ON the DISCRIMINANT LOCUS of a RANK N − 1 VECTOR BUNDLE on Pn−1

ON THE DISCRIMINANT LOCUS OF A RANK n − 1 VECTOR BUNDLE ON Pn−1

HIROTACHI ABO

ABSTRACT. In this paper, we show that if a vector bundle of rank n−1 on projective (n−1)- space is very ample and if its top Chern class is greater than one, then the discriminant locus of the vector bundle, the locus of singular sections of the vector bundle, is an irreducible hypersurface and that the degree of the hypersurface can be expressed as a function of invariants of the vector bundle. As applications, we study the eigendiscriminant locus (the locus of tensors whose eigenschemes are singular) and the discriminant locus of the generic graded matrix (the locus of singular determinantal schemes).

1. INTRODUCTION This paper is concerned with the classification of singular elements of the space of global sections of a vector bundle on projective space. A non-zero global section of a vector bundle of rank r on projective space defines a closed subscheme of the projective space called the zero scheme of the global section. The zero scheme of a global section of a rank r vector bundle has codimension at most r (if it is non-empty). The global sections of the vector bundle whose zero scheme is non-singular of codimension r form a Zariski open set of the projective space of the global sections of the vector bundle. We call the complement of such an open set the discriminant locus of the vector bundle. The discriminant locus of a vector bundle is very interesting in its own right, but it is also a very important object to study for several reasons. First, the complement of the discriminant locus of a vector bundle is related to a moduli problem in algebraic geometry [3, 31]. Second, the discriminant locus of a vector bundle helps classify varieties (or more generally, schemes) that appear as limits of a flat family of non-singular varieties [3]. Third, it relates closely to the problem of classifying the “dual defect” varieties, the varieties whose projective duals are not hypersurfaces [14, 30]. Last, the discriminant locus of a vector bundle has applications in linear and multilinear algebra [1, 2, 6]. In this paper, we study the geometric properties of the discriminant locus of a vector bundle of rank n − 1 on projective (n − 1)-space Pn−1. The discriminant locus of such a vector bundle generally is irreducible of codimension one. It is, in fact, an irreducible hypersurface if the vector bundle is a direct sum of (n − 1) very ample line bundles and if its top Chern class is greater than one (see, for example, the paper [7] by L. Busé and J. P. Jounaolou). There are, however, indecomposable vector bundles whose discriminant loci are not irreducible or have higher codimensions. Thus, classifying degenerate discriminant loci is a tempting, but challenging problem. The problem of classifying degenerate discriminant loci was extensively studied for line bundles, or more generally, linear systems of divisors (see, for example, [14, 23, 22, 21, 20, 19]), but the classification is still incomplete. With regards to vector bundles of higher

2010 Mathematics Subject Classiﬁcation. 13C10, 13C40, 13D02, 14A10, 14A15, 14F05, 14M12, 15A69. Key words and phrases. Vector bundles, discriminant, eigenvectors, tensors, determinantal varieties. 1 2 HIROTACHI ABO rank, the classiﬁcation problem has never systematically been explored. A main goal of this paper is to establish a large family of vector bundles whose discriminant loci are non- degenerate. More precisely, we prove that if the vector bundle is very ample (or if the tautological line bundle of the projective bundle associated with the vector bundle is very ample) and if the top Chern class of the vector bundle is greater than one, then the discriminant locus of the vector bundle has the expected geometric properties. Furthermore, we show that the degree of the discriminant hypersurface is expressible as a function of invariants of the vector bundle.

Theorem 1.1. If E is a very ample vector bundle of rank n − 1 on projective (n − 1)- space Pn−1 over an algebraically closed ﬁeld of characteristic zero and if its top Chern class cn−1(E ) of E is greater than one, then the discriminant locus of E is an irreducible hypersurface of degree     Xn−3 ^i+2    n−1+i    (1.1) 2cn− (E ) + 2  (−1) (i + 1) χ  E  (−n) − 2, 1     i=0

Vi+2 where χ E (−n) is the Euler characteristic of the (i + 2)nd exterior power of E twisted by −n. In particular, if d ≥ 2, then the discriminant locus of the tangent bundle n−1 n−1 TPn−1 (d−2) on P twisted by d−2 is an irreducible hypersurface of degree (n−1)n(d−1) . In their 2014 paper [7], L. Busé and J. P. Jounaolou proved that the discriminant locus of any direct sum of n−1 very ample line bundles on projective (n−1)-space is an irreducible hypersurface not only over an algebraically closed field of characteristic zero, but also over any commutative ring. They also gave a formula for the degree of the discriminant hypersurface of such a vector bundle (see [7, Proposition 3.9]). In cases where the ground ring is an algebraically closed field of characteristic zero, Theorem 1.1 generalizes their result to an arbitrary very ample vector bundle of rank n − 1. Our approach is also different from Busé and Jounaolou’s approach. Theirs is algebraic; ours is geometric. Indeed, the proof of Theorem 1.1 uses the geometry of the curve obtained as the dependency locus of two global sections of a vector bundle. The discriminant locus of a very ample vector bundle is a hypersurface precisely when it intersects the pencil of two generic global sections of the vector bundle. Moreover, the degree of the discriminant locus is equal to the intersection number of the discriminant locus and the pencil. The key step in the proof of Theorem 1.1 is to reduce the problem of finding such an intersection number to the problem of finding the degree of the ramification divisor of the finite morphism from the dependency locus of the two generic global sections to the pencil. Consequently, formula (1.1) follows from the Hurwitz-Riemann formula. Theorem 1.1 has applications to studying the loci of singular elements of two different spaces that parameterize zero-dimensional closed subschemes; namely, the first is the “eigendiscriminant locus,” the locus of tensors whose eigenschemes are singular, and the second is the discriminant locus of the generic graded matrix, the locus parameterizing graded matrices whose determinantal schemes are singular. Eigenvectors of tensors, extensions of eigenvectors of matrices, were introduced by L.- H. Lim [24] and L. Qi [26] independently in 2005 and have been studied in numerical multilinear algebra. Recently, the concept of eigenvectors of a tensor drew attention from the algebraic geometry community because algebraic geometry is proven to provide useful techniques for the tensor eigenproblem (see the excellent expository paper written by B. Sturmfels [28] for a more detailed algebro-geometric treatise of the spectral theory of tensors). ON THE DISCRIMINANT LOCUS OF A RANK VECTOR BUNDLE 3

The conditions for a non-zero vector to be an eigenvector of a tensor can be described by means of the ideal generated by the 2 × 2 minors of a n × 2 polynomial matrix. The scheme defined by such a determinantal ideal is called the eigenscheme of the tensor (see Definition 5.1). As shall be seen in Section 5, the eigenscheme of an n-dimensional tensor can be identified with the zero scheme of a global section of a certain very ample vector bundle, and thus the eigenscheme of a generic tensor is non-singular of codimension n − 1. The eigenscheme of a tensor could, however, be “degenerate,” which means that it is non- reduced or has a positive dimensional primary component. The eigendiscriminant locus is the locus of such degenerate tensors. In this paper, we show that the eigendiscriminant locus is an irreducible hypersurface in the projective space of tensors and calculate the degree of the eigendiscriminant hypersurface. The tensors whose eigenschemes are degenerate have a remarkable structure. For instance, the eigenscheme of a matrix is degenerate if and only if the matrix has a nilpotent Jordan structure (see [1]). Therefore, analyzing the eigendiscriminant hypersurface would enhance our understanding of tensors that have such a structural property. This result was stated in [2, Theorem 4.1], where the outline of the proof was presented. Our purpose is to give a more detailed proof of the result. A determinantal scheme is the scheme defined by the fixed order minors of a graded matrix, which is a matrix with polynomial entries satisfying certain degree conditions (see Section 6.1 for a more precise definition). The determinantal scheme defined by the (r + 1) × (r + 1) minors of an ` × m graded matrix whose entries are polynomials in n variables is expected to have codimension (` − r)(m − r) in Pn−1. Furthermore, the codimension of its singular locus is supposed to be (`−r+1)(m−r+1). Therefore, if (`−r+1)(m−r+1) ≥ n, then the determinantal scheme is expected to be non-singular. In this case, we wish to describe an algebraic condition(s) the graded matrices must satisfy, so that their determinantal loci are singular or have higher codimensions than expected. We call the locus of the graded matrices with such unexpected properties the determinantal locus of the generic graded matrix. In this paper, we prove that if ` = n + k − 1, m = k, and r = k − 1, then the discriminant locus of the generic graded matrix is an irreducible hypersurface. The polynomial that defines the discriminant hypersurface is multi-homogeneous with respect to the k + 1 sets of variables, each of which corresponds to a (n+k−1)×k submatrix of the (n+k−1)×(k+1) graded matrix. Such a submatrix induces a morphism between direct sums of line bundles whose cokernel is a very ample vector bundle of rank n − 1. We show that the degree of the discriminant hypersurface with respect to each such set of variables is equal to the degree of the discriminant hypersurface of the corresponding very ample rank n − 1 vector bundle. The formula for the degree of the discriminant hypersurface of the generic graded matrix is very complicated in general. However, if the entries of the graded matrices are linear, then it is surprisingly simple.

Theorem 1.2. The discriminant locus of the generic graded (n + k − 1) × (k + 1) matrix n+k−1 with linear entries is an irreducible hypersurface of degree 2(k + 1)k n−2 .

In the rest of this section, we outline how the paper is organized and give a brief sum- mary of each section. Section 2 studies the geometry of the curve which is the dependency locus of two generic global sections of a rank n − 1 vector bundle on Pn−1. The main purpose of Sec- tion 2 is to express the degree of the ramiﬁcation divisor in terms of data from the vector bundle. 4 HIROTACHI ABO

Section 3 is devoted to the proof of Theorem 1.1. We also explore several explicit examples of discriminant loci, including ones that are reducible or have higher codimensions. Many of these examples are based on the computations performed in Macaulay2 [15]. In Section 4, we apply Theorem 1.1 to a fundamental class of very ample vector bundles called “Schwarzenberger bundles.” A Schwarzenberger bundle, roughly speaking, is a very ample vector bundle of rank n − 1 on Pn−1 which has a minimal free resolution of length one. The Schwarzenberger bundle satisfies the desired Chern class requirement, and hence its discriminant locus is an irreducible hypersurface. The main goal of this section is to use the graded Betti numbers of the Schwarzenberger bundle to reformulate the degree of the discriminant hypersurface. Sections 5 and 6 are about applications. More precisely, Section 5 concerns the eigendiscriminant locus and Section 6 concerns the discriminant locus of the generic graded matrix. In these sections, the connections that link Schwarzenberger bundles to eigenvectors of tensors and determinantal schemes play a central role. In Sections 2–6, the characteristic of the field is assumed to be zero. Section 7 is devoted to extending several results obtained in Sections 3–6 to a field of positive characteristic.

