AB, f − BA) is identiﬁed with a cycle in Ext2 (M ,M ) ⊗ ker π#, where π : S → R is the natural inclusion. OX 0 0 OΣ n n+1 We write a basis e , . . . , e of Ext2 (M ,M ) and express (f − AB, f − BA) as a linear combination 1 t OX 0 0

(f − AB, f − BA) = gn+1,1(a1, . . . , ak, ξ1, . . . , ξd)e1 + ··· + gn+1,t(a1, . . . , ak, ξ1, . . . , ξd)et.

The elements gn+1,1, . . . , gn+1,t ∈ Rn+1 may be seen as coefficients in a Taylor-like expansion of the generators of the obstruction ideal In+1 which defines Sn+1. We construct In+1 by piecing together the coefficients from this and all previous iterations in the following way. For each 0 < i ≤ n + 1 and

1 ≤ j ≤ t, letg ˜i,j be a lifting of gi,j to R := C[a1, . . . , ak, ξ1, . . . , ξd], chosen so that the coeﬃcient of any term of degree greater than i is zero. Then set

n+1 n+1 ! X X In+1 := g˜i,1,..., g˜i,t . i=0 i=0

Finally, set

n+2 Sn+1 := Spec R/ In+1 + mR , where m is the ideal (a , . . . , a , ξ , . . . , ξ ). Observe that a change of basis of Ext2 (M ,M ) only R 1 k 1 d OX 0 0 changes the choice of generators of In+1, not In+1 itself. Thus Sn+1 is independent of the choice of basis of Ext2 (M ,M ). OX 0 0

0 0 Now we identify a particular solution (An+1,Bn+1) of (4.9). Given such a solution, all other solutions may be obtained by adding an element of Ext1 (M ,M ) ⊗ ker π# to (A0 ,B0 ), just as in §4.2.2. OX 0 0 OΣ n+1 n+1 Chapter 4. Deformations of Modules 69

Thus, ostensibly, we have a solution of the form

0 An+1 = An+1 + η1E1 + ··· + ηdEd,

0 Bn+1 = Bn+1 + η1F1 + ··· + ηdFd.

However the change of coordinates ξi 7→ ξi − ηi for i = 1 . . . d eliminates the extraneous parameters

0 0 η1, . . . , ηd. Thus it suﬃces to let (An+1,Bn+1) := (An+1,Bn+1). Summarizing, we have the following procedure:

1. Compute f¯ − AB and f¯ − BA. Let ω be the cycle in Ext2 (M ,M ) ⊗ ker π# associated to OX 0 0 OΣ this matrix pair.

2. Express ω in terms of a ﬁxed basis of Ext2 (M ,M ), e.g. ω = g e + ··· + g e , where g = OX 0 0 1 1 t t i

gi(a1, . . . , ak, ξ1, . . . , ξd) are elements of Rn+1. Letg ˜1,..., g˜t be liftings of g1, . . . , gt to R.

3. Let ~on+1 := ~on + (˜g1,..., g˜t) (with ~o0 = 0). Let I be the ideal generated by the entries of ~on+1

and set Sn+1 := Rn+1/I.

4. Find a particular solution (An+1,Bn+1) over Sn+1 to (4.9).

5. Return Sn+1 and the matrix pair (An+1,Bn+1).

The Macaulay 2 routine ExtendDeformation listed in Appendix A implements the above procedure. As input, the procedure ExtendDeformation takes the following parameters:

• OYxSAmbient, the coordinate ring of the ambient space of OY ×ΣS

• pIdeal, the existing obstruction ideal

• i, the current iteration number

• A and B, the existing matrix factorization;

• , the diﬀerentials of the complex Hom• (M ,M ); d OX 0 0

• , the -dimension of the module Ext1 (M ,M ); dimExt1 C OX 0 0

• , a vector representation of a -basis of Ext2 (M ,M ) Ext2Vect C OX 0 0

• obstruction, the generators of the obstruction ideal;

As output, ExtendDeformation returns a pair of (A, B) representing the lifted matrix factorization and the obstruction ideal generators. Chapter 4. Deformations of Modules 70

Remark 4.2.6. Suppose that Y and Σ are germs of affine varieties with graded coordinate rings and that the map Y → Σ is the germ of a graded homomorphism of the associated affine varieties. Suppose further that M0 is a graded module over X. Then the solutions obtained in each step of the algorithm outlined in this section may also be taken to be homogeneous with respect to the grading on R induced by the grading on the module Ext1 (M ,M ). This entails that the Taylor-like coefficientsg ˜ are also OX 0 0 i,j quasihomogeneous with respect to this grading and, for a given index j, of constant degree. Thus the generators of the resulting obstruction ideal I are quasihomogeneous polynomials. We may then take S, along with Y and Σ, to be an affine variety and M to be a graded module over the affine variety

S ×Σ Y . In the next two sections, we shall make use of this identiﬁcation in order to exploit the grading on OS and classify graded liftings of M0 to Y .

Remark 4.2.7. In principle, this process must continue ad inﬁnitum so as to construct a sequence

(S0,M0) → (S1,M1) → (S2,M2) → · · · of deformations forming a directed system of which we may then take the limit. If, however, it can be proved that for all orders n greater than some order N, the equations (4.9) and (4.10) have no obstruction and particular solution (An+1,Bn+1) = (0, 0), then the process may terminate at order N. This is true, for example, in the graded case of 4.2.6: for n >> 0, the degrees of the generators of ker π# are too large to contribute to either the obstruction ideal or the entries of the matrix factorization (A, B).

2 Example 4.2.8. We continue Example 4.2.4 to further deformations. We have S1 = Spec C[a, ξ]/(a, ξ) and a matrix factorization (z + ξ, z − ξ) corresponding to a module M1. Let us write the equation for the second-order deformation

2 2 a − ξ = zB2 + A2z.

The terms a2 − ξ2 do not vanish in Ext2 (M ,M ) ⊗ ker π#, and we obtain an obstruction a2 − ξ2. OX 0 0 2 2 3 2 2 3 A particular solution is then (0, 0). Thus S2 = Spec C[a, ξ]/(a − ξ , a , aξ , a ξ, ξ ) and we obtain the matrix factorization (z + ξ, z − ξ). In the next step, we obtain the equation

0 = zB3 + A3z.

There is no obstruction, and the particular solution is (0, 0). It is easy to verify that this situation continues for all higher orders, so the process in fact may terminate after the second-order deformation.

The resulting base space is S = Spec K[a, ξ]/(a2 − ξ2) and the resulting module is coker(z + ξ). Chapter 4. Deformations of Modules 71

In conclusion, we obtain the module coker(z + ξ) over the base space Spec C[a, ξ]/(a2 − ξ2). This is the Hilbert scheme of a point on Spec C[a, z]/(a2 − z2).

4.3 Rank one MCM modules over xn − yn−1

n n−1 We now address the question of ﬁnding rank one MCM modules over Γn = Spec K[x, y]/(x − y ). The following proposition, whose essential point is due to Wiegand [Wie94], shows that there are only ﬁnitely many such graded modules up to isomorphism and twist.

Proposition 4.3.1. Let R be a reduced graded ring of dimension one which is ﬁnitely generated as an algebra over K. Let M be a graded rank one maximal Cohen-Macaulay module over R. Let R¯ be the normalization of R. Then M may be embedded in R¯ as a graded R-submodule.

Furthermore, if R is a domain, then M may be embedded in R¯ in such a way that R ⊆ M ⊆ R¯, where the composition is the normalization map. In particular, in this case, there are up to twist only ﬁnitely many isomorphism classes of graded rank one maximal Cohen-Macaulay modules over R.

Proof. Let M be a graded rank one MCM module over R. Then M is torsion-free, so the natural map ∼ M → M ⊗R Q = Q, where Q is the total ring of fractions of R, is injective. Let T be the kernel of the ¯ ¯ ∼ 0 ¯ 0 ∼ ∼ map M ⊗R R → (M ⊗R R)⊗R¯ Q = M ⊗R Q and let M := (M ⊗R R)/T . Then M ⊗R¯ Q = M ⊗R Q = Q as Q-modules, so M 0 is of rank one and, since Spec R¯ is smooth, isomorphic to R¯. Since the map

∼ 0 M → M ⊗R Q = Q factors through M , we obtain an injection i : M → R¯.

