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Matrix Factorizations of the Classical Discriminant by Bradford Hovinen A

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Matrix Factorizations of the Classical Discriminant by Bradford Hovinen A Matrix Factorizations Of The Classical Discriminant by Bradford Hovinen A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto Copyright c 2009 by Bradford Hovinen Abstract 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 B´ezout,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 fx − y g. 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<j≤n is called the discriminant of f. Its utility lies in its ability to detect polynomials with repeated roots and in its many well-known arithmetic properties (c.f., e.g., [DF04, Chapter 14], [Art91], [Bou03, Chapter 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) 2 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<j≤n is still symmetric in α1; : : : ; αn, as well as its complement. There are two cases. Q In the first case, D1 = D2 = 1≤i<j≤n(αi − αj). Then, since the characteristic of K is not two, the permutation reversing α1 and α2 and leaving α3; : : : ; αn fixed reverses the sign of D1 and D2 and thus 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 ∆^ 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.
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