Koszul modules with vanishing resonance in algebraic geometry

Selecta Mathematica Pub Date : 2024-02-13 DOI:10.1007/s00029-023-00912-4

引用次数: 0

Abstract

We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace \(K\subseteq \bigwedge ^2 V\) , where V is a vector space. Previously Koszul modules of finite length have been used to give a proof of Green’s Conjecture on syzygies of generic canonical curves. We now give applications to effective stabilization of cohomology of thickenings of algebraic varieties, divisors on moduli spaces of curves, enumerative geometry of curves on K3 surfaces and to skew-symmetric degeneracy loci. We also show that the instability of sufficiently positive rank 2 vector bundles on curves is governed by resonance and give a splitting criterion.

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代数几何中具有消失共振的科斯祖尔模块

摘要我们讨论了与子空间 \(K\subseteq \bigwedge ^2 V\) 相关的有限长度 Koszul 模块的分级成分的均匀消失结果的各种应用，其中 V 是一个向量空间。在此之前，有限长度的科斯祖尔模块曾被用来证明关于一般典型曲线的协同性的格林猜想（Green's Conjecture on syzygies of generic canonical curves）。现在，我们将其应用于代数变体增厚同调的有效稳定、曲线模空间上的除数、K3 曲面上曲线的枚举几何以及倾斜对称退化位置。我们还证明了曲线上足够正的秩 2 向量束的不稳定性受共振支配，并给出了一个分裂准则。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Selecta Mathematica

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