Suppose now that R is a domain. Then R¯ is of the form K[t], so we may select an element m ∈ M with minimal degree. We may then construct a new embedding M,→ R¯ via x 7→ i(m)−1i(x). The element m has degree zero under this embedding, so it is a unit and thus the image of M contains the image of R in R¯.

For the final claim, we need only observe that, since R¯ is of the form K[t], there is a one-dimensional vector space of elements in each degree. After reducing modulo action by scalars, this amounts to one possible module generator per degree. Furthermore, R/R¯ , being a torsion module on a curve, is Artinian, and thus contains elements in only finitely many degrees. Hence, in view of the previous claim, up to isomorphism, there are finitely many possible generators of a torsion-free module over R, and the claim follows.

n−1 n The curve Γn admits a parametrization t 7→ (t , t ). Thus its normalization is the aﬃne line

Spec K[t] and K[x, y]/(xn − yn−1) embeds into K[t] via the map x 7→ tn−1 and y 7→ tn. Chapter 4. Deformations of Modules 72

t1 t2 t3 ··· tn−2 1 1 1 t n − t n − − n n . n t . - t - . t tn+1 tn+2 t2n−3 1 1 t n − − n .. n - t . t t2n+1 t3n−4 ......

2 tn −3n+1

Figure 4.1: Monomials in K[t] which are not in OΓn

To deﬁne a graded sub-OΓn -module of K[t] it suﬃces to specify the monomials in K[t] included therein, subject to the requirement that the resulting set is closed under action of OΓn . To contain OΓn , the set must contain all monomials of the form tnk+(n−1)l for k, l ≥ 0. In particular, it must contain 1.

n−1 The monomials of K[t] which are not in OΓn form a diagram as shown in Figure 4.1. The action of t carries a monomial down one row and to the left, while the action of tn carries a monomial down one row and to the right. The leftmost, respectively rightmost, monomials are carried via tn−1, respectively

n t to the image of OΓn in K[t].

We construct a graded submodule of K[t] by specifying its generators, namely, 1 and some monomials in K[t] \OΓn . The submodule generated by a given monomial is the diamond whose top vertex is the given monomial and which is closed under the action of tn and tn−1. The module which results from the selection of a set of such generators is torsion-free and thus automatically maximal Cohen-Macaulay.

k1 km Given a minimal set {t , . . . , t } (with k1 = 0) of generators of a module M as above, we compute a presentation matrix for M. For each 1 ≤ i ≤ m, let ri be such that 0 ≤ ri < n − 1 and ki − ri

ki kj is divisible by n − 1. Note that if ri = rj for some 1 ≤ i < j ≤ m, then one of t or t is in the submodule generated by the other, so the set of generators is not minimal. Now order the generators so that r1 < ··· < rm. In terms of Figure 4.1, this corresponds to the following. First augment the ﬁgure by adding the n monomials 1, tn, . . . , tn(n−1) to the left-hand side of the diagram and the n monomials

2 1, tn−1, . . . , t(n−1) to the right-hand side of the diagram. The resulting augmented diagram is shown in Figure 4.2. Then order the generators so that, in the order given, they appear on the diagram from left to right.

We know a priori that M is maximal Cohen-Macaulay, so its syzygy module has m generators. To

ﬁnd the syzygies, we consider the pairs (tk1 , tk2 ), (tk2 , tk3 ),..., (tkm−1 , tkm ), (tkm , tk1 ). For notational

ki ki+1 convenience, set km+1 := k1 = 0. A minimal relation on the generators t and t is given as follows. Chapter 4. Deformations of Modules 73

t0 t1 t2 ··· tn−3 tn−2 tn−1

tn tn+1 t2n−1 t2(n−1)

t2n t2n+1 ··· t3n−2 t3(n−1) ......

n(n−1) tt

Figure 4.2: Figure 4.1 augmented with monomials in OΓn

Write [tki ] for the generator of a minimal free cover of M which maps to tki . Let tj be the monomial

ki ki+1 of minimal degree in the intersection of the OΓn -submodules generated by t and t . In terms of

Figure 4.2, tj is the following. Start at the monomial tki (using the leftmost copy of t0 when i = 1) and move downwards and to the right on the diagram, and start at the monomial tki+1 (using the rightmost copy of t0 when i = m) and move downwards and to the left. Then tj is the monomial at which the two

j bi ki ai ki+1 paths meet. Thus t = y t = x t for some ai, bi > 0 — strict inequality is guaranteed since the generating set is minimal — and ybi · [tki ] − xai · [tki+1 ] is a syzygy on those generators.

Entering these syzygies into a matrix, we obtain

  b a  y 1 −x m      −xa1 yb2     .   a2 ..  .  −x     .. b   . y m−1       −xam−1 ybm 

We ﬁrst claim that this matrix presents a rank one MCM module over OΓn . Its determinant is equal to b a P P y − x , where a := i ai and b := j bj. It suﬃces then to show that a = n and b = n − 1, for then the determinant is −(xn − yn−1). This claim is easiest to see when visualized in Figure 4.2. The exponent a is the total number of steps downward and to the left among all of the syzygies, and the exponent b is the total number of steps downward and to the right. Connecting these steps into a continuous path from t0 on the left to t0 on the right, we see that the total number of steps downward and to the right must be n − 1 and the total number of steps downward and to the left must be n.

To see that the this matrix indeed presents M, we identify the degrees of the generators and show that they agree with those of M. The degree of the generator of the ﬁrst row may be taken to be zero.

Suppose that the degree of the generator in row i is ki. Then the degree of the generator in row i + 1 is Chapter 4. Deformations of Modules 74

bi ki ai ki+1 ki +nbi −(n−1)ai. Since ai and bi are chosen so that y t = x t , we have ki+1 = ki +nbi −(n−1)ai, as required.

Example 4.3.2. Let n = 5. Then x = t4 and y = t5. The table of monomials is as follows:

t1 t2 t3

t 5 4 t 5 4 t t - - t6 t7 (4.11)

t 5 4 t - t11.

Consider the submodule generated by 1, t, and t7. A minimal syzygy on 1 and t is y · [1] − x · [t] and has degree 5. A minimal syzygy on t and t7 has degree 11 and is therefore y2 · [t] − x · [t7]. Finally, a minimal syzygy on 1 and t7 has degree 12 and is therefore y · [t7] − x3 · [1]. The resulting presentation matrix is therefore   3  y 0 −x      −x y2 0  .      0 −x y 

Example 4.3.3. Again let n = 5 and consider now the submodule generated by 1 and t3. One syzygy, which results from starting at t3 and moving leftwards in (4.11), is y3 · [1] − x3 · [t3]. The other syzygy, which results from starting at t3 and moving rightwards, is y · [t3] − x2 · [1]. The resulting presentation matrix is    y −x3     . −x2 y3 

Example 4.3.4. Consider the case n = 4, namely the curve x4 − y3. Let R := K[x, y]/(x4 − y3). Then R is the K-linear span of the monomials 1, t3, t4, t6, t7, t8,... . The pool of generators for rank one maximal Cohen-Macaulay modules over R is thus t, t2 and t5. The modules thus obtained are listed in Table 4.3. We include in the table a realization of each module as an ideal of R.

The case n = 4 is the E6 singularity, one of the simple singularities. Among hypersurface singularities, or more generally Gorenstein singularities, simple singularities are characterized by the property that there exist only ﬁnitely many isomorphism classes of indecomposable maximal Cohen-Macaulay modules over them. In addition, all rank one maximal Cohen-Macaulay modules over a simple singularity are isomorphic to graded ones; see [Yos90, Theorem 15.14]. Thus all isomorphism classes of rank one maximal Cohen-Macaulay modules over K[x, y]/(x4 − y3) — not just the graded ones — are obtained in this manner. See [Yos90, 9.13] for more on the modules over the E6 singularity. Chapter 4. Deformations of Modules 75

Figure 4.3: Rank one maximal Cohen-Macaulay modules over x4 − y3 = 0

Name Generators Presentation Matrix Ideal   M0 1 0 (1) Rank one free module  y2 −x3 M 1, t   (x, y) 1 −x y   y2 −x2 M 1, t2   (x2, y) 2 −x2 y   y −x3 M 1, t5   (x, y2) 3 −x y2   y 0 −x2   M 1, t, t2 −x y 0  (x, y)2 Normalization 4    0 −x y 

We now consider the restriction of the open swallowtail Sn, deﬁned in Chapter 3, to Γn.