2. PRELIMINARIES In this section, we recall basic concepts and results from the theory of vector bundles. The main focus of this section is on the dependency locus of two generic global sections of a rank n − 1 vector bundle on projective (n − 1)-space Pn−1. The dependency locus of such global sections is a curve if the vector bundle is very ample. The goal of this section is to use invariants of the vector bundle to express the degree of the ramification divisor of the morphism from the curve to the pencil of the global sections. The Hurwitz-Riemann formula describes the degree of the ramification divisor in terms of the degree of the morphism and the genus of the curve. The degree of the morphism is the top Chern class of the vector bundle, and hence all we need to do is find the genus of the curve. To do so, it suffices to calculate the Euler characteristic of the curve. The key technical tool to find the Euler characteristic of the curve is the Eagon-Northcott complex, which enables one to compute a free resolution of the structure sheaf of the curve. 2.1. Degeneracy locus of a morphism between vector bundles. Let k be an algebraically closed field of characteristic zero and let Pn−1 be projective (n − 1)-space over k. If F is a coherent sheaf on Pn−1, then we write • kF for the direct sum of k copies of F , • χ(F ) for the Euler characteristic of F , • F (a) for F twisted by OPn−1 (a), • F ∨ for the dual sheaf of F , • ci(F ) for the ith Chern class of F for each i ∈ {1,..., n − 1}, • ct(F ) for the Chern polynomial of F , • Hi(F ) for the ith cohomology group of F for each i ∈ {0,..., n − 1}. If E is a vector bundle on Pn−1, then we denote by S iE the ith symmetric power of E and Vi by the ith exterior power of . If and are n−1 -modules, then Hom ( , ) E E F G OP OPn−1 F G denotes the set of morphisms of n−1 -modules from to . Let ϕ ∈ Hom ( , ). OP F G OPn−1 F G For each a ∈ Z, we write ϕ[a] for the morphism of OPn−1 -modules from F (a) to G (a) induced by ϕ and H0(ϕ) for the homomorphism from H0(F ) to H0(G ) associated with ϕ. Let f, g, r ∈ N with r < f ≤ g, let F and G be vector bundles on Pn−1 of ranks f and g respectively, and let ϕ : F → G be a morphism of OPn−1 -modules. The rth degeneracy ON THE DISCRIMINANT LOCUS OF A RANK VECTOR BUNDLE 5 locus n n−1 o Dr(ϕ) = p ∈ P rank(ϕ(p)) ≤ r . associated with ϕ has codimension at most ( f − r)(g − r). If F ∨ ⊗ G is globally generated and if ϕ ∈ Hom ( , ) is generic, then D (ϕ) either is empty or has codimension OPn−1 F G r ( f − r)(g − r). Furthermore, if Dr(ϕ) is not empty, then the singular locus of Dr(ϕ) is contained in Dr−1(ϕ). In particular, if ( f − r + 1)(g − r + 1) > n − 1, then Dr(ϕ) is non- singular (see, for example, [5]). n−1 If the codimension of D f −1(ϕ) in P is g − f + 1, then D f −1(ϕ) is locally Cohen- Macaulay (see, for example, [11, Theorem 14.4]), and the locally free resolution of its structure sheaf is obtained from the Eagon-Northcott complex of the dual of ϕ. More ∨ ∨ precisely, let c1 = c1(F ), let A = F (c1), and let B = G (c1). The Eagon-Northcott ∨ complex of ϕ [c1]: B → A ^f ^f (2.1) 0 → Kg− f → Kg− f −1 → · · · → K1 → B → A

i ∨ V f +i with Ki = S A ⊗ B is a locally free resolution of OD f −1(ϕ) (see, for example, [10] for a more detailed discussion about the Eagon-Northcott complex). V f Since A is isomorphic to OPn−1 , exact sequence (2.1) yields g− f X i (2.2) χ OD f −1(ϕ) = 1 − (−1) χ (Ki) . i=0 In particular, if the degeneracy locus is a curve, then its genus can be expressed as an alternating sum of Euler characteristics of Ki. 2.2. The dependency locus of global sections of a vector bundle. Let k, n, r ∈ N with n−1 1 ≤ k ≤ r ≤ n − 1, let E be a vector bundle of rank r on P , and let s1, s2,..., sk ∈ 0 H (E ). Deﬁne a morphism of sheaves ϕ : kOPn−1 → E by ϕ = (s1, s2,..., sk). We call the degeneracy locus Dk−1(ϕ) of ϕ the dependency locus of s1, s2,..., sk and denote it by 0 Z(s1 ∧ s2 ∧ · · · ∧ sk). If s ∈ H (E ), we conventionally call Z(s) the zero scheme of s. Suppose that the integers k, n, and r satisfy the following inequalities: ( 1 ≤ k ≤ r ≤ n − 1, (2.3) n − 1 < 2(r − k + 2). If E is globally generated and if the dependecy locus Z of k generic global sections of E is non-empty, then it is non-singular and has codimension r − k + 1 (see Subsection 2.1). i k−1+i Since S (kOPn−1 ) is isomorphic to the direct sum of k−1 copies of OPn−1 . using (2.2), we can formulate the Euler characteristic of OZ as follows:   Xr−k ! ^k+i  i k + i − 1  ∨ (2.4) χ (OZ) = 1 − (−1) χ  E  . k − 1   i=0 For each j ∈ {0, 1,..., n − 1}, the following equality follows from the Serre duality:  k+i   k+i   ^  ^   dim H j  E ∨ = dim Hn−1− j  E  (−n) ,      which implies  k+i   k+i   ^  ^   (2.5) χ  E ∨ = (−1)n−1χ  E  (−n) .      6 HIROTACHI ABO

Substituting (2.5) in (2.4) then yields    Xr−k ! ^k+i   n−1+i k + i − 1    (2.6) χ (OZ) = 1 − (−1) χ  E  (−n) . k − 1    i=0 In the remaining subsection, we focus on the case when r = n − 1 and k = 2. Since such r and k satisfy the required numeric conditions (2.3), if E is globally generated and if the dependency locus Z of two generic global sections s0 and s1 of E is non-empty, then it is non-singular of dimension 1. Moreover, if Z is a irreducible (and hence it is a curve), then, by (2.6), we can formulate the genus of Z as follows:    Xn−3 ^i+2   n−1+i    (2.7) g(Z) = 1 − χ(OZ) = (−1) (i + 1) χ  E  (−n) .    i=0 Proposition 2.1. Let E be a globally generated vector bundle of rank n − 1 on Pn−1. Suppose that the zero scheme of a generic global section of E is non-empty and that the dependency locus Z = Z(s0∧s1) of two generic global sections s0 and s1 of E is irreducible.

(1) If L is the pencil of s0 and s1, that is, n 0 1 o L = [λ0 s0 + λ1 s1] ∈ P H (E ) [λ0 : λ1] ∈ P , then, for each P ∈ Z, there exists a unique [s] ∈ L such that P ∈ Z(s). (2) The map Ψ : Z → L defined by Ψ(P) = [s] if and only if P ∈ Z(s) is a finite separable morphism of degree cn−1(E ). Furthermore, the degree of the ramification divisor R of Ψ is expressed as follows:     Xn−3 î+2    n−1+i    (2.8) deg(R) = 2cn− (E ) + 2  (−1) (i + 1) χ  E  (−n) − 2. 1     i=0 (3) Let P ∈ Z and let [s] = Ψ(P). The morphism Ψ is ramified at P if and only if Z(s) is singular at P.

Proof. (1) Since E is globally generated and since s0 and s1 are generic, we may assume Z(s0) ∩ Z(s1) = ∅ (see, for example, the proof of Theorem 2.6 in [9]). Let [s], [s0] ∈ L. We want to prove that Z(s) ∩ Z(s0) , ∅ if and only if s and s0 are linearly dependent. If s and s0 are linearly dependent, then Z(s) = Z(s0), and hence the intersection of Z(s) and Z(s0) is clearly non-empty. Conversely, suppose Z(s)∩Z(s0) , ∅. Let P ∈ Z(s)∩Z(s0). Assume for the contradiction that s and s0 are linearly independent. The pencil of s and s0 coincides with L. The zero scheme of any linear combination of s and s0 contains P, which contradicts the assumption that the intersection of Z(s0) and Z(s1) is non-empty. (2) The ﬁber of Ψ over a generic [s] ∈ L is non-singular of codimension n−1. Furthermore, it consists of cn−1(E ) distinct closed points. Therefore Ψ is a ﬁnite separable morphism of curves with deg(Ψ) = cn−1(E ). An application of the Hurwitz-Riemann formula (see, for example, Section 2 of Chap- ter IV in [17]) implies

(2.9) deg(R) = deg(Ψ)(2 − 2g(L)) + 2g(Z) − 2 = 2cn−1(E ) + 2g(Z) − 2. Thus, substituting (2.7) in (2.9) yields (2.8). (3) Let Q = [s]. The morphism Ψ : Z → L induces a local ring homomorphism Ψ # : OL,Q → OZ,P. If t is a local parameter for L at Q, then Ψ is unramiﬁed at P precisely when Ψ #(t) is a local parameter for Z at P. ON THE DISCRIMINANT LOCUS OF A RANK VECTOR BUNDLE 7

Considering Z(s) as a closed subscheme of Z, we obtain the following exact sequence:

0 → JZ,P → JZ(s),P → OZ,P → OZ(s),P → 0. # # The image Ψ (t) of t under Ψ can be thought of as a generator for the image of JZ(s),P. In particular, the zero scheme Z(s) is non-singular at P if and only if the image of JZ(s),P is a maximal ideal. The latter happens precisely when Ψ #(t) is a local parameter for Z at P. We therefore completed the proof.

3. THEDISCRIMINANTLOCUSOFAVECTORBUNDLE The zero scheme of a global section of a vector bundle is generally expected to be non- singular and its codimension is presumed to be the same as the rank of the vector bundle. There could, however, be global sections of the vector bundle whose zero schemes are singular or have lower codimensional primary components. In this section, we study the discriminant locus of a vector bundle, the locus of global sections of the vector bundle with such unexpected properties. The focus is set on the discriminant locus of a vector bundle of rank n − 1 on Pn−1. In most cases, the discriminant locus of a vector bundle of rank n − 1 on Pn−1 is an irreducible hypersurface. There do exist vector bundles whose discriminant loci are reducible or have higher codimensional components. The goal of this section is to prove that the discriminant locus of a very ample vector bundle of rank n − 1 on Pn−1 is irreducible. Furthermore, if the very ample vector bundle satisﬁes a certain positivity of the top Chern class, then one can show that the discriminant locus of the vector bundle is a hypersurface and that its degree can be expressed as a function of invariants of the vector bundle.

3.1. Definitions and examples. Let R = k[x0, x1,..., xn−1] be the homogeneous coordinate ring of Pn−1. If M is a graded R-module, then we write M˜ for the coherent sheaf associated with M. The set of graded R-module homomorphisms of degree zero from a graded R-module M to another graded R-module N is denoted by HomR(M, N). If E is a vector bundle of rank n − 1 on Pn−1, then we denote by E the finitely generated L 0 graded R-module d∈Z H (E (d)). Let q−1 M ϕ R(ai) −→ E −→ 0 i=0 be a minimal set of generators for E and let p−1 M ψ ∨ R(−bi) −→ E = HomR(E, R) −→ 0 i=0 be a minimal set of generators for E∨. The graded R-module E is projective, and hence it is reflexive. Therefore, one can compose the dual ψ∨ of ψ with ϕ and identify E with the image of ψ∨ ◦ ϕ:

∨ Lq−1 ψ ◦ϕ Lp−1 ··· / i=0 R(ai) / i=0 R(bi) / ··· 8

& E = (E∨)∨ 7

0 ' 0 8 HIROTACHI ABO

Lq−1 Lp−1 Give i=0 R(ai) and i=0 R(bi) free bases. Denote by A the matrix representation of ψ∨ ◦ ϕ with respect to these bases. The (i, j)-entry of A is a homogeneous polynomial of degree bi − a j, and hence E can be considered as the graded R-module generated by the columns A0,..., Aq−1 of A. Let s ∈ H0(E ) be arbitrary, and consider it as an element of E. There exists a homoge- Pq−1 neous polynomial fi of degree ai in R for each i ∈ {0, 1,..., q − 1} such that s = i=0 fi Ai. For each i ∈ {0, 1,..., p − 1}, let si be the ith entry of s. Write J(s) for the Jacobian matrix of s, that is,  ∂s /∂x ··· ∂s /∂x   0 0 0 n−1   . . .  J(s) =  . .. .  .   ∂sp−1/∂x0 ··· ∂sp−1/∂xn−1 Denote by I(s) the ideal of R generated by si and the (n − 1) × (n − 1) minors of J(s). Definition 3.1. We call a global section s of E singular if the subvariety of Pn−1 defined by I(s) is not empty. By the discriminant locus of E , we mean the subset ∆(E ) of P H0(E ) parameterizing singular global sections of E . More precisely, n o ∆(E ) = [s] ∈ P H0(E ) s is singular . Remark 3.2. The discriminant locus of a vector bundle of rank n − 1 on Pn−1 could be the empty set. For example, the zero scheme of every non-zero global section of the tangent n−1 bundle TPn−1 (−1) on P twisted by −1 consists of a single point, and thus it is non-singular. Therefore, ∆(TPn−1 (−1)) = ∅. Remark 3.3. Let s ∈ H0(E ). Then “s is singular” does not necessarily mean that “Z(s) is singular.” It could mean that Z(s) is non-singular, but has codimension less than n − 1. Below, we discuss such an example. 3 L 0 Example 3.4. Let TP3 be the tangent bundle on P , let T = d∈Z H (TP3 (d)), and let R = k[x0, x1, x2, x3]. The the graded R-module T is generated by the columns of the following matrix:  −x x 0 0   1 0   −x 0 x 0   2 0   0 −x x 0  A =  2 1  : 4R(1) −→ 6R(2).  −x x   3 0 0 0   0 −x 0 x   3 1  0 0 −x3 x2 0 Every element of H (TP3 ) can be written as a linear combination of the columns of A whose coefficients are linear forms in R. Note that the zero section of a generic global section of TP3 is non-singular of codimension 3. For example, if s = x1A1 − 2x2A2 − 3x3A3, then the ideal of Z(s) is

hx0 x1, −2x0 x2, −3x1 x2, −3x0 x3, −4x1 x3, −x2 x3i , and its decomposition into primary ideals is

hx0, x2, x3i ∩ hx1, x2, x3i ∩ hx0, x1, x3i ∩ hx0, x1, x2i. 0 Hence, Z(s) consists of four distinct points. Consequently, we have ∆(T) , P H (TP3 ). The discriminant of E is non-empty. For example, if s = −x1A0 + x2A2 + 2x3A3, then I(s) is generated by D 2 E x1, x0 x2 + x1 x2, x1 x2, 2x0 x3 + x1 x3, 2x1 x3, x2 x3 . ON THE DISCRIMINANT LOCUS OF A RANK VECTOR BUNDLE 9

The primary decomposition of I(s) is 2 I(s) = hx0, x1, x2i ∩ hx0, x1, x3, i ∩ hx1, x2, x3i. Hence, Z(s) is a zero-dimensional, but non-reduced. Therefore, s is singular. Let us consider s = x0A0 + x1A1. The ideal I of R generated by the entries of s is

hx0 x2, x0 x3, x1 x2, x1 x3i.

