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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