Proposition 4.3.5. Let S¯n denote the restriction of the open swallowtail Sn to Γn. Then the image of ¯ k Sn in K[t] is the sub-K-vector space generated by the monomials 1 and t for k ≥ n − 1. Equivalently, ¯ k Sn is generated as an OΓn module by 1 and t for n + 1 ≤ k ≤ 2n − 3.

Proof. We ﬁrst characterize the kernel of the universal derivation d : [t] → Ω1 . The module K K[t]/OΓn

1 1 a0 Ω is an O [t]-module and a quotient of the cyclic module Ω . Thus it isomorphic to [t]/t K[t]/OΓn K K[t] K for some a0 ≥ 0. Consider the Zariski-Jacobi sequence

i Ω1 ⊗ [t] - Ω1 - Ω1 - 0. Γn K K[t] K[t]/OΓn

n−2 We have that d(x) = (n − 1)t dt is the element of minimal degree in the image of i; hence a0 = n − 2.

Since d(ta) = ata−1 dt, the kernel of d consists of the K-span of monomials 1 and ta for a ≥ n − 1.

To prove the theorem, we claim that S¯n is the kernel of d. Starting with the exact sequence

d 1 0 - S - O ¯ - Ω - 0 n ∆n ∆¯ n/∆n

which deﬁnes Sn, we tensor with O∆n /(a1, . . . , an−2) to obtain an exact sequence

- -d 1 - S¯n [t] Ω 0. K K[t]/OΓn

¯ It suﬃces now to show that the kernel K of the map Sn → K[t] is zero, so suppose otherwise. The module

S¯n, being maximal Cohen-Macaulay, is torsion-free. Since K is a submodule of S¯n, it is also torsion-free.

Since Γn is one-dimensional, K is maximal Cohen-Macaulay over Γn. Therefore, it is generically free of Chapter 4. Deformations of Modules 76

¯ ¯ rank 1 and the map K,→ Sn is generically an isomorphism. This implies that the image of Sn → K[t] ¯ is generically zero. However, the characterization of ker d above shows that the map Sn → K[t] is not zero, so its image is a nonzero torsion submodule of K[t], which is impossible. Thus K is zero, whence the claim.

Proposition 4.3.5 allows one to compute easily the degrees of a minimal presentation matrix of Sn.

Proposition 4.3.6. The open swallowtail Sn is generated in degrees 0 and n + 1,..., 2n − 3. Its ﬁrst syzygy is generated in degrees 2n, . . . , 3n − 3.

Proof. The degrees of the generators of Sn and of its ﬁrst syzygy are preserved under projection to S¯n. The ﬁrst claim therefore follows immediately from Proposition 4.3.5. We now consider the second claim. Applying the procedure outlined above, we obtain the syzygies

y2 · [t0] − x · [tn+1],

y · [tn+i] − x · [tn+i+1] for i = 1, . . . , n − 4,

y · [t2n−3] − x3 · [t0].

These syzygies have degrees, respectively, 2n, . . . , 3n − 3 and the claim follows.

The proof of Proposition 4.3.6 also allows us to write a presentation matrix for the restriction S¯n of

Sn to Γn:  2 3  y −x      −x y       −x y  ¯   Sn = coker   .  ..   −x .     .   .. y       −x y 

4.4 Example: Graded modules over ∆4

In this subsection we compute all deformations of the graded rank one MCM modules over the E6 curve singularity Γ := Γ4 to the reduced discriminant ∆˜ 4. In doing so, we classify all graded rank one MCM modules over ∆˜ 4. The graded rank one MCM modules over Γ are given in Figure 4.3; we use the naming convention given therein. Chapter 4. Deformations of Modules 77

The equation of ∆˜ 4 is

3 4 2 2 2 3 2 4 256a4 − 27a3 + 144a2a3a4 − 128a2a4 − 4a2a3 + 16a2a4 = 0. (4.12)

To simplify the coeﬃcients, we apply the change of coordinates a2 7→ 6a2, a3 7→ 4a3, and a4 7→ 3a4. After this change and upon removing the resulting common factor of 6912, the equation becomes

˜ 3 4 2 2 2 3 2 4 D4 := a4 − a3 + 6a2a3a4 − 6a2a4 − 2a2a3 + 9a2a4 = 0. (4.13)

Observe that M0 is just the rank one free module over OΓ. This module is rigid: the only module over O which restricts to M is itself free of rank 1. Furthermore, M and M are dual to one another. ∆˜ 4 0 1 3

The dual of any lifting of M1 to ∆˜ 4 must restrict to M3 and vice versa, so we may consider only M1.

Deformation of M1

In our coordinates, the matrix factorization associated to M1 is

     3 2 3  a4 a3  a4 −a3   ,   . (4.14)  2   a3 a4 −a3 a4 

Direct computation in shows that Ext1 (M ,M ) is two-dimensional. As computed in Macaulay 2 OΓ 1 1 Macaulay 2, the base space S of a miniversal deformation is

3 4 2 2 2 3 2 4 Spec C[a2, ξ1, ξ2]/(ξ1 − ξ2 + 6a2ξ1ξ2 − 6a2ξ1 − 2a2ξ2 + 9a2ξ1), where {ξ , ξ } is a basis of Ext1 (M ,M )∗. Thus ξ has degree 4 and ξ has degree 3. The deformation 1 2 OΓ 1 1 1 2 module is the cokernel of

  3 2 2 3 3 a4 − ξ1 (a3 + a3ξ2 + a3ξ2 + ξ2 ) + 2a2(a3 + ξ2) − 6a2a4(a3 + ξ2)  . (4.15)  2 2 2 4 2  a3 − ξ2 (a4 + a4ξ1 + ξ1 ) − 6a2(a4 + ξ1) + 9a2 + 6a2ξ2 

The results of the Macaulay 2 computation immediately imply the following theorem, which is also proved using a theoretical argument in Section 4.5.

∼ Theorem 4.4.1. The module M0 has a miniversal deformation (S, M) where S = ∆˜ 4 and M is the ideal deﬁning the image of the diagonal embedding ∆˜ 4 → ∆˜ 4 ×Σ ∆˜ 4.

Proof. The map S → ∆˜ 4 is (ξ1, ξ2) 7→ (a4, a3), which by inspection transforms the ideal deﬁning S into Chapter 4. Deformations of Modules 78

Figure 4.4: The two graded sections of S → Σ

the ideal deﬁning ∆˜ 4, whence the ﬁrst claim.

0 As for the second claim, let D be the defining polynomial of ∆˜ 4 and D be the defining polynomial ˜ 0 ˜ of S so that, the ideal defining ∆4 ⊗Σ S in A := C[a2, a3, a4, ξ1, ξ2] is (D ,D). The variety ∆4 ⊗Σ S is a codimension one subvariety of Z := Spec A/(D − D0). Consider now the subvariety Ξ of Spec A defined by the ideal (ξ1 − a4, ξ2 − a3). It is a codimension two complete intersection and a subvariety of Z. It is therefore Cohen-Macaulay and thus its defining ideal in OZ is maximal Cohen-Macaulay over OZ . Consider now the presentation matrix of M, viewed as a matrix over A. Its determinant is D − D0, so it presents a rank one MCM module over OZ , which is by inspection the aforementioned defining ideal of Ξ. Since ∆˜ 4 ⊗Σ S is a codimension one subvariety of Z, the same matrix presents the defining ideal of the intersection of Ξ and ∆˜ ⊗ S as a module over O . The intersection of Ξ with ∆˜ ⊗ S is 4 Σ ∆˜ 4⊗ΣS 4 Σ just the image of the diagonal embedding. The claim follows.

We now investigate sections of the map S → Σ. Since ξ2 has odd degree, the graded sections have

2 ξ2 = 0. Since ξ1 has degree 4, ξ1 = αa2 for some α. Substituting those values into the equation describing S, we have

6 2 a2(α − 3) α = 0.

We conclude that there are two graded sections,

Γ1 : a 7→ (a2, 0, 0),

2 Γ2 : a 7→ (a2, 3a2, 0).

Figure 4.4 shows the two graded sections pictorially. Chapter 4. Deformations of Modules 79

Substituting Γ1 and Γ2 into (4.15), we obtain the modules

  a a3 + 2a a3 − 6a a a ˜  4 3 3 2 3 4 2 M1,1 = coker  ,  4 2 2  a3 9a2 − 6a4a2 + a4    a − 3a2 a3 + 2a a3 − 6a a a ˜  4 2 3 3 2 3 4 2 M1,2 = coker  .  2 2   a3 a4 − 3a4a2 

We determine that M˜ 1,1 and M˜ 1,2 are not isomorphic using Fitting ideals. We have

˜ 4 Fitt1(M1,1) = (a3, a4, a2),

˜ 2 Fitt1(M1,2) = (a3, 3a2 − a4).