Its primary decomposition is hx0, x1i ∩ hx2, x3i. Therefore, Z(s) is the union of two skew lines in P3, and hence it is non-singular. On the other hand, since the codimension of Z(s) is two, the Jacobian matrix J(s)P of s evaluated at each closed point P of Z(s) has rank two. Hence, all the 3 × 3 minors of J(s)x must vanish. As a result, we have I = I(s), and hence s is singular. Definition 3.5. Let ` = dim H0(E ). Fix a basis for H0(E ). We call the linear combination s of the basis elements whose coefficients are independent variables u0, u1,..., u`−1 the general global section of E . Proposition 3.6. The discriminant locus of a vector bundle of rank n − 1 on Pn−1 is a subvariety of P H0(E ). Proof. Let s be the general global section of E . Then I(s) is a bi-homogeneous ideal of n−1 0 R[u0, u1,..., u`−1], and thus it defines a subvariety Γ(E ) of P × P H (E ). Let π2 be the n−1 0 0 projection of P × P H (E ) to P H (E ). Then, by definition, we have π2(Γ(E )) = ∆(E ), and hence ∆(E ) is a subvariety of P H0(E ). Remark 3.7. Let E be a vector bundle of rank n − 1 on Pn−1, let s be the general global section of E , and let I(s) be the ideal of k[x0, x1,..., xn−1, u0, u1,..., u`−1] associated with s. Then the ideal I(∆(E )) of ∆(E ) is obtained by computing the nth projective elimination ideal, that is, the intersection of k[u0, u1,..., u`−1] and the saturation I(s): ∞ hx0, x1,..., xn−1i of I(s) by hx0, x1,..., xn−1i. Example 3.8. In this example, we use Remark 3.7 to find the ideal of the discriminant locus of E = 2OP2 (1). Every non-zero global section of E can be expressed as a column of two linear forms in R = k[x0, x1, x2]. The zero scheme of a generic global section of E , therefore, consists of a single point. Furthermore, a non-zero global section of E is singular if and only if its linear entries differ only by a non-zero constant. The following calculation re-confirms the observation that ∆(E ) parameterizes the global sections of E whose zero scheme is a line in P2. The general global section of E can be written as follows: ! u x + u x + u x s = 0 0 1 1 2 2 . u3 x0 + u4 x1 + u5 x2 The Jacobian matrix of s, therefore, is ! u u u J(s) = 0 1 2 . u3 u4 u5 Hence, the ideal I(s) is generated by the entries of s and the following quadratic forms in u0, u1,..., u5: −u1u3 + u0u4, −u2u3 + u0u5, −u2u4 + u1u5. ∞ The third projective elimination ideal k[u0, u1,..., u5]∩(I(s): hx0, x1, x2i ) is generated by the quadratic forms as given above. In particular, the discriminant locus ∆(E ) of E is a non-singular three-fold of degree three in P5, which is known as the Segre embedding of P1 × P2. 10 HIROTACHI ABO

Example 3.9. Let R = k[x0, x1, x2] and let E be the graded R-module generated by the columns of the following matrix:  − 2 − − 2   x1 + x0 x2 x1 x2 x2 0   2   x0 x1 x0 x2 0 −x2  A =  2  : 4R → 4R(2).  −x0 0 x0 x2 x1 x2   2 2  0 −x0 −x0 x1 −x1 + x0 x2 The sheaf E associated with E is a vector bundle of rank two because all the 3 × 3 minors of A vanish and the radical of the ideal generated by 2 × 2 minors of A is the irrelevant ideal. If s is the general global section of E , then, computing the third projective elimination ideal of I(s), we obtain the ideal of ∆(E ) D 2 2 3 3 2 2E I(∆(E )) = u1u2 − 4u0u2 − 4u1u3 + 18u0u1u2u3 − 27u0u3 . Thus, the discriminant locus ∆(E ) is a surface of degree 4 in P3.

2 This surface is the tangential surface of the twisted cubic deﬁned by u2 − 3u1u3, u1u2 − 2 9u0u3, and u1 − 3u0u2. In the next example, we demonstrate that the discriminant locus of a vector bundle E could help determine all possible decompositions of the zero schemes of global sections of E into primary components.

Example 3.10. Let R = k[x0, x1, x2]. The matrix    0 0 0 −x2 x1   2 2  A =  −x0 x1 −x1 −x1 x2 x0 0   2 2  −x0 x2 −x1 x2 −x2 0 x0 induces a graded R-module homomorphism 5R → 2R(2) ⊕ R(1) whose image, denoted E, is a projective module of rank two. The zero scheme of a generic global section of the vector bundle E associated with E consists of two distinct points. In the same way as in Example 3.9, we can show that the ideal of k[u0, u1,..., u4] defining ∆(E ) is D 2 2 2 2 E D 2 E u0u4 + 4u1u3u4 + 4u2u4, u0u3 + 4u1u3 + 4u2u3u4 = u0 + 4u1u3 + 4u2u4 ∩ hu3, u4i Therefore, the discriminant locus ∆(E ) consists of the irreducible quadric, non-singular 2 hypersurface Q defined by u0 + 4u1u3 + 4u2u4 = 0 and the plane P defined by u3 = u4 = 0. In particular, it is reducible. ON THE DISCRIMINANT LOCUS OF A RANK VECTOR BUNDLE 11

Below, we describe a possible decomposition into primary components for the zero scheme of a non-zero global section s of E . Let I be the ideal generated by the entries of s. If [s] lies on P, then we get  0    s = (−u x − u x − u x )  x  . 0 0 1 1 2 2  1  x2 The intersection of the primary ideals appearing in the above primary decomposition of I 2 is hu0, u1, u2i. Thus if [s] ∈ P \ Q, then u0 , 0, and hence I has a decomposition into primary ideals I = h−u0 x0 − u1 x1 − u2 x2i ∩ hx1, x2i, which implies that Z(s) consists of a line and a point, which is not on the line. If [s] ∈ P ∩ Q, then u0 = u3 = u4 = 0. Therefore, 2 I = hu1 x1 + u2 x2i ∩ hx1, x2i is a primary decomposition of I. Therefore, Z(s) consists of a line and a 0-dimensional subscheme of length 2 whose support lies on the line. Next, suppose that [s] lies on Q \ P. Then    −u3 x2 + u4 x1   2 2  s =  −u0 x0 x1 − u1 x1 − u2 x1 x2 + u3 x0  .  2 2  −u0 x0 x2 − u2 x1 x2 − u2 x2 + u4 x0

Since [s] < P, we may assume that (u3, u4) , (0, 0). If u3 , 0, then x2 = (u4/u3)x1. By substituting it in the second and third components of s, one can obtain 2 2 2 4(−u0 x0 x1 − u1 x1 − u2 x1 x2 + u3 x0) = (u0 x1 − 2u3 x0) ; 2 2 2 4(−u0 x0 x2 − u2 x1 x2 − u2 x2 + u4 x0) = u4(u0 x1 − 2u3 x0) . 2 Thus, I = h−u3 x2 + u4 x1, (u0 x1 − 2u3 x0) i. Similarly, one can show that if u3 = 0, then 2 I = hx1, (u0 x2 − 2u4 x0) i. Consequently, if [s] lies on Q, but not on P, then Z(s) is a zero-dimensional subscheme of length two, which is supported at a single point.

3.2. Discriminant locus of a very ample vector bundle. In the section, we focus on the discriminant locus of a very ample vector bundle. We ﬁrst show that the very ampleness of a vector bundle implies the irreducibility of the discriminant locus of the vector bundle. Let E be a vector bundle of rank n − 1 on Pn−1. Suppose that H0(E ) has dimension `. 0 If {s0, s1,..., s`−1} is a basis for H (E ), then every global section s of E can be written as P`−1 s = i=0 ui si with u0, u1,..., u`−1 ∈ k. n−1 Since E is locally free, there exists an open covering {Uλ}λ∈Λ of P such that the restriction E |Uλ of E to Uλ is isomorphism to (n − 1)OUλ . Thus, for every λ ∈ Λ and for ∈ { ··· − } λ λ ··· λ ∈ 0 every j 0, 1, , ` 1 , there exist s j,1, s j,2, , s j,1 H (Uλ, OUλ ) such that  `−1 `−1 `−1  X X X  s| =  u sλ , u sλ . ··· , u sλ  Uλ  i j,1 i j,2 i j,n−1 j=0 j=0 j=0 n−1 For the sake of simplicity, write fi for the ith coordinate of s|Uλ . Let P ∈ P be arbitrary and let λ ∈ Λ such that P ∈ Uλ. The Jacobian matrix of s with respect to the regular system (y1,..., yn−1) of parameters for the stalk OPn−1,P of OPn−1 at P is ! ∂ fi JP = . ∂y j 1≤i, j≤n−1 12 HIROTACHI ABO

Thus the global section s of E is singular if and only if the following two conditions are satisﬁed:

(3.1) f1(P) = ··· = fn−1(P) = 0;

(3.2) rank JP = n − 1.

Let P(E ) be the projective bundle associated with E with projection π : P(E ) → Pn−1 and let OPn−1 (E )(1) be the tautological line bundle on P(E ). Since OP(E )(1) is deﬁned to be the inverse image π∗E of E by π, we have 0 0 ∗ 0 H (OP(E )(1)) = H (π E ) = H (E ).

If E is very ample, then OPn−1 (E )(1) is very ample, and hence the rational map Φ corresponding to the linear subspace |OPn−1 (E )(1)|, which sends a point Q of P(E ) to the hyperplane of global sections of OPn−1 (E )(1) vanishing at Q, is a closed embedding of P(E ) 0 ∨ 0 ∨ in the projective space PH (OPn−1 (E )(1)) associated with the dual H (OPn−1 (E )(1)) of 0 H (OPn−1 (E )(1)). −1 If λ ∈ Λ such that P ∈ Uλ, then we can identify π (Uλ) with the relative projective n−2 n−2 n−2 space P = U × P . Thus, there exists a point ξ = [ξ : ξ : ··· : ξ − ] of P such Uλ λ 1 2 n 1 −1 0 ∨ −1 n−1 that Q = (P, ξ), and we see that Φ|π (Uλ) : π (Uλ) → PH (OP (E )(1)) is given by

n−1 n−1 n−1  X X X  (P, ξ) 7−→  sλ (P)ξ : sλ (P)ξ : ··· : sλ (P)ξ  .  0,i i 1,i i `−1,i i i=1 i=1 i=1 In particular, if Y is the image of P(E ) under Φ, then the afﬁne cone over the projective tangent space TΦ(P,ξ)(Y) to Y at (P, ξ) can be expressed as the row space of the matrix obtained by stacking the following two matrices vertically:

Pn−1 λ Pn−1 λ  ∂ s (P)ξi s (P)ξi   i=1 0,i ··· i=1 `−1,i   sλ (P) ··· sλ (P)   ∂ξ1 ∂ξ1   0,1 `−1,1   . .   . .  J =  . .  =  . .  1  . .   . .   Pn−1 λ Pn−1 λ   λ λ   ∂ i=1 s0,i(P)ξi i=1 s`−1,i(P)ξi  s P ··· s P  ···  0,n−1( ) `−1,n−1( ) ∂ξn−1 ∂ξn−1 and  Pn−1 sλ P ∂ Pn−1 sλ P ξ  λ λ ∂ i=1 0,i( )ξi i=1 `−1,i( ) i  Pn−1 ∂s0,i Pn−1 ∂s`−1,i   ···   i (P)ξi ··· i (P)ξi   ∂y1 ∂y1   =1 ∂y1 =1 ∂y1   . .   . .  J =  . .  =  . .  . 2  . .   . .   Pn−1 λ Pn−1 λ   λ λ   ∂ s (P)ξn−1 ∂ s (P)ξ   ∂s ∂s   i=1 0, j i=1 `−1,i i   Pn−1 0,i Pn−1 `−1,i  ··· (P)ξi ··· (x)ξi ∂yn−1 ∂yn−1 i=1 ∂yn−1 i=1 ∂yn−1 Lemma 3.11. Let E be a very ample vector bundle of rank n − 1 on Pn−1, let s ∈ H0(E ) be 0 non-zero, and let L ∈ H (OP(E )(1)) correspond to s. The global section s is singular if and ∨ 0 only if the projective dual L to [L] ∈ PH (OP(E )(1)) contains a projective tangent space to Y.

