Generalized determinantal representation of hypersurfaces

1. Problem Statement and Motivation

The paper addresses the classical problem of determinantal representations of hypersurfaces. Traditionally, this involves expressing a homogeneous polynomial $F$ (defining a hypersurface) as the determinant of a matrix of linear forms (or polynomials of specific degrees).

The authors aim to generalize this concept from homogeneous polynomials to sections of the determinant line bundle of a vector bundle. Specifically, they investigate whether a general hypersurface (or section of a line bundle) on a projective variety $X$ can be realized as the degeneracy locus of a set of sections of a vector bundle $E$.

Key Definitions Introduced:

• $H$-abundant Bundle: A vector bundle $E$ of rank $d$ on a smooth projective variety $X$ with an ample line bundle $H$ is called $H$-abundant if, for sufficiently large $m$, a general member $D$ of the linear system $|\det(E(mH))|$ is the degeneracy locus of $d$ sections of $E(mH)$, up to an automorphism of $X$.
• Abundant/Semi-abundant: $E$ is abundant if it is $H$-abundant for all ample $H$; it is semi-abundant if it is $H$-abundant for some ample $H$.

The central question is: Which vector bundles are abundant, and on which varieties do such bundles exist?

2. Methodology

The authors employ a combination of algebraic geometry and linear algebra techniques:

• Exact Sequences and Syzygies: They construct specific vector bundles using short exact sequences (e.g., syzygy bundles, kernels of evaluation maps) and analyze their cohomology groups ($H^0$) to determine the existence of sections.
• Grassmannian Maps: They define rational maps from Grassmannians of subspaces of sections, $\text{Gr}(d, H^0(E(mH)))$, to the projective space of the determinant line bundle, $\mathbb{P}(H^0(\det E(mH)))$.
• Dominance and Surjectivity: The core of their proofs relies on showing that these rational maps are dominant. This implies that the image covers a dense open set of the target space, meaning "almost all" hypersurfaces can be represented this way.
• Tangent Map Analysis: To prove dominance, they analyze the tangent maps of these rational maps. They demonstrate surjectivity of the tangent map at specific points (generically point-wise linearly independent subspaces), which implies the map is a submersion and thus dominant.
• Riemann-Roch and Dimensional Analysis: For the non-existence results (Theorem 1.6), they use the asymptotic Riemann-Roch theorem to compare the dimensions of the spaces of sections of the bundle versus the space of automorphisms, deriving constraints on the dimension of the underlying variety.
• Linear Algebra Applications: They translate geometric surjectivity results into linear algebraic statements regarding the generation of ideals by specific sets of polynomials.

3. Key Contributions and Results

A. Existence of Abundant Bundles on $\mathbb{P}^2$

The paper provides several new examples of indecomposable abundant vector bundles on the projective plane $\mathbb{P}^2$, extending previous results which mostly focused on split (decomposable) bundles.

• Theorem 1.1: The rank-2 bundle $N$, obtained as a quotient of $\mathcal{O}_{\mathbb{P}^2}^{\oplus 2} \oplus \mathcal{O}_{\mathbb{P}^2}(1)$ by a specific subbundle, is abundant.
• Theorem 1.2: The syzygy bundles $M_k$ (duals of the kernels of evaluation maps for $\mathcal{O}_{\mathbb{P}^2}(k)$) are abundant for all $k \geq 1$. These are stable, indecomposable bundles.
• Corollaries:
  • The tangent and cotangent bundles of $\mathbb{P}^2$ are abundant.
  • Corollary 1.4 (Main Geometric Result): For any degree $d \geq 1$, there exists an indecomposable rank-2 bundle $E_d$ on $\mathbb{P}^2$ such that almost all curves of degree $d$ arise as the degeneracy loci of two holomorphic sections of $E_d$ (up to automorphism).

B. Generalization to Higher Ranks

• Theorem 1.5: Extends the result to bundles $E_r$ (duals of kernels of specific surjections) for ranks $2 \leq r \leq 4$, proving they are abundant.

C. Constraints on Varieties (Non-Existence)

The authors prove that the class of varieties admitting abundant vector bundles is highly restrictive.

• Theorem 1.6: If a variety $X$ of dimension $n$ admits a semi-abundant vector bundle, then $n \leq 2$.
  • If $n=2$, the Kodaira dimension $\kappa(X)$ must be $-\infty$ or $0$ (with $X$ minimal).
  • If $E$ is abundant (for all ample $H$), then $-K_X$ must be rationally effective (i.e., $X$ is a Fano or similar type).
  • Significance: This rules out the existence of such bundles on varieties of dimension $\geq 3$ or on surfaces of general type.

D. Examples on Other Varieties

• Theorem 1.7:
  • The trivial rank-2 bundle on $\mathbb{P}^1 \times \mathbb{P}^1$ is semi-abundant.
  • Trivial bundles on curves embedded in $\mathbb{P}^2$ or $\mathbb{P}^1 \times \mathbb{P}^1$ are semi-abundant.
  • Restrictions of syzygy bundles to curves in $\mathbb{P}^2$ are semi-abundant.

E. Linear Algebraic Application

• Theorem 4.1: The geometric results are applied to commutative algebra. They prove that for generic sets of homogeneous polynomials satisfying disjointness conditions, a specific linear map induced by multiplication is surjective. This provides an upper bound for the "Castelnuovo-Mumford regularity" or the power of the maximal ideal contained in the ideal generated by these polynomials.

4. Significance and Impact

1. Extension of Determinantal Theory: The paper successfully moves the theory of determinantal representations beyond scalar polynomials to the realm of vector bundles, providing a robust framework for understanding how vector bundles generate hypersurfaces.
2. Indecomposable Examples: Prior to this work, most known abundant bundles were split (direct sums of line bundles). The construction of stable, indecomposable abundant bundles (like $M_k$ and $N$) on $\mathbb{P}^2$ is a significant advancement, showing that "complex" bundles can still generate "generic" geometric objects.
3. Dimensional Rigidity: Theorem 1.6 provides a strong negative result, establishing that the phenomenon of "abundance" is intrinsically linked to low-dimensional geometry ($n \leq 2$) and specific curvature properties (Fano-like or Calabi-Yau-like). This guides future research by narrowing the search space for such bundles.
4. Bridge to Commutative Algebra: By linking the surjectivity of geometric tangent maps to the generation of ideals in polynomial rings, the authors demonstrate a practical application of algebraic geometry to problems in commutative algebra (specifically regarding effective Nullstellensatz bounds).

In summary, this paper establishes a comprehensive theory of "abundant" vector bundles, proving their existence on $\mathbb{P}^2$ and curves, proving their non-existence on higher-dimensional or general type varieties, and leveraging these geometric insights to solve algebraic problems regarding polynomial ideals.