In fact, it follows from Theorem 4.4.1 that M˜ is the ideal (a , a ) ⊆ O and M˜ is the ideal 1,1 3 4 ∆˜ 4 1,2 deﬁning the self-intersection locus on ∆˜ 4. In particular, the restriction of the open swallowtail S4 deﬁned in Chapter 3 to ∆˜ is the O -dual of M˜ . 4 ∆˜ 4 1,2

Deformation of M2

In our coordinates, the matrix factorization associated to M2 is

     2 2 2  a4 a3  a4 −a3  ,  .  2 2  2  a3 a4 −a3 a4 

In this case Ext1 (M ,M ) is four-dimensional. The base space S is Spec [a , ξ , ξ , ξ , ξ ]/I, where I OΓ 2 2 C 2 1 2 3 4 is the ideal generated by

3 3 4 2 2a2ξ2 + 2ξ1ξ2 + ξ2 − 6a2ξ2ξ3 − 9a2ξ4 − 6a2ξ1ξ4 − 6a2ξ2 ξ4

2 2 2 2 2 3 2 3 + 12a2ξ3ξ4 − 3ξ3 ξ4 + 6a2ξ2ξ4 − 3ξ2ξ3ξ4 − ξ1ξ4 − ξ2 ξ4 ,

3 2 2 4 2 2 3 2 2 2 3 2a2ξ1 + ξ1 + ξ1ξ2 − 9a2ξ3 − 6a2ξ1ξ3 + 6a2ξ3 − ξ3 − 6a2ξ1ξ2ξ4 + 6a2ξ1ξ4 − 3ξ1ξ3ξ4 − ξ1ξ2ξ4 .

Here ξ1 has degree 6, ξ2 has degree 3, ξ3 has degree 4, and ξ4 has degree 1.

The resulting deformation module is the cokernel of

  a4 − a3ξ4 − ξ3 β  , (4.16)  2  a3 − a3ξ2 − ξ1 γ Chapter 4. Deformations of Modules 80 where

2 2 3 3 2 2 2 3 β = a3 + a3ξ2 + ξ1 − 6a2a3ξ4 − 6a2ξ3 + ξ2 + 2a2 − 6a2ξ2ξ4 − a3ξ4 + 6a2ξ4 − 3ξ3ξ4 − ξ2ξ4 , (4.17)

2 2 2 2 2 2 2 2 4 γ = a4 + 6a2a3 + a4ξ3 + a3a4ξ4 − 6a2a4 + a3ξ4 + 2a3ξ3ξ4 + ξ3 − 6a2a3ξ4 − 6a2ξ3 + 9a2. (4.18)

As before, we ﬁrst eliminate parameters with odd degree, namely, we set ξ2 and ξ4 to zero. We obtain the ideal

0 3 2 4 2 2 3 I = 2a2ξ1 + ξ1 − 9a2ξ3 − 6a2ξ1ξ3 + 6a2ξ3 − ξ3

3 2 2 3 = (a2 + ξ1 − 3a2ξ3) − (a2 + ξ3) .

3 2 By degree considerations, any section must be of the form ξ1 = αa2 and ξ3 = βa2. Substituting these into I0, we ﬁnd that the coeﬃcients α and β must satisfy

(α − 3β + 1)2 = (β + 1)3.

This deﬁnes a cuspidal cubic, parametrized by t 7→ (t3 + 3t2 − 4, t2 − 1). The singularity occurs at t = 0.

Thus we obtain a family of sections

3 2 3 2 2 Γt : a2 7→ a2, (t + 3t − 4)a2, 0, (t − 1)a2, 0 indexed by a parameter t ∈ C. The corresponding family of modules is

  a − (t2 − 1)a2 a2 + (t − 2)2(t + 1)a3 ˜  4 2 3 2  M2,t = coker  . (4.19)  2 2 3 2 2 2 2 2 2 4 a3 − (t − 1)(t + 2) a2 a4 + 6a2a3 + (t − 7)a2a4 + (t − 4) a2

Proposition 4.4.2. The modules in the family {M˜ 2,t} are pairwise non-isomorphic.

˜ ˜ Proof. We ﬁrst claim that M2,s M2,t when s 6= ±t. Were this not the case for some pair s, t, then the

ﬁrst Fitting ideal of M˜ 2,s and of M˜ 2,t would be equal, and by inspection of the (1, 1) entry of each, both would then contain a4. But this is only the case when s and t are ±1, contradicting the hypothesis.

Now we consider the case that s = −t 6= 0. In this case, the (1, 1) and (2, 2) entries of the presentation matrices As and At of M˜ 2,s and respectively M˜ 2,t are identical, but the (2, 1) and (1, 2) entries are not.

3 Let cs, respectively ct be the the coeﬃcients of the a2 terms in the (2, 1) entries of As, respectively At. Chapter 4. Deformations of Modules 81

Then

2 2 cs − ct = (t − 1)(t + 2) − (s − 1)(s + 2)

= (t − 1)(t + 2)2 + (t + 1)(t − 2)2

= 2t3.

3 A row operation applied to As to set the coeﬃcient of a2 in the (2, 1) entry equal to ct would introduce a nonzero coeﬃcient of a2a4 in the (2, 1) entry. Inspection shows that no row or column operation can

3 eliminate this a2a4 term without also changing the coeﬃcient of the a2 term. However, the coeﬃcient of a2a4 in the (2, 1) entry of At is zero. Thus As and At are not equivalent and M˜ 2,s and M˜ 2,t are not isomorphic.

If we specialize (4.19) to t = 0, we obtain the following module:

  a + a2 a2 + 4a3 ˜  4 2 3 2  M2,0 = coker  . (4.20)  2 3 2 2 2 4 a3 + 4a2 a4 + 6a2a3 − 7a2a4 + 16a2

1 ˜ ˜ We have that ExtO ˜ (M2,0, M2,0) has Krull dimension 1. Its degree-zero graded piece has dimension 2 ∆4 as a C-vector space.

Take now t = 1, which corresponds to the section a2 7→ (a2, 0, 0, 0, 0) and is a smooth point of the family. We obtain the module

  a a2 + 2a3 ˜  4 3 2  M2,1 = coker  . (4.21)  2 2 2 2 4 a3 a4 + 6a2a3 − 6a2a4 + 9a2

1 ˜ ˜ In this case, Macaulay 2 computation shows that ExtO ˜ (M2,1, M2,1) has Krull dimension 0 and its ∆4 degree-zero graded piece has dimension 1 as a C-vector space.

0 0 ˜ 0 Now let us consider the following map. Let S := Spec C[a2, ξ1, ξ3]/I and deﬁne φ : ∆4 → S via 2 0 ˜ a2 7→ a2, ξ1 7→ −a3, and ξ3 7→ a4. Thus φ is a double cover of S by ∆4. The image of the section associated to t = 0 in S0 is deﬁned by the ideal

2 3 a2 = (a2 + ξ3, 4a2 − ξ1).

∗ 2 3 2 We have φ (a2) = (a2 + a4, 4a2 + a3), which is the ideal deﬁning the caustic in ∆4. Resolving this ideal, Chapter 4. Deformations of Modules 82 we see that   2 2 3  a4 + a a + 4a  ∗ ∼  2 3 2  ∼ ˜ φ (a2) = coker   = M2,2.  2 3 2 2 2 4 a3 + 4a2 a4 + 6a2a3 − 7a2a4 + 16a2

We conjecture that this holds throughout the family. More speciﬁcally:

Conjecture 4.4.3. For each t ∈ C let at be the ideal deﬁning the image in S of the associated section ∗ ∼ of S → Σ. Then φ (at) = M˜ 2,t.

As in the deformation of M1 above, we see here that the ideal deﬁning the caustic on ∆˜ 4 is maximal

Cohen-Macaulay. The results in Chapter 3 imply that the ideal deﬁning the caustic of ∆˜ n is not maximal Cohen-Macaulay for n > 5, but it is maximal Cohen-Macaulay for 3 ≤ n ≤ 5.