Proof. Let Q ∈ P(E ), let P = π(Q), and let λ ∈ Λ with P ∈ Uλ. Identify Q with (P, ξ) ∈ −1 ∨ π (Uλ). The hyperplane L contains TΦ(P,ξ)(Y) if and only if the the column vector u of the coefﬁcients of L satisﬁes the following two conditions:

(3.3) J1u = 0;

(3.4) J2u = 0. ON THE DISCRIMINANT LOCUS OF A RANK VECTOR BUNDLE 13

It is immediate to see that the ith coordinate of J1 u equals fi(P), and hence condition (3.1) is equivalent to condition (3.3). The jth coordinate of J2 u is

n−1 λ n−1 λ X ∂s (P) X ∂s − (P) u 0,i ξ + ··· + u ` 1,i ξ 0 ∂y i `−1 ∂y i i=1 j i=1 j     `−1 ∂sλ `−1 ∂sλ X k,1  X k,n−1  =  uk (P) ξ1 + ··· +  uk (P) ξn−1  ∂y   ∂y  k=0 j k=0 j n−1 ! X ∂ f = i (P) ξ . ∂y i i=1 j

Therefore J2 u = 0 precisely when the homogeneous linear system in ξ1, ξ2, . . . , ξn−1 whose coefﬁcient matrix is JP has a non-trivial solution, from which the equivalence between conditions (3.2) and (3.4) follows.

Proposition 3.12. The discriminant locus ∆(E ) of a very ample vector bundle E of rank n − 1 on Pn−1 is irreducible.

n−1 0 Proof. For each i ∈ {1, 2}, let πi be the projection from P ×PH (E ) to its ith factor. Let s be the general global section of E and let Γ(E ) be the subvariety of Pn−1 × PH0(E ) defined by I(s). Since ∆(E ) is the image of Γ(E ) under π2, it suffices to show the irreducibility of Γ(E ). Let Q ∈ Pn−1 be arbitrary and let P ∈ P(E ) such that π(P) = Q. By Lemma 3.11, a hyperplane H contains tangent space to Y at Φ(Q) if and only if the corresponding global sec- S tion s of E is singular. Therefore, one can consider the fiber of π1 over Q as P∈π−1(Q) FP, where n 0 ∨ o FP = [H] ∈ PH (OP(E )(1)) H contains the tangent space to Y at Φ(P) .

0 ∨ Let ` = dim H (OP(E )(1)) . The hyperplanes containing a tangent space to Y form a linear subspace of dimension ` − 1 − dim Y − 1 = ` − 2n + 1. In particular, FP is irreducible. Furthermore, π−1(Q) irreducible because it is isomorphic to Pn−2. Therefore, −1 all the ﬁbers π1 (Q) are irreducible by the characterization of irreducibility (see, for example, Theorem 11.12 in [16]). They also have the same dimension, namely, ` − n − 1. The irreducibility of Γ(E ) therefore follows from the above-mentioned characterization of irreducibility.

The discriminant locus of a vector bundle of rank n − 1 on Pn−1 generally is expected to have codimension one. There are, however, vector bundles whose discriminant loci do not have codimension one. As Example 3.8 suggests, even the discriminant locus of a very ample vector bundle is not necessarily a hypersurface. In the following theorem, we give a sufﬁcient condition for the discriminant locus of a very ample vector bundle to be one codimensional. A formula for the degree of its discriminant hypersurface will also be presented.

Theorem 3.13. The discriminant locus ∆(E ) of a very ample vector bundle E of rank n−1 n−1 on P with cn−1(E ) ≥ 2 is an irreducible hypersurface of degree     Xn−3 ^i+2    n−1+i    (3.5) deg(∆(E )) = 2cn− (E ) + 2  (−1) (i + 1) χ  E  (−n) − 2. 1     i=0 14 HIROTACHI ABO

Proof. The irreducibility of ∆(E ) follows from Proposition 3.12. In order to prove that the codimension of ∆(E ) is one, we need to show that the intersection of ∆(E ) and the pencil of two generic sections of E n 0 1o L = [λ0 s0 + λ1 s1] ∈ P H (E ) [λ0 : λ1] ∈ P is non-empty. Since E is very ample, it is globally generated. Therefore, the dependency locus Z of s0 and s1 is non-singular and has dimension 1 (see Subsection 2.1). It is also irreducible due to the connectedness theorem of Fulton and Lazersfeld [12]. The morphism Ψ : Z → L as given in Subsection 2.2 is a finite, separable morphism of degree cn−1(E ), and we showed that the degree of its ramification divisor R can be written in terms of the Euler characteristics of exterior powers of E (see (2.8)). As the morphism Ψ : Z → L is finite, the global section λ0 s0 + λ1 s1 is singular if and only if the zero scheme of λ0 s0 +λ1 s1 is singular of dimension zero. In Proposition 2.1 (3), we proved that the latter condition is equivalent to the condition that [λs0 +µs1] is a branch point of Ψ. Therefore, the discriminant locus ∆(E ) is of codimension one if and only if the degree of the ramification divisor R of Ψ is positive.

In Subsection 2.2, we expressed the degree of R as a function of the top Chern class cn−1(E ) of E and the genus g(Z) of Z:

deg(R) = 2cn−1(E ) + 2g(Z) − 2 ≥ 2cn−1(E ) − 2, from which it follows that deg(R) > 0 if cn−1(E ) ≥ 2. We thus proved that ∆(E ) is a hypersurface. The degree of ∆(E ) equals the number of intersection points of ∆(E ) and L. Since Q ∈ L is a branch point of Ψ if and only if there exists at least one ramification point of Φ over Q, the degree of ∆(E ) is bounded above by deg(R). We thus obtain the inequality     Xn−3 î+2    n−1+i    deg(∆(E )) ≤ 2cn− (E ) + 2  (−1) (i + 1) χ  E  (−n) − 2 1     i=0 by (2.8). This inequality is an equality if and only if the number of branch points of Ψ is P equal to deg(R) = P∈Z(eP − 1), where eP is the ramification index of Ψ at P. The latter occurs precisely when the zero scheme of the global section sQ of E corresponding to each branch point Q ∈ L has a single singularity, and the singularity is an ordinary double point. Therefore, it is sufficient to show the existence of a pencil L of global sections of E such that

(1) the zero schemes Z(sQ) of global sections sQ in L are non-singular for all Q in some open subset U of L; (2) if Q < U, then Z(sQ) has an ordinary double point as an only singularity. ON THE DISCRIMINANT LOCUS OF A RANK VECTOR BUNDLE 15

Recall that a global section of E is singular if and only if the corresponding hyperplane contains the tangent space to Y. The existence of a pencil of global sections of E satisfying (1) and (2), therefore, follows from the existence of a Lefschetz pencil for Y.

Example 3.14. If a0, a1,..., an−2 be positive integers, at least one of which is greater than Ln−2 or equal to two, then the top Chern class of E = i=0 OPn−1 (ai) satisﬁes the following inequality: Yn−2 cn−1(E ) = ai ≥ 2. i=0 Since E is a direct sum of very ample line bundles (see [29, Proposition 1.2]), it is also very ample. Therefore, by Theorem 3.13, the discriminant locus ∆(E ) is an irreducible hypersurface. The degree of ∆(E ) was presented by L. Busé and J. Jounaolou in [7] as a polynomial of a0,..., an−1. In this example, we use (3.5) to reformulate deg ∆(E ). For a ﬁxed i ∈ {0, 1,..., n − 3}, let

Λi+2 = {(aλ0 , aλ1 ,..., aλi+1 ) | 0 ≤ λ0 < λ1 < ··· < λi+1 ≤ n − 2}. Pi+1 If a = (aλ0 , aλ1 ,..., aλi+1 ) ∈ Λi+2, then we write |a| for j=0 aλ j . The (i + 2)nd exterior power of E is the direct sum of invertible sheaves: ^i+2 M E = OPn−1 (|a|).

a∈Λi+2 Since |a| − n > −n, we have ! |a| − 1 χ (O n−1 (|a| − n)) = . P n − 1

Vi+2 The Euler characteristic of ( E )(−n) is  i+2     ! ^    M  X |a| − 1 χ  E  (−n) = χ  OPn−1 (|a| − n) = .      n − 1 a∈Λi+2 a∈Λi+2 Therefore, the degree of ∆(E ) is   Yn−2 Xn−3 X !  n−1+i |a| − 1  deg ∆(E ) = 2 ai + 2  (−1) (i + 1)  − 2.  n − 1  i=0 i=0 a∈Λi+2 If n = 3, then ! a0 + a1 − 1 deg ∆(E ) = 2a a + 2 − 2 = a2 + 4a a + a2 − 3a − 3a . 0 1 2 0 0 1 1 0 1

In particular, if a0 = 2 and a1 = 1, then deg ∆(E ) = 4. The rest of this example is devoted to finding the polynomial defining ∆(E ) for this case. If s ∈ H0(E ) is non-zero, then the zero scheme of s is defined by the ideal generated by a quadratic form Q and a linear form L:

2 2 2 Q(x0, x1, x2) = u0 x0 + u1 x0 x1 + u2 x0 x2 + u3 x1 + u4 x1 x2 + u5 x2 L(x0, x1, x2) = v0 x1 + v1 x1 + v2 x2. 16 HIROTACHI ABO

Since L(x0, x1, x2) is non-zero, at least one of v0, v1, and v2 is non-zero. Without loss of 2 2 generality, we may assume v0 = 1. The coefﬁcients of x1, x1 x2, and x2 in the bivariate quadratic form q(x1, x2) obtained from Q(x0, x1, x2) by substituting x0 = −v1 x1 − v2 x2 are 2 u0v1 − u1v1 + u2, 2u0v1v2 − u1v2 − u2v1 + u4, 2 u0v2 − u2v2 + u5 respectively. The polynomial obtained from the discriminant of q(x1, x2) by homogenizing using v0 2 2 2 2 2 u1v2 − 2u1u2v1v2 + u2v1 − 4u0u3v2 + 4u2u3v0v2 + 4u0u4v1v2 2 2 2 2 −2u1u4v0v2 − 2u2u4v0v1 + u4v0 − 4u0u5v1 + 4u1u5v0v1 − 4u3u5v0, therefore, deﬁnes ∆(E ). Example 3.15. In this example, we give a closed formula for the degree of the discriminant locus of a very ample vector bundle E of rank two on P2 whose second Chern class is greater than or equal to 2. If E is a direct sum OP2 (a0)⊕OP2 (a1) of two line bundles, then a0, a1 ≥ 1 and a0 +a1 ≥ 2. Therefore, the degree of ∆(E ) has already been calculated in Example 3.14, and hence we may assume that E is indecomposable. If E is indecomposable, then it has a minimal free resolution of length one: (3.6) 0 → F → G → E → 0, where Mk−1 Mk+1 F = OP2 (ai); G = OP2 (bi). i=0 i=0 By (3.6) and the Whitney sum formula, we have ct(G ) = ct(F )ct(E ). Thus, the Chern classes of E can be expressed as follows: Xk+1 Xk−1 c1(E ) = c1(G ) − c1(F ) = bi − ai i=0 i=0 c2(E ) = c1(F ) (c1(F ) − c1(G )) − c2(E ) + c2(G ) k−1  k−1 k+1  k−2 k−1 k k+1 X X X  X X X X = a  a − b  − a a + b b . i  i i i j i j i=0 i=0 i=0 i=0 j=i+1 i=0 j=i+1 Therefore, by Theorem 3.13, we obtain  2   ^   deg ∆(E ) = 2c (E ) + 2χ  E  (−3) − 2 2   

= 2c2(E ) + 2χ (OP2 (c1(E ) − 3)) − 2  k−1  k−1 k+1  k−2 k−1 k k+1  X X X  X X X X  = 2  a  a − b  − a a + b b   i  i i i j i j i=0 i=0 i=0 i=0 j=i+1 i=0 j=i+1 − ! Pk+1 b − Pk 1 a − 1 +2 i=0 i i=0 i − 2. 2 from which one can calculate the degree of the discriminant hypersurface of a very ample 2 vector bundle E of rank two on P with c2(E ) ≥ 2 as soon as the shape of the minimal free resolution of E is determined. ON THE DISCRIMINANT LOCUS OF A RANK VECTOR BUNDLE 17

4. THE DISCRIMINANT LOCUS OF A CERTAIN VECTOR BUNDLE In Example 3.14, we discussed the discriminant loci of directs sums of line bundles, which are the simplest examples of vector bundles. In this section, we study the discriminant loci of the next simplest examples of vector bundles, namely, vector bundles which admit free resolutions of length one. Such vector bundles form an important class of vector bundles, which include suitable twists of the so-called “Schwarzenberger bundles,” the vector bundles of rank n − 1 on projective (n − 1)-space which were ﬁrst discovered by R. L. E. Schwarzenberger [27].

