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Generic Green's Conjecture and Generic Geometric Syzygy Conjecture in Positive Characteristic

Yi Wei

TL;DR

This work proves generic Green's Conjecture for canonical curves in positive characteristic when $p\ge\frac{g+4}{2}$, covering both even and odd genus via a K3–Lazarsfeld–Mukai bundle framework. It also establishes a substantial case of Schreyer's Geometric Syzygy Conjecture for the last linear syzygy space on general even-genus curves by showing that $K_{k-1,1}(C,\omega_C)$ is generated by rank-$k$ syzygies arising from $W_{k+1}^1(C)$ and the associated Veronese map. A central innovation is the construction and deformation of families of Lazarsfeld–Mukai bundles over versal (and lattice-polarized) K3-surface families, together with a key regularity assumption yielding a finite flat incidence, enabling a precise description of the last syzygy space. The odd-genus case is treated via lattice-polarized K3 deformation to transfer vanishing from the geometric generic fiber to general curves, using upper semicontinuity of Koszul cohomology and compatible LM-bundle data.

Abstract

We study the syzygies of canonical curves of genus $g\geq 3$ over an algebraically closed field $\mathbb{F}$ of characteristic $p>0$. We provide a new proof of generic Green's Conjecture for $p\geq\frac{g+4}{2}$. Using the techniques from the even-genus case, we establish a significant case of the Geometric Syzygy Conjecture for the last syzygy space of a general even-genus canonical curve (assuming $p>g$). In characteristic 0, it was shown in prior work that this case implies the full conjecture.

Generic Green's Conjecture and Generic Geometric Syzygy Conjecture in Positive Characteristic

TL;DR

This work proves generic Green's Conjecture for canonical curves in positive characteristic when , covering both even and odd genus via a K3–Lazarsfeld–Mukai bundle framework. It also establishes a substantial case of Schreyer's Geometric Syzygy Conjecture for the last linear syzygy space on general even-genus curves by showing that is generated by rank- syzygies arising from and the associated Veronese map. A central innovation is the construction and deformation of families of Lazarsfeld–Mukai bundles over versal (and lattice-polarized) K3-surface families, together with a key regularity assumption yielding a finite flat incidence, enabling a precise description of the last syzygy space. The odd-genus case is treated via lattice-polarized K3 deformation to transfer vanishing from the geometric generic fiber to general curves, using upper semicontinuity of Koszul cohomology and compatible LM-bundle data.

Abstract

We study the syzygies of canonical curves of genus over an algebraically closed field of characteristic . We provide a new proof of generic Green's Conjecture for . Using the techniques from the even-genus case, we establish a significant case of the Geometric Syzygy Conjecture for the last syzygy space of a general even-genus canonical curve (assuming ). In characteristic 0, it was shown in prior work that this case implies the full conjecture.

Paper Structure

This paper contains 15 sections, 13 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Koszul Cohomology and Syzygies
  4. Green's Conjecture on Syzygies of Canonical Curves
  5. The Geometric Syzygy Conjecture
  6. Brill-Noether Theory
  7. Lazarsfeld-Mukai Bundle
  8. Symmetric, divided and exterior powers
  9. Key Assumption (\ref{['key-assumption']}) and (\ref{['key-assumption-2']})
  10. Family of Lazarsfeld-Mukai Bundles
  11. Even Genus Curves: Green's Conjecture & the Geometric Syzygy Conjecture
  12. Odd Genus Curves: Green's Conjecture
  13. Deformation of Lattice Polarized K3 Surfaces
  14. Vanishing Theorem for K3 surfaces
  15. Green's Conjecture for Odd Genus $g=2k+1$

Key Result

Theorem 1

Let $\mathbb{F}$ be any algebraically closed field with $\mathop{\mathrm{char}}\nolimits(\mathbb{F})=p>0$. Let $C$ be a general smooth curve over $\mathbb{F}$ of even genus $g=2k$ or odd genus $g=2k+1$ for $k\geq 2$. Assume $p\geq \frac{g+4}{2}$. Then $K_{k,1}(C,\omega_C)=0$. In other words, $C$ sat

Theorems & Definitions (29)

  • Theorem 1: Generic Green's Conjecture in char $p$
  • Conjecture 1: The Geometric Syzygy Conjecture
  • Theorem 2: The Geometric Syzygy Conjecture for Even Genus
  • Proposition 1
  • Theorem 3
  • Proposition 2
  • Remark 1
  • Definition 2.1
  • Lemma 2.1: EL12*Prop 3.2
  • Lemma 2.2: KM24*Prop 1.3
  • ...and 19 more