Deformation of M4

The matrix factorization associated to M4, in our coordinates, is

    2 2 3 2 −a4 0 a   a a a a4  3   4 3 3       a −a 0 , a a a2 a3  . (4.22)  3 4   3 4 4 3         2 2   0 a3 −a4  a3 a3a4 a4 

We have that Ext1 (M ,M ) is six-dimensional. The base space S is Spec [a , ξ , . . . , ξ ]/I, where I OX 4 4 C 2 1 6 is generated by

4 3 2 2 2 2 2 9a2 − 2a2ξ6 − 6a2ξ2 + 6a2(ξ4 + ξ2ξ6 + ξ3ξ6) − (3ξ4 + 2ξ4ξ5 + ξ5 )ξ6 − (ξ2 + 2ξ3)ξ6

2 2 + ξ2 + ξ2ξ3 + ξ3 + ξ1(ξ4 − ξ5),

3 2 2 2 2a2(ξ4 + ξ5) + 6a2ξ1 − 6a2((ξ2 + ξ3)(ξ4 + ξ5) + ξ1ξ6) + (ξ4 + ξ4ξ5 + ξ5 )ξ5

3 2 + ξ4 + 2(ξ2 + 2ξ3)(ξ4 + ξ5)ξ6 + ξ1ξ6 − ξ1(2ξ2 + ξ3),

3 2 2 2 2 2a2(ξ5 + ξ3ξ6) + 6a2(ξ2 + ξ3)ξ3 + 6a2((ξ2 + ξ3)(ξ4 − ξ5 − ξ3ξ6) + (ξ4 + ξ5)ξ1ξ6)

4 2 2 2 + ξ5 + 2(ξ2 + ξ3)(ξ5 − ξ4 )ξ6 + ξ3(2ξ4 + 3ξ5)ξ5ξ6 + (ξ2 + 2ξ3)ξ3ξ6

2 2 + 2(ξ4 + ξ5)ξ1ξ6 + (ξ1ξ4 − ξ2ξ3 − ξ1ξ5)(ξ2 + ξ3) − ξ1 ξ6.

The degrees of the parameters are listed in Table 4.1. Chapter 4. Deformations of Modules 83

Parameter Degree ξ1 5 ξ2 4 ξ3 4 ξ4 3 ξ5 3 ξ6 2

Table 4.1: Parameter degrees for versal deformation of M4

The deformation module is the cokernel of

  −a − 6a ξ − ξ + 6a2 + ξ2 2(ξ − 3a )(ξ + ξ ) − ξ θ   4 2 6 2 2 6 6 2 4 5 1     , (4.23)  a3 − ξ4 −a4 − ξ3 ξ1       −ξ6 a3 − ξ5 −a4 + ξ2 + ξ3 where

2 2 2 3 θ := a3 − 6a2a4 + a3(ξ4 + ξ5) + a4ξ6 + ξ4 + ξ4ξ5 + ξ5 + ξ3ξ6 + 2a2.

As above, to ﬁnd graded sections, we ﬁrst immediately set parameters of odd degree, namely ξ1, ξ4,

2 2 and ξ5, to zero. By examining degrees, we set ξ2 := αa2, ξ3 := βa2, and ξ6 := γa2. The resulting ideal is generated by

4 2 2 2 2 a2(2βγ − αβ − 6βγ − β + αγ − α − 6αγ + 6α + 2γ − 9),

6 2 2 2 a2β(2βγ − αβ − 6βγ + 6β + αγ − α − 6αγ + 6α + 2γ).

2 Subtracting the product of a2β and the ﬁrst generator from the second generator, we obtain

6 2 6 2 a2β(β + 6β + 9) = a2β(β + 3) .

Thus the primes associated to this ideal are

(a2),

(β, αγ2 − α2 − 6αγ + 6α + 2γ − 9),

(β + 3, αγ2 − α2 − 6αγ − 6γ2 + 9α + 20γ − 18).

Hence there are two allowable values of β: 0 and −3. These yield two families in the parameters α and Chapter 4. Deformations of Modules 84

γ:

β = 0 : αγ2 − α2 − 6αγ + 6α + 2γ − 9 = 0, (4.24)

β = −3 : αγ2 − α2 − 6αγ − 6γ2 + 9α + 20γ − 18 = 0. (4.25)

We consider these families separately. First we consider the family with β = 0. The curve deﬁned by (4.24) has a singularity at (α, γ) = (−1, 4). If we apply a change of coordinates (α0, γ0) = (α + 1, γ − 4), which moves the singularity to the origin, we obtain the equation α0γ02 − α02 + 2α0γ0 − γ02 = 0.

This has parametrization 1 t 7→ (t − 1)2, (t − 1)2 , t where t ∈ C∗. The singularity is at t = 1. Thus the family β = 0 has sections

1 Γ : a 7→ a , 0, (t − 1)2 − 1 a2, 0, 0, 0, (t − 1)2 + 4 a . 0,t 2 2 2 t 2

The associated modules are

  1 2 2 1 2 3 −a4 + t2 (1 − 2t)a2 0 a3 + t (t − 4t + 1)a2a4 + 2a2   ˜   M4,0,t = coker  a3 −a4 0 .    2   (t+1) 2   − t a2 a3 −a4 + t(t − 2)a2 

The 2 × 2 minors of M˜ 4,0,t are homogeneous and, among them, the lower-left minor has minimal degree. Thus it is a minimal generator of the Fitting ideal F1(M˜ 4,0,t). We see therefore that F1(M˜ 4,0,t)

2 2 2 2 (t+1) ˜ ˜ (s+1) (t+1) contains a3 − t a2a4. In particular, M4,0,s M4,0,t when s 6= t , showing that there are many nonisomorphic modules in this family. Even more is true, as the following proposition shows.

Proposition 4.4.4. The modules in the family corresponding to β = 0 are pairwise non-isomorphic.

Proof. Two such modules with presentation matrices, say, Ar and As, are isomorphic if and only if there exist invertible homogeneous matrices U and V with Ar = UAsV . Since there are no elements of degree one in O , an examination of the degrees of the entries in this family indicates that the only ∆˜ 4 nonzero entries of U and V are along the diagonal and in the (1, 3) position. A nonzero entry in the top-right corner of U or V corresponds to a row or column operation which must destroy the zero entries in the (1, 2) or (2, 1) position. Thus the entries in the (1, 3) position of U and V are also zero, and both Chapter 4. Deformations of Modules 85 matrices are diagonal. Inspection shows that no distinct members of the family can be related by the action of diagonal matrices, whence the claim.

At the singularity t = 1, we obtain

  −a − a2 0 a2 − 2a a + 2a3  4 2 3 2 4 2   M˜ = coker  . 4,0,1  a3 −a4 0     2   −4a2 a3 −a4 − a2 

1 Macaulay 2 computation shows that Ext (M˜ 4,0,1, M˜ 4,0,1) has Krull dimension 1 and its degree zero O∆4 part has dimension 2.

We now turn our attention to the family β = −3. The curve deﬁned by (4.25) has a singularity at (α, γ) = (2, 1). Using the same technique as for β = 0 above, we obtain the following parametrization

1 t 7→ (t + 2)2 + 2, (t + 2)2 + 1 . t

The singularity is at t = −2. This leads to the following family of sections

1 Γ : a 7→ a , 0, (t + 2)2 + 2 a2, −3a2, 0, 0, (t + 2)2 + 1 a . −3,t 2 2 2 2 t 2

The associated modules are

  1 2 2 1 2 1 3 −a + 2 (t + 4)(3t + 4)a 0 a + (t − t + 4)a a − (t + 3)(3t + 4)a   4 t 2 3 t 2 4 t 2   M˜ = coker  2 . 4,−3,t  a3 −a4 + 3a2 0     1 2   − t (t + 1)(t + 4)a2 a3 −a4 + (t + 1)(t + 3)a2 

Proposition 4.4.5. The modules in the family corresponding to β = −3 are pairwise non-isomorphic.

Proof. The argument is precisely the same as with β = 0.

By direct comparison of the matrix presentations, we see that M˜ 4,−3,−2 is the normalization of ∆˜ 4. We may also show that the families β = 0 and β = −3 are genuinely diﬀerent.

Proposition 4.4.6. The modules in the family associated to β = 0 are not isomorphic to any of the modules in the family associated to β = −3.