Proposition 4.1. Let k ∈ N and let a0, a1,..., ak−1, b0, b1,..., bn+k−2 ∈ Z such that ( ak−1 ≤ ak−2 ≤ · · · ≤ a0 < bn+k−2 ≤ bn+k−3 · · · ≤ b0, (4.1) bn+k−2 = 1. The set of sheaf morphisms Mk−1 nM+k−2 OPn−1 (ai) −→ OPn−1 (bi) i=0 i=0 whose cokernel is a vector bundle of rank n − 1 is non-empty. Lk−1 Ln+k−2 Proof. Deﬁne a sheaf morphism ϕ from i=0 OPn−1 (ai) to i=0 OPn−1 (bi) by  xb0−a0   0   . .   . ..    ϕ =  xbn−1−a0 xbk−1−ak−1   n−1 0   . .   .. .     bn+k−2−ak−1  xn−1 The radical of the ideal generated by the k × k minors of ϕ is the irrelevant ideal of R, and hence the cokernel of ϕ is a vector bundle of rank n − 1. Deﬁnition 4.2. A vector bundle E of rank n − 1 on Pn−1 is called a Schwarzenberger bundle if there exist integers a0,..., ak−1, b0,..., bn+k−2 satisfying (4.1) and a morphism of OPn−1 -modules  k−1 n+k−2  M M   n−1 n−1  ϕ ∈ Hom n−1  OP (ai), OP (bi) OP   i=0 i=0 whose cokernel is E . Proposition 4.3. Any Schwarzenberger bundle is very ample.

n−1 Proof. Let E be a Schwarzenberger bundle on P . There exist integers a0, a1, ··· , ak−1, b0, b1,..., bn+k−2 that satisfy (4.1) and a sheaf morphism Mk−1 nM+k−2 ϕ : OPn−1 (ai) −→ OPn−1 (bi) i=0 i=0 whose cokernel coincides with E . The codomain of ϕ is very ample because it is a direct sum of very ample line bundles (see [29, Proposition 1.2 (2)]). As E is a quotient of a very ample vector bundle, it is also very ample (see [29, Proposition 1.2 (1)]. Proposition 4.4. The Chern classes of a Schwarzenberger bundle E on Pn−1 are all positive. Furthermore, the top Chern class of E is greater than or equal to two. 18 HIROTACHI ABO

Proof. Proposition 2.2 in [4] (also see [13, Proposition 10 (2)]) implies that the top Chern class of an ample vector bundle on a projective variety is positive if the rank of the vector bundle is greater than or equal to the dimension of the projective variety. Furthermore, the restriction of a very ample vector bundle to a hyperplane is again very ample (see [29, Proposition 1.2 (5)]). The Chern classes of E , therefore, are all positive. Assumption (4.1) implies that E (−1) is ﬁnitely generated, and hence its restriction to any linear subspace is also ﬁnitely generated. Thus, it follows from [13, Proposition 10 (1)] that ci(E (−1)) ≥ 0 for all i ∈ {1, 2,..., n − 1}. By the Chern class identity n−1 n−1 X n−1−i (4.2) cn−1(E (a)) = a + a ci(E ), i=1 (see, for example, [11, Example 2.2.2]), we obtain Xn−1 cn−1(E ) = 1 + ci(E (−1)). i=1 By the Whitney sum formula, one gets

ct(G (−1)) = ct(F (−1))ct(E (−1)). In particular, we have nX+k−2 Xk−1 c1(E (−1)) = c1(G (−1)) − c1(F (−1)) = (bi − 1) − (a j − 1). i=0 j=0

By assumption (4.1), we have bi − 1 ≥ 0 for each i ∈ {0, 1,..., n + k − 2} and a j − 1 < 0 for each j ∈ {0, 1,..., k − 1}. Hence, one obtains c1(E (−1)) > 0, which proves the desired Pn−1 inequality cn−1(E ) = 1 + i=1 ci(E (−1)) > 1. n−1 Corollary 4.5. If E is a Schwarzenberger bundle on P , then cn−1(E (a)) ≥ 2 for all non-negative integers a.

Proof. The corollary is an immediate consequence of (4.2) and Proposition 4.4. Theorem 4.6. If E is a Schwarzenberger bundle on Pn−1 and if a is non-negative integer, then the discriminant locus ∆(E (a)) of E (a) is an irreducible hyperplane of P H0(E ) with degree     Xn−3 ^i+2    n−1+i    (4.3) deg(∆(E (a))) = 2cn− (E (a)) + 2  (−1) (i + 1) χ  E (a) (−n) − 2. 1     i=0 Proof. This theorem immediately follows from Theorem 3.13 and Corollary 4.5. Remark 4.7. In Theorem 4.6, we showed that the degree of the discriminant locus of a twisted Schwarzenberger bundle can be expressed in terms of the top Chern class and the Euler characteristics of exterior powers of the twisted Schwarzenberger bundle. The top Chern class of the twisted Schwarzenberger bundle can explicitly be written by using the Whitney sum formula and (4.2) as a function of the twist a, the Chern classes of F , and the Chern classes of G . The Euler characteristic of an exterior power of the twisted Schwarzenberger bundle can be calculated using a general result such as the Riemann-Roch theorem. In this remark, we alternatively use a locally free resolution of the exterior power of the twisted Schwarzenberger bundle to compute its Euler characteristic. ON THE DISCRIMINANT LOCUS OF A RANK VECTOR BUNDLE 19

Let a0, a1,..., ak−1, b0, b1,..., bn+k−2 ∈ Z satisfy (4.1), let a be a non-negative integer with a < bn+k−2, and let E be the Schwarzenberger bundle obtained via the following short exact sequence: (4.4) 0 → F → G → E → 0, Lk−1 Ln+k−2 where F = i=0 OPn−1 (ai) and G = i=0 OPn−1 (bi). For each i ∈ {0, 1,..., n − 3}, the (i + 2)nd exterior power of E (a) has a locally free resolution of type î+1 î+2 î+2 (4.5) 0 → S i+2A → S i+1A ⊗ B → · · · → A ⊗ B → B → E (a) → 0, where A = F (a) and B = G (a). We therefore have  i+2   i+2  i− j+2   ^   X  ^   χ  E (a) (−n) = (−1) jχ S jA ⊗ B (−n) .       j=0 Vi− j+2 As a result, it suffices to express the Euler characteristic of S jA ⊗ B twisted by −n as a function of a0, a1,..., ak−1, b0, b1,..., bn+k−2. For each j ∈ {0, 1,..., i + 2}, let

Λ = {(aλ0 , aλ1 ,..., aλ j−1 ) | 0 ≤ λ0, λ1, . . . , λ j−1 ≤ k − 1} and let

Ξ = {(bξ0 , bξ1 ,..., bξi− j+2 ) | 0 ≤ ξ0 < ξ1 < ··· < ξi− j+2 ≤ n + k − 2}.

If a = (aλ0 , aλ1 ,..., aλ j−1 ) ∈ Λ and b = (bξ0 , bξ1 ,..., bξi− j+2 ) ∈ Ξ, then we write |a| and |b| for aλ0 + aλ1 + ··· + aλ j−1 and bξ0 + bξ1 + ··· + bξi− j+2 respectively. Vi− j+2 The vector bundles S jA and B are direct sums of line bundles: i− j+2 j M ^ M S A = OPn−1 ( ja + |a|), B = OPn−1 ((i − j + 2)a + |b|). a∈Λ b∈Ξ We thus have i− j+2 j ^ M M S A ⊗ B = OPn−1 ((i + 2)a + |a| + |b|), a∈Λ b∈Ξ and hence we obtain     i−^j+2   X X  j   χ S A ⊗ B (−n) = χ (O n−1 ((i + 2)a + |a| + |b| − n)) .    P a∈Λ b∈Ξ Combining this with  (i+2)a+|a|+|b|−1 if (i + 2)a + |a| + |b| ≥ n  n−1  n−1n−1−(i+2)a−|a|−|b| χ (OPn−1 ((i + 2)a + |a| + |b| − n)) = (−1) if (i + 2)a + |a| + |b| ≤ 0  n−1  0 otherwise,

Vi− j+2 one can express the Euler characteristic of S jA ⊗ B (−n) as a polynomial in a0, a1,..., ak−1, b0, b1,..., bn+k−2, and a. Remark 4.7 indicates that the formula for the degree of the discriminant hyeprsurface of a Schwarzenberger bundle, in general, requires a large number of binomial coefﬁcients to express. It is, therefore, inconvenient to write all this down in most cases. There are, however, instances, where the degree formula has a very simple form. To end this section, we discuss two such examples. 20 HIROTACHI ABO

Example 4.8. Let k ≥ 1, let a0 = a1 = ··· = ak−1 = 0, and let b0 = b1 = ··· = bn+k−2 = 1. Assume that a = 0. If E is a Schwarzenberger bundle obtained via short exact sequence (4.4), then, by the Whitney sum formula, we have ! n + k − 1 cp(E ) = cp(G ) = i for each p ∈ {1, 2,... n − 1}. For each j ∈ {0, 1,..., i − j + 2}, we have |a| = 0 and |b| = i − j + 2. Therefore,    i−^j+2   j  S F ⊗ G  (−n) = |Λ| |Ξ| O n−1 (i − j − n + 2)   P ! ! k + j − 1 n + k − 1 = O n−1 (i − j − n + 2). j i − j + 2 P

Note that ( 0 if j ∈ {0, 1,..., i + 1} χ( n−1 (i − j − n + 2)) = OP (−1)n−1 if j = i + 2.

We thus have  i+2   ! ^   k + i + 1 χ  E  (−n) = (−1)n+i+1 .    i + 2

Therefore, it follows from Theorem 4.6 that     Xn−3 ^i+2    n−1+i    deg ∆(E ) = 2cn− (E ) + 2  (−1) (i + 1) χ  E  (−n) − 2 1     i=0 ! n−3 ! n + k − 1 X k + i + 1 = 2 + 2 (i + 1) − 2. n − 1 i + 2 i=0

One can use several well known binomial coefﬁcient identities to simplify the second term of the above formula as follows:

n−3 ! n−3 ! n−3 ! X k + i + 1 X k + i + 1 X k + i + 1 2 (i + 1) = 2 (i + 2) − 2 i + 2 i + 2 i + 2 i=0 i=0 i=0 n−3 ! n−1 ! ! ! X k + i + 1 X k − 1 + j k − 1 k  = 2 k − 2  − −  i + 1  j 0 1  i=0 j=0 n−2 ! ! " ! # X k + j k  k + n − 1 = 2k  −  − 2 − (k + 1)  j 0  n − 1 j=0 " ! # " ! # k + n − 1 k + n − 1 = 2k − 1 − 2 − (k + 1) n − 2 n − 1 " ! ! # k + n − 1 k + n − 1 = 2 k − + 1 . n − 2 n − 1 ON THE DISCRIMINANT LOCUS OF A RANK VECTOR BUNDLE 21

Therefore, we obtain ! " ! ! # n + k − 1 n + k − 1 n + k − 1 deg ∆(E ) = 2 + 2 k − + 1 − 2 n − 1 n − 2 n − 1 ! n + k − 1 = 2k n − 2

= 2kcn−2(E ). If k = 1, then E is the tangent bundle on Pn−1. The discriminant hypersurface of the tangent bundle on Pn−1 was studied in [1], and its degree was proved to be twice the (n − 2)nd Chern class of the tangent bundle. Formula (4.6) can, therefore, be considered as a generalization of their result.

