In this article we extend the notion of determinantal representation of hypersurfaces to the determinantal representation of sections of the determinant line bundle of a vector bundle. We give several examples, and prove some necessary conditions for existence of determinantal representation. As an application, we show that for any integer $d \geq 1,$ there is an indecomposable vector bundle $E_d$ of rank $2$ on $\mathbb{P}^2$ such that almost all curves of degree $d$ of $\mathbb{P}^2$ arise as the degeneracy loci of a pair of holomorphic sections of $E_d$, upto an automorphism of $\mathbb{P}^2$. We use this result to obtain a linear algebraic application.
Deep Dive into Generalized determinantal representation of hypersurfaces.
In this article we extend the notion of determinantal representation of hypersurfaces to the determinantal representation of sections of the determinant line bundle of a vector bundle. We give several examples, and prove some necessary conditions for existence of determinantal representation. As an application, we show that for any integer $d \geq 1,$ there is an indecomposable vector bundle $E_d$ of rank $2$ on $\mathbb{P}^2$ such that almost all curves of degree $d$ of $\mathbb{P}^2$ arise as the degeneracy loci of a pair of holomorphic sections of $E_d$, upto an automorphism of $\mathbb{P}^2$. We use this result to obtain a linear algebraic application.
Throughout we work over the field C of complex numbers. For a vector space W of dimension n over C and an integer 0 < r < n we denote by Gr(r, W ) the Grassmannian variety of r-dimensional subspaces of W.
Determinantal representation of homogeneous polynomials have been studied for quite some time in literature, see for example [3], [4], [6], [5], [11], [12] and [14]. We want to extend the notion of determinantal representation of homogeneous polynomials to the determinantal representation of sections of the determinant line bundle of a vector bundle. For a smooth projective variety X with an ample line bundle H and a vector bundle E on X of rank d ≥ 2, call E to be H-abundant if for all m » 0, and D a general member of the complete linear system given by the line bundle det E(mH), there is an automorphism ϕ of X such that ϕ * D is the degeneracy loci of d sections of E(mH). Call E to be (semi-)abundant, if E is H-abundant for (some) all ample H. Of course, if E and H are homogeneous vector bundles, the automorphism ϕ is not needed.
The main result of [6] shows that split vector bundles on P 1 and P 2 are abundant. One of the main goals of this paper is to give several examples of indecomposable abundant vector bundles on P 2 . Theorem 1.1. Let N be the rank two vector bundle on P 2 obtained by taking the quotient of O 2 P 2 ⊕ O P 2 (1) by the subbundle (x, y, z 2 )O P 2 (-1). Then N is abundant.
Theorem 1.2. For k ≥ 1, let M k be the syzygy bundle of O P 2 (k), that is, the dual M * k is the kernel of the evaluation map H 0 (P 2 , O P 2 (k)) ⊗ k O P 2 → O P 2 (k). Then M k is abundant for all k.
Note that M k is indecomposable, it is in fact a stable bundle.
Corollary 1.3. The tangent and cotangent bundle of P 2 are abundant.
Corollary 1.4. For any d ≥ 1 there is an indecomposable vector bundle E d of rank 2 on P 2 such that almost all curves of degree d in P 2 arise as the degeneracy loci of a pair of holomorphic sections of E d , up to an automorphism of P 2 .
Next we show the bundles considered in [13] are abundant.
Theorem 1.5. For integer 2 ≤ r ≤ 4, let E r be the dual of the kernel of the surjection
where x 0 , x 1 , x 2 is the standard basis of H 0 (O P 2 (1)). Then E r is abundant. Now we show that the class of varieties possessing a (semi-)abundant vector bundle is very restrictive. Theorem 1.6. Suppose there is an semi-abundant vector bundle on X and dimX = n. Then n ≤ 2. If n = 2, then either κ(X) = -∞, or κ(X) = 0 and X is minimal. If n = 2 and there is an abundant vector bundle on X, then -K X is rationally effective, that is, H 0 (X, O(-mK X )) ̸ = 0 for some m > 0.
Finally, we give more examples of varieties possessing (semi-)abundant bundles.
Theorem 1.7.
(1) The trivial vector bundle of rank 2 on P 1 × P 1 is semi-abundant.
(2) If C is a smooth projective curve which can be embedded in either
For n ≥ 0, we have an exact sequence
of vector bundles on P 2 , where the first map is given by f → (f.x, f.y, f.z 2 ). Since H 1 (O P 2 (n -1)) = 0, we conclude that the map
is surjective. Thus every section of N (n) comes from a section of
Definition 2.1. For a vector bundle F on P 2 a two dimensional subspace V of H 0 (P 2 , F (n)) is called generically point-wise linearly independent (GPLI, for short), if there is a point p ∈ P 2 and v 1 , v 2 ∈ V such that the tangent vectors v 1 (p) and v 2 (p) are independent.
Example 2.2. The sequence (1) gives an exact sequence of vector spaces :
The image of sections (X n+1 , 0, 0) and (Y n+1 , 0, 0) in H 0 (T P 2 (n + 1)) are linearly independent but not generically point-wise linearly independent.
Remark 2.3. a) If v 1 and v 2 are two linearly independent section of N (n) such that the subspace
, then the complement of U is the base locus of the linear system determined by the subspace
. which is defined exactly on the open set U. This map can be described as follows: If [V ] ∈ U and (f 1 , f 2 , f 3 ) and (g 1 , g 2 , g 3 ) are two sections of the vector bundle O P 2 (n) 2 ⊕ O P 2 (n + 1) that maps to a basis of the vector space V then using sequence (1) we can identify the curve defined by ∧ 2 (V ) with the zero locus of the curve defined by the determinant of the matrix
Hence it suffices to show that for a general curve C of degree 2n + 2 in P 2 , there is an automorphism ϕ of P 2 such that ϕ * (C) is the zero locus of the determinant of a matrix of the form
where g 1 , g 2 , f 1 , f 2 are all homogeneous polynomials of degree n, g 3 and f 3 are homogeneous of degree n+1. Now [6, Main Theorem] says that a general plane curve C of degree 2n+2 is the zero locus of the determinant of a matrix of the form
where g i and f i are as above, l and m are homogeneous linear polynomials, and Q is a homogeneous quadratic polynomial. As C is general we can assume that l and m are linearly independent, and Q is not in the ideal (l, m) of k[x, y, z]. So, replacing C by ϕ * (C) for an automorphism ϕ of P 2 , we may assume that l = x and m =
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