Proof. Proceeding along the same lines as with Propositions 4.4.4 and 4.4.5, were a module M4,0,t in the family β = 0 isomorphic to one M4,−3,t0 in the family β = −3 it would be possible to apply row and Chapter 4. Deformations of Modules 86

column operations to the matrix of M4,0,t to obtain the matrix M4,−3,t0 . The order of the generators and syzygies is prescribed, since no two generators and no two syzygies have the same degree. Thus the only available row and column operations are scaling and the addition of a multiple of one row or column to another. Consider now the (2, 2) entry in each of the matrices. Independently of t and t0, these entries are −a and, respectively, −a + 3a2. Neither is a scalar multiple of the other. Furthermore, since O 4 4 2 ∆˜ n has no element of degree one, no degree-preserving row or column operation can aﬀect the (2, 2) entry of either matrix. Thus the two modules cannot be isomorphic, whence the claim.

˜ 4.4.1 A classiﬁcation of rank one graded MCM modules over ∆4

We conclude this section by putting together the results above into the following theorem which classiﬁes all of the graded rank one MCM modules for ∆˜ 4.

Theorem 4.4.7. Every graded rank one MCM module over ∆˜ 4 is isomorphic to a twist of one of the following:

• the free module O ; ∆˜ 4

• the ideal (a , a ) and its O -dual; 3 4 ∆˜ 4

• the restriction of the open swallowtail S to ∆˜ and its O -dual, the ideal deﬁning the self- 4 4 ∆˜ 4

intersection locus of ∆˜ 4;

˜ • the modules of the family {M2,t | t ∈ C}, where

  a − (t2 − 1)a2 a2 + (t − 2)2(t + 1)a3 ˜  4 2 3 2  M2,t = coker  ;  2 2 3 2 2 2 2 2 2 4 a3 − (t − 1)(t + 2) a2 a4 + 6a2a3 + (t − 7)a2a4 + (t − 4) a2

˜ ∗ • the modules of the family {M4,0,t | t ∈ C }, where

  1 2 2 1 2 3 −a4 + t2 (1 − 2t)a2 0 a3 + t (t − 4t + 1)a2a4 + 2a2   ˜   M4,0,t = coker  a3 −a4 0 ;    2   (t+1) 2   − t a2 a3 −a4 + t(t − 2)a2 

and Chapter 4. Deformations of Modules 87

˜ ∗ • the modules of the family {M4,−3,t | t ∈ C }, where

M˜ 4,−3,t =   1 2 2 1 2 1 3 −a + 2 (t + 4)(3t + 4)a 0 a + (t − t + 4)a a − (t + 3)(3t + 4)a   4 t 2 3 t 2 4 t 2   coker 2 .  a3 −a4 + 3a2 0     1 2   − t (t + 1)(t + 4)a2 a3 −a4 + (t + 1)(t + 3)a2 

Furthermore, all of the modules in the list above are pairwise non-isomorphic.

4.5 Structure of Maximal Cohen-Macaulay Modules

In this section, we present some results relating the deformation of a module over a hypersurface singularity X with that of its maximal Cohen-Macaulay approximation (c.f. [Buc], [AB89], or [BH93, §3.1]). In the case that X is a curve singularity, this yields a strong connection between deformations of maximal Cohen-Macaulay modules on X and deformations of Artinian modules. In many cases, deformations of Artinian modules can be constructed explicitly. Given a module M over X, denote by MCM(M) its maximal Cohen-Macaulay approximation.

The deformation theory of M0 as a module is equivalent to the deformation theory of an OX -free resolution of M0 as a complex (c.f. [BF, §5.5]) in the following sense. Let F• be a free resolution of

M0. If (S, F•) is a deformation of F•, then F• has homology M only in degree zero, and (S, M) is a deformation of M0. Conversely, given a deformation (S, M) of M0, an OY ×ΣY -free resolution of M gives rise to a deformation of F• as a complex. These constructions are functorial and deﬁne inverse morphisms of ﬁbrations in groupoids between the deformation theory of modules and its image in the deformation theory of complexes.

Given a deformation (S, M) of an OX -module M0 of inﬁnite projective dimension, we may construct a deformation (S, N ) of MCM(M0) in the following way. Let (S, F•) be the associated deformation of complexes. Let c be the codepth of M0 so that the c-th syzygy of M0 is maximal Cohen-Macaulay over

OX . From [Eis80], we see that F• is two-periodic beyond homological degree c. In particular, MCM(M0) is isomorphic to the d-th syzygy of M0, where d = c if c ≡ 0 mod 2 and d = c+1 otherwise. Let F≥d be the truncation of F• at homological degree c. Then F≥c is exact except in homological degree c and its homology N in degree c deﬁnes a deformation (S, N ) of MCM(M0). Call this deformation MCM(S, M).

The following proposition gives a criterion for whether the above construction, applied to a miniversal deformation of M0, yields a versal, or even miniversal, deformation of MCM(M0). For OX -modules M,N Chapter 4. Deformations of Modules 88 and for i > 0, denote by Exti (M,N) the module Exti (MCM(M), MCM(N)). There exist natural OX OX maps η : Exti (M,N) → Exti (M,N) for i > 0, c.f. [Buc]. i OX OX

Proposition 4.5.1. Suppose that η1 is surjective and that η2 is an isomorphism. Then, if a = (S, M) is a miniversal deformation of M0, MCM(a) is a versal deformation of MCM(M0). If η1 is also injective, then the resulting deformation of MCM(M0) is even miniversal.

Proof. Suppose that a = (S, M) is a deformation of M0 and let a := MCM(a). Let I be a ﬁnitely-

0 generated OS-module. We have a map α : ExΣ(a, I) → ExΣ(a, I) which takes the class of a to the class

0 of MCM(a ). The maps α, η1, and η2 give rise to a map of the associated Kodaira-Spencer sequences

Ext1 (M ,M ) ⊗ I - Ex (a, I) - Ex (S, I) - Ext2 (M ,M ) ⊗ I OX 0 0 Σ Σ OX 0 0 w w η1 ⊗ id α w η2 ⊗ id . w ? ? w ? Ext1 (M ,M ) ⊗ I - Ex (a, I) - Ex (S, I) - Ext2 (M ,M ) ⊗ I OX 0 0 Σ Σ OX 0 0

Since η2 is an isomorphism, η2 ⊗ id is too. An easy diagram chase (left to the reader) shows that, given that η2 ⊗ id is bijective and η1 ⊗ id is surjective, α is surjective. The ﬁrst claim now follows from the following general result.

Proposition 4.5.2. For a deformation a, the following are equivalent.

1. a is formally versal.

2. ExΣ(a, C) = 0.

3. ExΣ(a, I) = 0 for every ﬁnite OS-module I.

Proof. See [BF, Proposition 3.4.15].

If η1 is also injective, then it is an isomorphism. In that case, the tangent spaces at zero of miniversal deformations of M0 and MCM(M0) are of the same dimension, so the resulting versal deformation of

MCM(M0) is of minimal dimension and is therefore miniversal.

In the case that X is a curve singularity, the hypothesis of Proposition 4.5.1 is automatically satisﬁed.

Proposition 4.5.3. Suppose that X is a curve singularity. Then η1 is surjective and η2 is an isomorphism.

Proof. From [Buc, Corollary 6.3.4], we see that ηi is surjective for i = m and an isomorphism for i > m, where m is the codepth of M0. Since X is a curve singularity, the codepth of m is at most one, whence the claim. Chapter 4. Deformations of Modules 89

The upshot is that we can construct a versal deformation of a rank one maximal Cohen-Macaulay module M0 on X in the following way. First embed M0 as an ideal in OX . Then deform its residue

0 ring, which is Artinian, to obtain a miniversal deformation (S, M ) of OX /M0. Then the deformation

0 (S, M), where M is the ﬁrst syzygy of M , is a versal deformation of M0.

In the case X is a plane curve and M0 is the maximal ideal of OX , we obtain even miniversality.

Proposition 4.5.4. Suppose that X is a plane curve and M = C, the residue ﬁeld of OX . Then η1 is an isomorphism.