Example 4.9. Let `, d ∈ N with d ≥ 2, let a0 = −` + 1, let b0 = ··· = bn = 1, and let a = d − 2. By using the Whitney sum formula, one can show that the Schwarzenberger bundle obtained via short exact sequence (4.4) has the the pth Chern class ! p n cp(E ) = ` p for each p ∈ {1, 2,..., n − 1}. In particular, it follows from the Chern class identity (4.2) that n−1 X n−i−1 cn−1(E (d − 2)) = (d − 1)` . i=0 Let i ∈ {0, 1,..., n − 3}. For each j ∈ {0, 1,..., i + 2}, we have |a| = j(−` + 1) and |b| = i − j + 2. Therefore, i− j+2 ! j ^ n S A ⊗ B = O n−1 ((i + 2)(d − 1) − j`). i − j + 2 P We thus have    î+2   Xi+2 ! !    j n χ  E (d − 2) (−n) = (−1) χ OPn−1 ((i + 2)(d − 1) − j` − n) .    i − j + 2 j=0 In particular, if (i + 2)(d − 1) − j` − n ≥ 0 for each i, j, then ! (i + 2)(d − 1) − j` − 1 χ (O n−1 ((i + 2)(d − 1) − j` − n)) = , P n − 1 and hence  i+2   i+2 ! ! ^   X n (i + 2)(d − 1) − j` − 1 χ  E (d − 2) (−n) = (−1) j .    i − j + 2 n − 1 j=0 Therefore, the degree of the discriminant hypersurface of E (d − 2) is n−2 n−3 i+2 ! ! X − − X − X n (i + 2)(d − 1) − j` − 1  2 (d − 1)`n i 1 + 2  (−1)n i+1(i + 1) (−1) j  − 2.  i − j + 2 n − 1  i=0 i=0 j=0 If ` = 1, then the formula as given above can significantly be simplified: (4.6) deg ∆(E (d − 2)) = n(n − 1)(d − 1)n−1. This equality was proved by Mauel Kauers. His proof is presented in Corollary 4.2 of [2]. Note that if d = 2, then E is the tangent bundle on Pn−1. Therefore, formula (4.6) can be viewed as another generalization of Theorem 6.1 of [1]. 22 HIROTACHI ABO

5. EIGENVECTORSOFTENSORSANDEIGENDISCRIMINANTS The concept of eigenvectors of tensors, an extension of the concept of eigenvectoes of linear operators, were introduced by L. H. Lim [24] and L. Qi [26] independently in 2005. The conditions for a non-zero vector to be an eigenvector of a tensor can be described in terms of polynomials. The ideal generated by these polynomials defines the “eigenscheme of tensors,” which is a scheme whose underlying variety is formed by its eigenvectors. In general, the eigenscheme of a tensor is reduced of dimension zero. The purpose of this section is to study the locus of tensors whose eigenschemes are not reduced of dimension zero. We call such a locus the “eigendiscriminant locus.” The key observation is that the eigenscheme of a tensor is the zero scheme of a global section of a certain twist of a Schwarzenberger bundle. This observation allows one to relate the eigendiscriminant locus to the discriminant locus of the twisted Schwarzenberger bundle and to derive properties of the eigendiscrimiant locus from those of the discriminant locus of the twisted Schwarzenberger bundle. 5.1. Definitions and examples. Let V be an n-dimensional vector space over k with basis n−1 {e0,..., en−1}. Denote by [v] the point of P = PV corresponding to v. Given a positive integer d, we write V⊗d for the tensor product of k copies of V. Let X ⊗d A = ai1···id ei1 ⊗ · · · ⊗ eid ∈ V 0≤i1,...,id≤n−1 be an order k tensor. Define n−1 n−1 n−1  X X X  (5.1) Axd−1 =  ··· a x ··· x  e ∈ V ⊗ S d−1(V∗),  j i2···ik i2 ik  j j=0 i2=0 ik=0 and let Avd−1 be the vector of V obtained from Axd−1 by evaluating at v ∈ V.

Pn−1 ` Pn−1 ` Definition 5.1. Let v = i=0 vi ei ∈ V. For a fixed ` ∈ {1, 2,..., d − 1}, let v = i=0 vi ei. A vector v ∈ V is called an `th eigenvector of A if Avd−1 = λv` for some λ ∈ k (or equivalently if Avd−1 ∧ v` = 0). Remark 5.2. Let A ∈ V⊗d be an order d tensor. (1) A non-zero vector v ∈ V is an eigenvector of A if and only if the coefficients of d−1 ` ei ∧ e j in Av ∧ v vanish. (2) Since each coordinate function of Axd−1 is homogeneous, a non-zero vector v is an eigenvector of A if and only if every non-zero multiple of v is an eigenvector of A. Facts (1) and (2) lead us to the following definition. Definition 5.3. Let A ∈ V⊗d and let v ∈ V be an eigenvector of A. We call [v] ∈ Pn−1 an `th eigenpoint of A. Pn−1 Let R = k[x0,..., xn−1] and let x = i=0 xi ei. The set of eigenpoints of A, which is called the `th eigenvariety of A, is defined by the ideal generated by the coefficients of d−1 ` n−1 ei ∧ e j in Ax ∧ x . The closed subscheme of P defined by the ideal is called the `th eigenscheme of A and denoted by ZA.

Example 5.4. Let V be a three-dimensional vector space, let {e0, e1, e2} be a basis for V, P2 and let x = i=0 xi ei. ⊗3 2 2 (1) If A = e0 ⊗ e0 ⊗ e0 ∈ V , then Ax = x0 e0, and hence the coefﬁcients of e0 ∧ e1, 2 2 2 e0 ∧ e2, and e1 ∧ e2 in Ax ∧ x are x0 x1, x0 x2, and 0 respectively. Thus the ﬁrst ON THE DISCRIMINANT LOCUS OF A RANK VECTOR BUNDLE 23

2 2 eigenscheme ZA of A is deﬁned by the ideal I = hx0 x2, x0 x1i. The decomposition 2 of I into primary ideals is hx0i ∩ hx1, x2i. Therefore, the eigenscheme ZA of A consists of a double line L and a point P which does not lie on the double line.

2 2 2 (2) If A = e0 ⊗ e0 ⊗ e0 + e1 ⊗ e1 ⊗ e1, then Ax = x0 e0 + x1 e1. Thus, the ideal of the 2 2 2 2 ﬁrst eigenscheme ZA of A is generated by x0 x1 − x0 x1, x0 x2, and x1 x2. The primary decomposition of this ideal is

2 2 hx0, x2i ∩ hx1, x2i ∩ hx0 − x1, x2i ∩ hx0, x1i.

Thus, the eigenscheme ZA of A consists of three points P1, P2, and P3 on the line L deﬁned by x2 = 0 and a quadruple point P0 which does not lie on L.

L P1 P3 P2

P2 2 P2 (3) If A = i=0 ei ⊗ ei ⊗ ei, then Ax = i=0 xi ei, and hence the eigenscheme ZA of A is deﬁned by the ideal generated by

x0 x1(x0 − x1), x0 x2(x0 − x2), x1 x2(x1 − x2). Note that the cubics as given above vanish if and only if

P0 : x0 = x1 = 0, P1 : x0 = x2 = 0, P2 : x1 = x2 = 0, P3 : x0 = x1 − x2 = 0, P4 : x1 = x0 − x2 = 0, P5 : x2 = x0 − x1 = 0, P6 : x0 − x1 = x0 − x2 = 0.

P5 P6 P4

P1 P3 P2

Therefore, the eigenpoints of A are [e0], [e1], [e2], [e0 + e1], [e0 + e2], [e1 + e2], and [e0 + e1 + e2]. In particular, the eigenscheme ZA associated with A is reduced of dimension 0 and consists of seven eigenpoints.

The above example indicates that the eigenscheme of a tensor could be non-reduced. It could also have a positive dimensional primary component. However, if the eigenscieme is non-reduced of dimension 0, then there exists a formula for the number of distinct closed ⊗d points of the eigenscheme. More precisely, if A ∈ V such that ZA is non-reduced of 24 HIROTACHI ABO dimension zero, then ZA consists of n−1 ( n−1 X i n−1−i n` if d = ` + 1 (5.2) ρ(n, d, `) = (d − 1) ` = n n (d−1) −` otherwise. i=0 d−1−` distinct closed points in Pn−1 (see [8, 25, 2] for the proof of (5.2)). The tensors in V⊗d whose eigenschemes are non-reduced of dimension zero form an open set of the projective space PV⊗d associated with V⊗d. In this section, we study the complement of such an open set. ⊗d Deﬁnition 5.5. An order d tensor A ∈ V is called regular if the `th eigenscheme ZA of A is reduced of codimension n − 1. We call the subset of PV⊗d n ⊗d o ∆n,d,` = [A] ∈ PV ZA is non-regular the `th eigendiscriminant locus. We call the ﬁrst eigendiscriminant locus just the eigendiscriminant locus and denote it by ∆n,d instead of ∆n,d,1. 5.2. Eigendiscriminant and the discriminant of a Schwarzenberger bundle. Let K• α α be the co-chain complex {K , ∂ }0≤α≤n−1 of sheaves α α ^ K = V ⊗ OPn−1 ((α − 1)` + 1),

α Vα Vα+1 where the coboundary operators ∂ : V ⊗ OPn−1 ((α − 1)` + 1) → V ⊗ OPn−1 (α` + 1) are given by n α X i ` ∂ (ei1 ∧ · · · ∧ ei` ) = (−1) xi ei ∧ ei1 ∧ · · · ∧ ei` . i=0 For the sake of simplicity, we write ∂ for ∂1. Let E be the cokernel of ∂: ∂ (5.3) 0 −→ OPn−1 (−` + 1) −→ V ⊗ OPn−1 (1) −→ E −→ 0. By deﬁnition, the sheaf E is a Schwarzenberger bundle. Lemma 5.6. For each order d tensor A ∈ V⊗d, there exists, up to a non-zero scalar multiple, a unique global section s of E (d − 2) whose zero scheme is the `th eigenscheme of A. Moreover, the order d tensor A is non-regular if and only if s is singular.

Proof. Tensoring (5.3) with OPn−1 (d − 2) and taking cohomology, we obtain the following exact sequence:

H0 d− 0 0 (∂[ 2]) 0 0 −−−−−−→ H (OPn−1 (d − ` − 1)) −−−−−−→ H (V ⊗ OPn−1 (d − 1)) −−−−−−−−→ H (E (d − 2)). By Theorem 5.1 of Chapter III in [17], we have  0 d−`−1 ∗  H (OPn−1 (d − ` − 1)) = S (V );  1  H (OPn−1 (d − ` − 1)) = 0;  0 d−1 ∗  H (V ⊗ OPn−1 (d − 1)) = V ⊗ S (V ). Moreover, the following set equality holds: n − ⊗ o im H0(∂[d − 2]) = Axd 1 ∧ x`, A ∈ V d . Therefore, we have n − ⊗ o H0(E (d − 2)) = Axd 1 ∧ x` A ∈ V d . This proves the ﬁrst assertion of the lemma. ON THE DISCRIMINANT LOCUS OF A RANK VECTOR BUNDLE 25

The second assertion of the lemma immediately follows from the deﬁnitions of the regularity of a tensor and the singularness of a global section of E (d − 2).

Theorem 5.7. The eigendiscriminant locus ∆n,d,` is an irreducible hypersurface of degree

(5.4) deg(∆n,d,`) = 2γ(n, d, `) + 2ρ(n, d, `) − 2, where n−1  j ! ! X − X n j(d−1) − k` − 1  γ(n, d, `) = (−1)n 1+ j( j−1)  (−1)k   j−k n − 1  j=2 k=0 and Xn−1 ρ(n, d, `) = (d − 1)i`n−1−i. i=0 Proof. Let π˜ : V⊗d → H0(E (d − 2)) be the onto linear transformation deﬁned by π˜(A) = Axd−1 ∧ x` for each A ∈ V⊗d, let K be the kernel of π˜, and let π : PV⊗d \ PK → PH0(E (d − 2)) be the linear projection induced by π˜.