Proof. Let m be the maximal ideal of X. Applying the functor Hom(−, OX ) to the short exact sequence

- - - - 0 m OX C 0, we obtain a short exact sequence

0 - O - Hom(m, O ) - Ext1 ( , O ) - 0. (4.26) X X OX C X

Since X is Gorenstein of dimension one, we also have Ext1 ( , O ) ∼ . Thus MCM( ) = Hom(m, O ). OX C X = C C X

∗ Write m := Hom(m, OX ). Applying the functor Hom(−, C) to (4.26), we obtain the exact sequence

a b 0 - - Hom(m∗, ) - - Ext1 ( , ) - Ext1 (m∗, ) - 0. C C C OX C C OX C

Since the C-vector space dimension of Hom(m∗, C) is the minimal number of generators of m∗, which is two, the map a is zero and the map b is therefore an isomorphism. Furthermore, applying the functor Hom(m∗, −) to (4.26), we obtain an exact sequence

c Ext1 (m∗, O ) - Ext1 (m∗, m∗) - Ext1 (m∗, ) - Ext2 (m∗, O ). OX X OX OX C OX X

However, Ext1 (m∗, O ) = 0 = Ext2 (m∗, O ) since m∗ is maximal Cohen-Macaulay. Thus the map OX X OX X c is also an isomorphism, and the claim follows.

Furthermore, a miniversal deformation of the residue ring of M0, which is just the residue ﬁeld of

OX , is the germ of the Hilbert scheme of one point on Y . Thus the base space of this deformation is isomorphic again to Y and its deformed module is the structure sheaf of the diagonal embedding

Y → Y ×Σ Y . The following corollary, of which Theorem 4.4.1 is a special case, is then immediate. Chapter 4. Deformations of Modules 90

Corollary 4.5.5. The base space in a miniversal deformation of the maximal ideal of OX is isomorphic to Y . The deformed module M is the ideal deﬁning the diagonal embedding Y → Y ×Σ Y .

Given an arbitrary maximal Cohen-Macaulay modules M0 on X, all which remains to ﬁnd a versal deformation of M0 is to construct a miniversal deformation of its associated Artinian module. A complete description of the structure of versal deformations of Artinian modules on curve singularities would then yield a structure theorem for the maximal Cohen-Macaulay modules on the classical discriminant, characterizing such modules in terms of subvarieties of the base spaces of these deformations. Chapter 5

Conclusion

We began this study with the question, “Do there exist nontrivial determinantal formulae for the classical discriminant which are not equivalent to the classical ones?” The results answered the question very much in the aﬃrmative. In Chapter 3, we constructed explicitly a determinantal formula for the classical discriminant in each degree, and which, in degrees greater than three, is nontrivial and inequivalent to the classical formulae. Along other lines, we showed in Chapter 4 the existence of entire families of inequivalent determinantal formulae for the degree 4 discriminant. Such families appear even after restricting attention to the “simplest” formulae, namely those whose matrices have entries which are quasihomogeneous with respect to the weights introduced in Chapter 1. Thus the structure of determinantal formulae of the classical discriminant is indeed rich.

The main techniques used here involve realizing determinantal formulae as the presentation matrices of rank one maximal Cohen-Macaulay modules over the discriminant hypersurface. There appear in the literature other approaches to the explicit construction of determinantal formulae for discriminants and resultants, such as twisted Koszul complexes (c.f. [GKZ94]) and the Bernstein-Gel’fand-Gel’fand correspondence (c.f. [ESW03], [Khe05]). All of these techniques are limited to particular cases of resultants, and one might wonder whether a more systematic study of the MCM module structure over resultant hypersurfaces might yield a more general technique.

The general philosophy underlying the results in this thesis and in the literature is that the structure of the determinantal formulae of a hypersurface becomes more complicated as the codimension of singular locus of the variety falls. Namely, a result of Grothendieck proved in [Gro05] shows that a germ the codimension of whose singular locus is at least four has, up to isomorphism, only one rank one maximal Cohen-Macaulay module. We have studied the situation at nearly the other extreme, where the singular locus lies in codimension one.

91 Chapter 5. Conclusion 92

The results give rise to some avenues of future research.

• The open swallowtail construction for other varieties. Of course, for any variety, one may deﬁne the open swallowtail in a manner analogous to Deﬁnition 3.1.4. This obviously is only interesting when the variety is not itself normal. The question which arises is whether the resulting module has any reasonable properties. It would be interesting to check whether this generalized open swallowtail is maximal Cohen-Macaulay and non-free for, say, discriminants in versal deformations of isolated complete intersection singularities, or for resultant or multi-dimensional discriminant hypersurfaces.

• Construction of moduli spaces of MCM modules. There is great interest from the string theory community in understanding the structure of maximal Cohen-Macaulay modules over certain hypersurfaces. The techniques in Chapter 4 could be used to compute such moduli spaces. Perhaps more compellingly, the theory of Section 4.5 could be used to construct such moduli spaces from abstract theory, giving much greater insight into the structure of these modules.

• Other determinantal formulae. The results of Section 4.4 suggest that there is in general a ﬁnite set of “distinguished modules” over the classical discriminant the deformations of which give rise to all other graded rank one maximal Cohen-Macaulay modules and which are strongly connected with the geometry of the discriminant. One of these is the open swallowtail deﬁned in Chapter 3. There may exist explicit constructions for other such modules, which would yield new determinantal formulae. Furthermore, these formulae may encode information about the root structure of a polynomial which is not present in the known formulae.

• The open swallowtail in positive characteristic. The results particularly in Chapter 3 assume that the base ﬁeld has characteristic zero. It is possible that analogous results hold in positive characteristic.

• Resultants and multi-dimensional discriminants. Finally, as suggested above, it may be possible using the theory of maximal Cohen-Macaulay modules to construct explicitly determinantal formulae for other polynomials, such as resultants and multi-dimensional discriminants in more cases than have been discovered to date. Appendix A

Macaulay 2 package for Module

Deformations

The following Macaulay 2 [GS] package implements the deformation algorithm described in Chapter 4. It is also available at

http://www.math.utoronto.ca/hovinen/ModuleDeformations.m2 as of the time of publication of this thesis.

-- ModuleDeformations.m2 -- -- Copyright 2007-08 Bradford Hovinen -- Licensed under the GNU GPL version 3.0 newPackage( "ModuleDeformations", Version => "1.0", Date => "18 March 2008", Authors => {{Name => "Bradford Hovinen", Email => "[email protected]", HomePage => "http://www.math.utoronto.ca/hovinen/"}}, Headline => "Computing versal deformations of maximal Cohen-Macaulay modules", DebuggingMode => true ) export {deformMCMModule}

93 Appendix A. Macaulay 2 package for Module Deformations 94

-- ShiftRowDegs(M,d) -- -- Given a homogeneous matrix of degree e, return a new matrix of -- degree d+e with row degrees shifted by -d

ShiftRowDegs = (M, d) -> ( map(target M**(ring M)^{-d}, source M, M, Degree=>(d + degree M)) )

-- ShiftColDegs(M,d) -- -- Given a homogeneous matrix of degree e, return a new matrix of -- degree d+e with col degrees shifted by d

ShiftColDegs = (M, d) -> ( map(target M, source M**(ring M)^{d}, M, Degree=>(d + degree M)) )

-- HomBoundaryMaps -- -- Given a matrix factorization corresponding to an MCM module M, -- compute the boundary maps in the complex Hom^*(M,M). Since the -- resolution of M is two-periodic, there are two such maps. They are -- returned as (2n^2)x(2n^2) matrices, where n is the rank of M. The -- first map is the even-degree differential, and the second is the -- odd-degree differential. -- -- The matrices should be defined over the hypersurface ring over -- which they represent an MCM.

HomBoundaryMaps = (A, B) -> ( ((ShiftRowDegs(Hom(source A,A) | Hom(A,source B), -degree A)) || (ShiftRowDegs(-Hom(B,source A) | -Hom(source B,B), -degree B)),

ShiftColDegs ((ShiftColDegs(Hom(source A,B) || -Hom(B,source B), -degree B)) | (ShiftColDegs(Hom(A,source A) || -Hom(source B,A), -degree A)), degree B + degree A)) ) Appendix A. Macaulay 2 package for Module Deformations 95

-- VectToMatPair -- -- Convert a matrix pair represented as a column vector to a pair of -- matrices with the correct degrees.

VectToMatPair = (v, source1, target1, deg1, source2, target2, deg2) -> ( M := reshape (source1, target1 ++ target2, transpose v);

(map (target1, source1, M_[0], Degree=>deg1), map (target2, source2, M_[1], Degree=>deg2)) )

-- MatPairToVect -- -- Given a pair of matrices, represent it as single column vector with -- the correct degrees. The row degrees are prescribed.

MatPairToVect = (M1, M2) -> ( n := numgens target M1; N := 2 * n^2; reshape ((ring M1)^N,(ring M1)^1, M1 | M2) )

-- ComputeExtBasisVectors -- -- Given even and odd differentials of complex, compute a basis of -- Ext^1. Return the result as columns of a matrix.