By Lemma 5.6, the restriction π|∆n,d,` of π to ∆n,d,` is a morphism onto ∆(E (d − 2)). Let ⊗d [s] ∈ ∆(E (d − 2)) be arbitrary, and fix an A ∈ V such that π|∆n,d ([A]) = [s]. Then the fiber of π|∆n,d,` over [s] is −1 n ⊗d o (π|∆n,d,` ) ([s]) = [A + B] ∈ PV B ∈ K , and hence it is irreducible and has dimension dim V⊗d − dim H0(E (d − 2)). Therefore, it follows from Theorem 4.6 that ∆n,d,` is irreducible and has dimension ⊗d 0 dim ∆n,d,` = dim ∆(E (d − 2)) + dim V − dim H (E (d − 2)) = dim PH0(E (d − 2)) − 1 + dim V⊗d − dim H0(E (d − 2)) = dim H0(E (d − 2)) − 2 + dim V⊗d − dim H0(E (d − 2)) = dim V⊗d − 2 = dim PV⊗d − 1. Thus, it only remains to show that equality (5.4) holds. To do so, it suffices to show that ⊗d the number of the intersection of ∆n,d,` and a generic line in PV equals the right side of (5.4). Let A, B ∈ V⊗d be generic and let n o L = [λA + µB] [λ : µ] ∈ P1 .

By deﬁnition, the line L intersects ∆n,d,` at [λA + µB] if and only if the eigenscheme of λA + µB is non-regular. By Lemma 5.6, the latter condition is equivalent to the condition that the global section of E (d − 2) corresponding to λA + µB is singular. We thus have

L ∩ ∆n,d,` = 2cn−1(E (d − 2)) + 2g(Z) − 2, where Z is the curve obtained as the dependency locus of the global sections of E (d − 2) corresponding to A and B. Therefore, (5.4) follows from Example 4.9. The following corollary is an immediate consequence of (4.6) and Theorem 5.7. 26 HIROTACHI ABO

Corollary 5.8 ([2]). The eigendiscriminant locus ∆n,d is an irreducible hypersurface of degree n(n − 1)(d − 1)n−1.

6. DISCRIMINANTOFTHEGENERICGRADEDMATRIX A determinantal scheme is a closed subscheme of a projective space deﬁned by the ideal generated by the minors of a ﬁxed size of a matrix with polynomial entries. If the entries of the matrix are generic, then the associated determinantal scheme is non-singular and has the expected codimension. In this section, we study the locus of polynomial matrices whose associated determinantal schemes are singular or do not have the expected codimensions. We call such a locus the discriminant locus of the generic graded matrix. The focus of this section is on the discriminant locus of the generic (n + k − 1) × (k + 1) graded matrix.

6.1. Definitions and examples. Let R = k[x0, x1,..., xn−1] and let A be an ` × m “graded matrix,” that is, an ` × m matrix whose (i, j)-entry is a homogeneous polynomial in R with degree bi − c j for each i ∈ {0, 1, . . . , ` − 1} and j ∈ {0, 1,..., m − 1}. Then the ideal of R generated by the (r + 1) × (r + 1) minors of A, denoted Ir(A), is homogeneous. It, therefore, defines a closed subscheme of Pn−1, which is called the rth determinantal scheme associated with A and denoted by Zr(A). We call the reduced induced structure on Zr(A) the rth determinantal variety associated with A and denote it by Vr(A). The determinantal varieties form an important class of varieties; many fundamental classes of varieties such as Segre varieties and Veronese varieties are determinantal. The determinantal varieties are important not only in algebraic geometry, but also in other areas of mathematics because many mathematically formulable problems can be expressed as polynomials equations that take the form of minors of a matrix. One such example is the generalized eigenvector problem of two matrices. Example 6.1 (Generalized eigenvectors of two matrices). Let A and B be n × n matrices with entries from k. A non-zero vector v ∈ kn is called a generalized eigenvector of A and B if there exist λ ∈ k \{0} such that Av = λBv. In other words, a non-zero vector v ∈ kn is a generalized eigenvector of A and B if Av and Bv are linearly dependent, or equivalently, if the n × 2 matrix (Av | Bv) obtained by concatenating Av and Bv horizontally has rank one or less. In particular, the locus of common eigenvectors of A and B [v] ∈ P(kn) | v is a generalized eigenvector of A and B is defined by the 2 × 2 minors of the n × 2 matrix C = (Ax | Bx) with linear entries. It is therefore the first determinantal variety associated with C.

` m Let b = (b0, b1,..., b`−1) ∈ Z , let c = (c0, c1,..., cm−1) ∈ Z , and let M(b, c) be the set of graded matrices whose (i, j)-entries have degree bi − c j > 0. The determinantal scheme of a graded matrix of M(b, c) can be interpreted as the degeneracy locus of a sheaf mor- Lm−1 L`−1 Lm−1 phism from i=0 OPn−1 (ci) and i=0 OPn−1 (bi). The tensor product of i=0 OPn−1 (−ci) L`−1 and i=0 OPn−1 (bi) is very ample. Thus, as was discussed in Subsection 2.1, the codimension of Zr(A) for a generic choice of an element A of M(b, c) is (` − r)(m − r) for each r ≤ min{`, m} − 1. In particular, if (` − r + 1)(m − r + 1) ≥ n, then Zr−1(A) is empty, and hence Zr(A) is non-singular. The rth determinantal scheme associated with A ∈ M(b, c) is singular or has higher codimension than (` − r)(m − r) if and only if there exists a point [v] ∈ Pn−1 that lies in the ON THE DISCRIMINANT LOCUS OF A RANK VECTOR BUNDLE 27 subvariety defined by the sum of Ir(A) and the Jacobian ideal Jr(A) of the generators of Ir(A). We say that A is singular if the variety defined by Ir(A) + Jr(A) is not empty. Note that, for each j ∈ {0, 1,..., m − 1}, the jth column A j of A can be considered as an b −c ∗ element of the direct sum W j of the ` vector spaces S i j (V ). The ideal Ir(A) + Jr(A) is multi-homogeneous with respect to the set of variables of the homogeneous coordinate ring Qm−1 of each PW j. We therefore consider the subvariety of the multi-projective space j=0 PW j    m−1   Y  ∆n,r(b, c) = [A j] ∈ PW j (A0 | A1 | · · · | Am−1) is singular ,  j∈{0,1,...,m−1}   j=0  instead of the locus of points in PM(b, c) parametrizing singular graded matrices. This is a reasonable setup for classifying singular graded matrices because if one of the columns n−1 of A is zero, then Zr(A) = P . Furthermore, if B ∈ M(b, c) and if its jth column is a non-zero scalar multiple of A j for each j, then Zr(A) = Zr(B). In this section, we focus on the discriminant locus of a kth determinantal scheme for ` = n+k−1 and m = k+1. In this case, the determinantal scheme associated with a generic graded matrix in M(b, c) is reduced of codimension n − 1. Thus, the discriminant locus ∆n,k(b, c) parameterizes graded matrices whose determinantal schemes are non-reduced or have positive dimensional primary components. The goal of this section is twofold; namely, the first is to show that ∆n,k(b, c) is an irreducible hypersurface, and the second is to give a formula for its degree.

Example 6.2 (Generalized eigenvectors revisited). Let A = (ai j)0≤i, j≤n−1, B = (bi j)0≤i, j≤n−1 be non-zero n × n matrices with entries from k and let C = (Ax | Bx). If B is invertible, then a non-zero vector v of kn is a generalized eigenvector of A and B if and only if it is an −1 eigenvector of B A. This means that V1(C) is the union of projectivizations of eigenspaces −1 of B A. Furthermore, the determinantal scheme Z1(C) of C is reduced of codimension n − 1 if and only if B−1A has n distinct eigenvalues. Therefore, the determinantal locus of the generic n × 2 graded matrix with linear entries is deﬁned by the discriminant of the characteristic polynomial padj(B)A(t) of the product of the adjugate adj(B) of B and A. For example, if n = 2, then the coefﬁcients of t and 1 in padj(B)A(t) are

tr(adj(B)A) = −a00b11 + a01b10 + a10b01 − a11b00;

det(adj(B)A) = a01a10b01b10 − a01a10b00b11 − a00a11b01b10 + a00a11b00b11 respectively. Therefore, the discriminant of padj(B)A(t) is the bi-homogeneous polynomial of bi–degree (2, 2): 2 2 2 2 a00b11 − 2a00a01b10b11 + a01b10 − 2a00a10b01b11 − 2a01a10b01b10 2 2 + 4a01a10b00b11 + a10b01 + 4a00a11b01b10 − 2a00a11b00b11 − 2a01a11b00b10 2 2 − 2a10a11b00b01 + a11b00.

Thus, the discriminant locus ∆2,1((1, 1), (0, 0)) is a hypersurface of multi-degree (2, 2).

n+k−1 6.2. Determinantal scheme as a degeneracy locus. Let b = (b0, b1,..., bn+k−2) ∈ Z k+1 and let c = (c0, c1,..., ck) ∈ Z . Then the determinantal scheme of a graded matrix A ∈ 0 0 0 0 M(b, c) depends only on the difference bi −c j. More precisely, if b = (b0, b1,..., bn+k−2) ∈ n+k−1 0 0 0 0 k+1 0 0 Z and if c = (c0, c1,..., ck) ∈ Z , then M(b, c) = M(b , c ) precisely when bi − c j = 0 0 bi − c j for all i, j. One can, therefore, normalize b and c as follows: ( ck ≤ ck−1 ≤ · · · ≤ c0 < bn+k−2 ≤ bn+k−3 ≤ · · · ≤ b0, (6.1) bn+k−2 = 1. 28 HIROTACHI ABO

Let b and c satisfy (6.1) and let A ∈ M(b, c). Since A is a graded matrix, it can be interpreted as a graded homomorphism from H = R(c0) ⊕ · · · ⊕ R(ck) to G = R(b0) ⊕ · · · ⊕ R(bn+k−2). If H = H˜ and let G = G˜, then there exist natural isomorphisms Mk (6.2) Hom (H, G) = Hom ( , ) = H0( (−c )). R OPn−1 H G G i i=0 Furthermore, if ϕ ∈ Hom ( , ) corresponds to A, then the determinantal scheme OPn−1 H G associated with A and the degeneracy locus of ϕ are equal. Thus, it makes sense to call a morphism ϕ : H → G of OPn−1 -modules singular whenever the corresponding graded matrix A is singular. Through isomorphisms (6.2), one can identify ∆n,k(b, c) with  k   Y  (6.3) ([ϕ ] ∈ PH0( (−c )) (ϕ , ϕ , . . . , ϕ ) is singular .  i i∈{0,...,k} G i 0 1 k   i=0 

This alternative description (6.3) of ∆n,k(b, c) is used to relate ∆n,k(b, c) to the discriminant hypersurface of a Schwarzenberger bundle.

6.3. Main result. Consider the morphisms of OPn−1 -modules from H to G satisfying the following condition:

(∗) For a given subset i = {i0, i1,..., ik−1} of k elements of {0, 1,..., k}, the (k − 1)st degeneracy locus of

ϕi = (ϕi0 , ϕi1 , . . . , ϕik−1 ): H −→ G . is empty.

Let ϕ satisfy (∗) for a subset i = {i0, i1,..., ik−1} of k elements of {0, 1,..., k}. For the Lk−1 ∈ { − } n−1 sake of simplicity, let ap = cip for each p 0, 1,..., k 1 and let Fi = p=0 OP (ap). As ϕ satisﬁes (∗), the cokernel of the corresponding sheaf morphism ϕi, denoted Ei, is a vector bundle of rank n − 1. Moreover, it is a Schwarzenberger bundle:

ϕi (6.4) 0 −→ Fi −→ G −→ Ei −→ 0.

We call Ei the ith Schwarzenberger bundle associated with ϕ. n−1 If ik ∈ {0, 1,..., k}\ i, then we let ak = cik . Tensoring (6.4) by OP (−ak) and taking cohomology, we obtain the following exact sequence: H0(ϕ [−a ]) 0 i k 0 0 0 −−−−−−→ H (Fi(−ak)) −−−−−−−−→ H (G (−ak)) −−−−−−→ H (E (−ak)) −−−−−−→ 0. 0 Let sϕ = ϕik + im(H (ϕi[−ak])). Then ϕ corresponds to (Ei(−ak), sϕ). Conversely, if E is a Schwarzenberger bundle obtained via υ 0 −→ Fi −→ G −→ E −→ 0, 0 0 0 if s ∈ H (E (−ak)) is non-zero, and if ϕik ∈ H (G (−ak)) with s = ϕik + im(H (υ[−ak]), then we set ϕip to be the pth component of υ for each p ∈ {0, 1,..., k − 1}. The morphism n−1 ϕ = (ϕi0 , ϕi1 , . . . , ϕik ): H → G of OP -modules then satisﬁes (∗). Furthermore, if is the ith Schwarzenberger bundle associated with ϕ ∈ Hom ( , ) Ei OPn−1 H G 0 and if s ∈ H (Ei(−ak)) corresponds to ϕ, then, since 0 s = ϕik + im(H (ϕi[−ak])), n−1 the global section s vanishes at a point P ∈ P if and only if ϕi0 (P), ϕi1 (P), . . . , ϕik (P) are linearly dependent. Therefore, we have D (ϕ) = Z(s). In particular, if ϕ ∈ Hom ( , ) k OPn−1 H G satisﬁes (∗), then [ϕ] ∈ ∆n,k(b, c) if and only if [sϕ] ∈ ∆(Ei(−ak)). ON THE DISCRIMINANT LOCUS OF A RANK VECTOR BUNDLE 29

n+k−1 Theorem 6.3. For each j ∈ {0, 1,..., k}, let i j = {0, 1,..., k}\{ j}. If b ∈ Z and k+1 c ∈ Z satisfy condition (6.1), then the ∆n,k(b, c) is an irreducible hypersurface in Qk 0 p=0 PH (G (−ap)) of multi-degree deg ∆(Ei0 (−a0)), deg ∆(Ei1 (−a1)),..., deg ∆(Eik (−ak)) .