ComputeExtBasisVectors = (A,B,i) -> ( d := HomBoundaryMaps (A,B); ExtModule := trim(ker d_i/image d_(1-i)); ExtMatrix := super (basis (ExtModule));

-- For some reason, this computation returns the direct sum of -- two modules, only the first of which is the correct one. ExtMatrix_{0..numgens source ExtMatrix // 2 - 1} ) Appendix A. Macaulay 2 package for Module Deformations 96

-- ComputeExtBasis -- -- Given even and odd differentials of complex, compute a basis of -- Ext^1. Return the result as pairs of matrices. Also return the -- degrees of the generators.

ComputeExtBasis = (A,B,i) -> ( ExtBasisVect := ComputeExtBasisVectors(A,B,i); ExtBasisVect = map (ambient target ExtBasisVect, source ExtBasisVect, ExtBasisVect); ExtDegs := -((degrees ExtBasisVect)_1); ExtBasis := (i -> VectToMatPair (ExtBasisVect_{i}, source A, target A, degree A + (degrees ExtBasisVect)_1_i, source B, target B, degree B + (degrees ExtBasisVect)_1_i)) \ toList(0..numgens source ExtBasisVect - 1);

(ExtBasis, ExtDegs) )

-- LiftExt2Basis -- -- Lift a given basis of Ext2 to the given ring

LiftExt2Basis = (Ext2Vect, R) -> ( map(coker substitute(presentation target Ext2Vect, R), substitute(source Ext2Vect, R), substitute(map(ambient target Ext2Vect, source Ext2Vect, Ext2Vect), R)) ) Appendix A. Macaulay 2 package for Module Deformations 97

-- ConstructAmbientRing -- -- Construct the ambient product ring in quotients of which all -- computations take place

ConstructAmbientRing = (OY,OSigma,Ext1degs,pIdeal) -> ( dimExt1 := length Ext1degs;

-- We pull out the variables coming from Sigma and treat them -- separately for term ordering purposes SigmaVars := if (pIdeal == 0) then {} else flatten entries mingens pIdeal; YVars := select (flatten entries vars OY, (v -> not member(v, SigmaVars))); SigmaVarDegs := SigmaVars / (v -> degree v); YVarDegs := YVars / (v -> degree v);

OYAmbientxSAmbient := QQ[YVars, SigmaVars, xi_1..xi_dimExt1, Degrees=>(YVarDegs | SigmaVarDegs | Ext1degs), MonomialOrder=>{Weights => Eliminate(length YVars)}]; OYAmbientxSAmbient/(substitute (ideal OY, OYAmbientxSAmbient)) )

-- FirstDeformation -- -- Deform a module into its first infinitesimal neighbourhood.

FirstDeformation = (OYxSAmbient,pIdeal,A,B,d,Ext1,Ext2Vect) -> ( dimExt1 := length Ext1;

use OYxSAmbient; OYxS1 := OYxSAmbient / ((substitute(pIdeal, OYxSAmbient) + ideal {xi_1..xi_dimExt1})^2);

At := substitute (A, OYxS1); Bt := substitute (B, OYxS1); lhs := -At*Bt; vlhs := map(substitute(target d_1,OYxS1),, MatPairToVect (lhs, -lhs)); psolvect := vlhs // substitute (d_1, ring vlhs);

psol := VectToMatPair (psolvect, source At, target At, degree At, source Bt, target Bt, degree Bt);

use OYxS1;

A = fold (plus, ((j -> (xi_(j+1)_OYxS1 * substitute ((Ext1_j)_0, OYxS1))) \ (0 .. dimExt1 - 1))) + psol_0 + At; B = fold (plus, ((j -> (xi_(j+1)_OYxS1 * substitute ((Ext1_j)_1, OYxS1))) \ (0 .. dimExt1 - 1))) + psol_1 + Bt;

(A, B) ) Appendix A. Macaulay 2 package for Module Deformations 98

-- ExtendDeformation -- -- Given a deformation, extend it by one more infinitesimal -- neighbourhood.

ExtendDeformation = (OYxSAmbient,pIdeal,i,A,B,d,dimExt1,Ext2Vect,obstruction) -> ( use OYxSAmbient; OYxSi := OYxSAmbient / ((pIdeal + ideal {xi_1..xi_dimExt1})^(i+1)); At := substitute (A, OYxSi); Bt := substitute (B, OYxSi);

lhs1 := -Bt*At; lhs2 := -At*Bt;

-- Check whether we are done. If the left-hand side modulo the -- existing obstruction is done, then a particular solution is -- zero and no further computation is necessary.

obstruction = obstruction / (f -> substitute (f, OYxSi)); obsIdeal := ideal obstruction; if (all (flatten flatten entries lhs1, f -> (f % obsIdeal == 0)) and all (flatten flatten entries lhs2, f -> (f % obsIdeal == 0))) then return (At, Bt, obstruction, false);

-- This computation must take place in the ambient ring of O_Y, -- since otherwise relations in O_Y may screw up the degrees in -- the particular solution. use ambient OYxSAmbient; OYAmbientxSi := (ambient OYxSAmbient) / ((lift(pIdeal,ambient OYxSAmbient) + ideal {xi_1..xi_dimExt1})^(i+1));

Ext2VectOYAmbientxSi := LiftExt2Basis(Ext2Vect,OYAmbientxSi);

vlhs := map(substitute(target d_1, OYAmbientxSi),, substitute(MatPairToVect (lhs1, -lhs2),OYAmbientxSi)); vlhsobs := map(target Ext2VectOYAmbientxSi,, vlhs % substitute (d_1, OYAmbientxSi)); obsi := vlhsobs // Ext2VectOYAmbientxSi;

obstruction = flatten entries (substitute(obsi,OYxSi))_0;

rawpsol := substitute(substitute(vlhs,OYAmbientxSi) // substitute (d_1, OYAmbientxSi), OYxSi);

psol := VectToMatPair (rawpsol, source At, target At, degree At, source Bt, target Bt, degree Bt);

(At + psol_0, Bt + psol_1, obstruction, false) ) Appendix A. Macaulay 2 package for Module Deformations 99

-- deformMCMModule -- -- Deform a maximal Cohen-Macaulay module on a hypersurface into a -- module over the hypersurface ring defined by the given polynomial -- up to the given maximal degree of the indeterminate. Return the -- resulting matrix factorization and the obstruction. deformMCMModule = method(Options => {DegreeLimit => 10}) deformMCMModule(Module,RingMap) := Module => o -> (M0,phi) -> ( A0 := syz syz presentation M0; if (numrows A0 != numcols A0) then error "M0 is not maximal Cohen-Macaulay"; B0 := map(source A0,target A0,syz A0, Degree=>(degrees source mingens ideal target phi)_0); OY := target phi; OSigma := source phi; pIdeal := phi(ideal vars OSigma);

d := HomBoundaryMaps (A0, B0); Ext2BasisVect := ComputeExtBasisVectors (A0, B0, 0);

dimExt2 := numgens source Ext2BasisVect; obstruction := toList (dimExt2 : 0_(OY/pIdeal));

(Ext1, Ext1degs) := ComputeExtBasis (A0, B0, 1);

OYxSAmbient := ConstructAmbientRing (OY, OSigma, Ext1degs, pIdeal); (A,B) := FirstDeformation (OYxSAmbient, pIdeal, A0, B0, d, Ext1, Ext2BasisVect);

pIdeal = substitute(pIdeal, OYxSAmbient);

for i from 2 to o.DegreeLimit do ((A,B,obstruction,done) = ExtendDeformation (OYxSAmbient, pIdeal, i, A, B, d, length Ext1, Ext2BasisVect, obstruction); if done then break);

OYAmbientxSAmbient := ambient OYxSAmbient; OYAmbientxS := (OYAmbientxSAmbient) / ideal (obstruction / (f -> substitute (f, ambient OYAmbientxSAmbient))); coker substitute (A, OYAmbientxS) ) Appendix A. Macaulay 2 package for Module Deformations 100

-- deformMCMModule -- -- Deform a maximal Cohen-Macaulay module on a hypersurface. Return -- the resulting matrix factorization and the obstruction. deformMCMModule(Module) := Module => o -> (M0) -> ( R=ring M0; OSigma=QQ[]; phi=map(R,OSigma,matrix(R,{{}})); return deformMCMModule(M0,phi,o) ) Bibliography

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