Proof. The morphisms ϕ : H → G of OPn−1 -modules satisfying (∗) for a subset i = ◦ Qk 0 {i0, i1,..., ik−1} of {0, 1,..., k} form an open dense subset ∆n,k(b, c) of p=0 PH (G (−ap)):    Yk   0  ([ϕi ], [ϕi ],..., [ϕi ]) ∈ PH (G (−ap)) ϕ is singular and satisfies (∗) for i .  0 1 k   p=0  ◦ It is therefore enough to show that ∆n,k(b, c) is irreducible of codimension 1. Qk 0 Qk−1 0 Consider the projection πi from p=0 PH (G (−ap)) to p=0 PH (G (−ap)). For each ◦ 0 ([ϕi0 ], [ϕi1 ],..., [ϕik ]) ∈ ∆n,k(b, c) , let s = ϕik + im(H (ϕi[−ak])). Then [s] ∈ ∆(E (−ak)), and the fiber of πi over πi([ϕi0 ], [ϕi1 ],..., [ϕik ]) is n o −1 0 πi (πi([ϕi0 ], [ϕi1 ],..., [ϕik ])) = [([ϕi0 ], [ϕi1 ],..., [ϕik−1 ], [s + φ])] φ ∈ im(H (ϕi[−ak])) , 0 which is isomorphic to the cone over ∆(Ei(−ak)) with vertex P im(H (ϕi[−ak])). Therefore, it is irreducible of dimension −1 0 dim πi (πi([ϕi0 ], [ϕi1 ],..., [ϕik ])) = dim ∆(Ei(−ak)) + dim P im(H (ϕi[−ak])) + 1 0 0 = dim PH (Ei(−ak)) − 1 + dim H (Fi(−ak)) 0 0 = dim H (Ei(−ak)) − 2 + dim H (Fi(−ak)) 0 = dim H (G (−ak)) − 2 0 = dim PH (G (−ak)) − 1. We therefore obtain   Yk−1  ◦  0  −1 dim ∆n,k(b, c) = dim  PH (G (−ap)) + dim π (πi([ϕi ], [ϕi ],..., [ϕi ]))   i 0 1 k p=0   Yk−1   0  0 = dim  PH (G (−ap)) + dim PH (G (−ak)) − 1   p=0   Yk   0  = dim  PH (G (−ap)) − 1.   p=0 Qk 0 Consequently, ∆n,k(b, c) is an irreducible hypersurface in p=0 PH (G (−ap)). Qk 0 Since ∆n,k(b, c) is a hypersurface in p=0 PH (G (−ap)), it is defined by a single multi- homogeneous polynomial Dn,k(b, c) that is homogeneous with respect to the set of the vari- Qk 0 ables in the coordinate ring of each factor of p=0 PH (G (−ap)). The degree of Dn,k(b, c) 0 with respect to the set of the variables in the coordinate ring of PH (G (−ak)) equals the de- −1 gree of πi ([ϕi]), which is the same as the degree of ∆(Ei(−ak)). Therefore, we completed the proof. The formula for the degree of the discriminant hypersurface of any twist of the Schwarzen- berger bundle given in Theorem 4.6 and Remark 4.7 allows one to explicitly calculate the degree of ∆n,k(b, c) can explicitly be calculated by using Theorem 6.3 if b and c are given. As an example, we describe the degree of the discriminant hypersurface of the generic (n + k − 1) × (k + 1) graded matrix with linear entries in the following corollary. 30 HIROTACHI ABO

Corollary 6.4. For given k ∈ N, b = (1, 1,..., 1) ∈ Zn+k−1, and c = (0, 0,..., 0) ∈ Zk+1, the discriminant locus ∆n,k(b, c) is an irreducible hypersurface of multi-degree ! !! n + k − 1 n + k − 1 2k ,..., 2k ∈ Nk+1. n − 2 n − 2

In particular, the total degree of ∆n,k(b, c) is ! n + k − 1 deg ∆ (b, c) = 2(k + 1)k . n,k n − 2 Proof. This corollary is an immediate consequence of Example 4.8 and Theorem 6.3.

7. DISCRIMINANTLOCUSOFAVECTORBUNDLEINPOSITIVECHARACTERISTIC The purpose of this section is to extend Theorems 3.13 and 4.6 to a field of positive characteristic. Most arguments we made to prove the aforementioned theorems are still valid in positive characteristic. However, several general results used in the proofs of Theorems 3.13 and 4.6, such as the Bertini type theorem for vector bundles discussed in Subsection 2.2, do not necessarily hold in positive characteristic as they are presented. Therefore, we either use a positive characteristic analog of such a general result or give an alternative proof (if no such analog is available). We thus need to modify the statements of Theorems 3.13 and 4.6 accordingly. Let k be an algebraically closed field of positive characteristic and let Pn−1 be projective (n − 1)-space over k. If E is a vector bundle of rank n − 1 on Pn−1 such that E (−1) is globally generated, then the zero scheme of a generic global section of E is either empty or non-singular of codimension n − 1 (see [18]). Furthermore, the proof given by L. Ein in [9, Lemma 2.5] can be used to prove that the dependency locus Z of two global sections of E is non-singular of codimension n − 2. Suppose that Z is irreducible, and let Ψ be the finite morphism from Z to the pencil L of the two global sections of E defined in Proposition 2.1. As in the proof of theorem 3.13, the positivity of the degree of the ramification divisor R of Ψ implies that the discriminant locus ∆(E ) of E is a hypersurface. Moreover, we have deg ∆(E ) ≤ deg R with equality if P and only if the number of branch point of Ψ is the same as P∈Z(ep − 1). Furthermore, the latter occurs if and only if the zero scheme of the global section sQ corresponding to each branch point Q ∈ L has a single singularity, and the singularity is an ordinary double point. Such a pencil L of global sections of E exists due to the existence of a Lefschetz pencil. P We thus have deg ∆(E ) = P∈Z(eP − 1). If the characteristic of k is not two, then Ψ : Z → L is tamely ramified, and hence P we have deg R = P∈Z(eP − 1). If the characteristic of k is two, then the stalk (ΩZ/L)P of the sheaf ΩZ/L of relative differentials of Z over L at P has length greater than or equal to ep − 1 = 2. We therefore obtain the following theorem: Theorem 7.1. Let k be an algebraically closed field of positive characteristic p, let E be n−1 a very ample vector bundle of rank n − 1 on P with cn−1(E ) ≥ 2 such that E (−1) is globally generated, and let     Xn−3 î+2    n−1+i    δ(E ) = cn− (E ) +  (−1) (i + 1) χ  E  (−n) − 1. 1     i=0 If the degeneracy locus of two generic sections of E is irreducible, then the discriminant locus ∆(E ) of E is a hypersurface. Furthermore, the equality deg(∆(E )) = 2δ(E ) holds if p , 2; otherwise, we have the inequality deg(∆(E )) ≤ δ(E ). ON THE DISCRIMINANT LOCUS OF A RANK VECTOR BUNDLE 31

We explored a small number of examples in characteristic two, for all of which the equality deg(∆(E )) = δ(E ) holds. This leads us to the following question. Question 7.2. Does the equality deg(∆(E )) = δ(E ) always hold in characteristic two? n−1 Remark 7.3. If E is a direct sum of n−1 very ample line bundles on P with cn−1(E ) ≥ 2, then Question 7.2 was affirmatively answered by L. Busé and J. P. Jouanolou in [7]. Corollary 7.4. Let k be an algebraically closed field of positive characteristic p and let a be a positive integer. If E is a Schwarzenberger bundle on Pn−1, then the discriminant locus ∆(E (a)) of E (a) is a hypersurface. Furthermore, with the same notation as in The- orem 7.1, the equality deg ∆(E (a)) = 2δ(E (a)) holds if p , 2; otherwise, we have the inequality deg ∆(E (a)) ≤ δ(E (a)). Proof. It follows from condition (4.1) that E (b − 1) is globally generated for every non- negative integer b. By Corollary 4.5, the top Chern class of E (a) is greater than or equal to two for every non-negative integer a. The zero scheme of a generic global section of E (a) therefore is non-singular of codimension n − 1. In particular, if s0 and s1 are generic global sections of E , then their dependency locus Z = Z(s0 ∧ s1) is non-singular of codimension n − 2. In Theorem 4.6, we used the connectedness theorem of W. Fulton and R. Lazarsfeld [12] to show that Z is connected (and hence irreducible). The author does not, however, know whether their theorem has already been extended to a field of positive characteristic. We therefore prove the connectedness of Z differently. Since E is a Schwarzenberger bundle on Pn−1, there exist a positive integer k and integers a0, a1,..., ak−1, b0, b1,..., bn+k−2 satisfying (4.1) such that k−1 n+k−2 M ϕ M ψ 0 −→ OPn−1 (ai) −→ OPn−1 (bi) −→ E −→ 0 i=0 i=0 Ln+k−2 is a short exact sequence. For the sake of simplicity, we write G for i=0 OPn−1 (bi). 0 For each i ∈ {0, 1,..., k − 1}, let ϕ ∈ H ( (−a )) = Hom ( n−1 (a ), ) such that i G i OPn−1 OP i G ϕ = (ϕ0, ϕ1, . . . , ϕk−1). The map H0(ψ[a]) : H0(G (a)) → H0(E (a)) is surjective for every non-negative a. For 0 each i ∈ {0, 1}, let ϕk+i be an element of H (E (a)) corresponding to si and let ak+i = a. Consider ψ as an element of Hom ( n−1 (−a), ), and let i OPn−1 OP G Mk+1 ψ˜ = (ψ0, ψ1, . . . , ψk+1): OPn−1 (ai) → G . i=0 ˜ ˜ Lk+1 The kth degeneracy locus Dk(ψ) of ψ is the same as Z. Let F = i=0 OPn−1 (ai) and let Pk+1 ∨ ∨ c1 = i=0 ai. We write A for F (c1) and B for G (c1). For each i ∈ {1, 2,..., n − 3}, let i ∨ V f +i Ki = S A ⊗ B. As was discussed in Subsection 2.1, the Eagon-Northcott complex ∨ of (ψ˜) [c1]: A → B ^k+2 ^k+2 (7.1) 0 → Kn−3 → Kn−4 → · · · → K1 → B → A is a locally free resolution of OZ. j Since each Ki is a direct sum of lines bundles, the cohomology group H (Ki) vanishes for every j ∈ {1, 2,..., n − 2}. Thus, taking cohomology of exact sequence (7.1), we 0 obtain dim H (OZ) = 1, which proves the connectedness of Z. The curve Z therefore is irreducible, and hence the corollary follows from Theorem 7.1. 32 HIROTACHI ABO

Remark 7.5. Theorems 4.6 and 7.4 imply that Theorem 5.7 holds as stated if the characteristic of the ﬁeld is other than two. If the characteristic of the ﬁeld is two, then the degree of the eigendiscriminant hypersurface is at most the half of the right side of equality (5.4). The same theorems also imply that Theorem 6.3 holds in any characteristic.

ACKNOWLEDGEMENT This paper grew out of joint work with A. Seigal and B. Sturmfels. The author owes a debt of gratitude for the collaboration. The author is particularly grateful to B. Sturmfels for suggesting the problem discussed in Section 6. The author would like to thank C. Peterson for sharing with the author his idea of using the Hurwitz-Riemann formula to ﬁnd the degree of the discriminant hypersurface of a vector bundle. The author would also like to thank G. Smith. This paper has greatly beneﬁted from a collaboration with him. Finally, the author would like to express his gratitude to D. Faenzi, H. Schenck, and A. Woo for useful discussions.

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