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The minimal resolution property for points on general curves

Gavril Farkas, Eric Larson

TL;DR

The paper delivers an essentially complete solution to the Minimal Resolution Property for general curves embedded by Brill–Noether linear series with degree $d\ge 2r$, giving an explicit Betti-table description for large point sets and proving that MRP is governed by the strong Raynaud condition on the kernel bundle $M_V$. It develops a robust degeneration framework for Raynaud-type conditions on nodal curves, establishes stability and strong semistability of kernel bundles via elliptic-base constructions, and uses this to prove Butler’s conjecture for general BN curves in arbitrary characteristic. A two-pronged induction—one anchored on elliptic curves and one on degenerations involving rational components—propagates the strong Raynaud condition through smoothing, thereby achieving MRP and strong (semi)stability results across broad ranges of $d$, $g$, and $r$. The work also connects syzygies of points to moduli-space phenomena, yielding results in positive characteristic, such as precise strong semistability of kernel bundles and implications for Hilbert–Kunz multiplicities, with canonical curves exhibiting robust stability behavior.

Abstract

We present an essentially complete solution to the Minimal Resolution Conjecture for general curves, determining the shape of the minimal resolution of general sets of points on a general curve C of degree d>2r-1 in P^r. Our methods also provide a proof (valid in arbitrary characteristic) of the strong version of Butler's Conjecture on the stability of syzygy bundles on a general curve of every genus at least 3, as well as of the Frobenius semistability in positive characteristic of the syzygy bundle of a general curve in the range d>2r-1.

The minimal resolution property for points on general curves

TL;DR

The paper delivers an essentially complete solution to the Minimal Resolution Property for general curves embedded by Brill–Noether linear series with degree , giving an explicit Betti-table description for large point sets and proving that MRP is governed by the strong Raynaud condition on the kernel bundle . It develops a robust degeneration framework for Raynaud-type conditions on nodal curves, establishes stability and strong semistability of kernel bundles via elliptic-base constructions, and uses this to prove Butler’s conjecture for general BN curves in arbitrary characteristic. A two-pronged induction—one anchored on elliptic curves and one on degenerations involving rational components—propagates the strong Raynaud condition through smoothing, thereby achieving MRP and strong (semi)stability results across broad ranges of , , and . The work also connects syzygies of points to moduli-space phenomena, yielding results in positive characteristic, such as precise strong semistability of kernel bundles and implications for Hilbert–Kunz multiplicities, with canonical curves exhibiting robust stability behavior.

Abstract

We present an essentially complete solution to the Minimal Resolution Conjecture for general curves, determining the shape of the minimal resolution of general sets of points on a general curve C of degree d>2r-1 in P^r. Our methods also provide a proof (valid in arbitrary characteristic) of the strong version of Butler's Conjecture on the stability of syzygy bundles on a general curve of every genus at least 3, as well as of the Frobenius semistability in positive characteristic of the syzygy bundle of a general curve in the range d>2r-1.
Paper Structure (19 sections, 26 theorems, 83 equations, 1 table)

This paper contains 19 sections, 26 theorems, 83 equations, 1 table.

Table of Contents

  1. Introduction
  2. Syzygies of points on curves and the Raynaud condition
  3. The Raynaud condition under degeneration.
  4. Stability conditions for nodal curves.
  5. General vector bundles on rational and elliptic curves
  6. Elementary modification of kernel bundles
  7. Stability in exact sequences on nodal curves
  8. The Weak Raynaud condition and the Ideal Generation property
  9. The Raynaud condition for Elliptic Curves
  10. The proof of the Minimal Resolution Property
  11. The proof of Theorem \ref{['thm:main']}, respectively Theorem \ref{['thm:semistab-strong']}
  12. Butler's Conjecture
  13. Corank $1$ subbundles of the kernel bundle for $d \leq 2r$
  14. Main inductive argument: $d \leq 2r$ and $(d, g) \neq (2r, r + 1)$
  15. If inequality \ref{['L-contrib']} is strict
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $C\subseteq \mathbb{P}^r$ be a general Brill--Noether curve of genus $g \geq 1$ and degree $d\geq 2r$. Then the Minimal Resolution property holds for $C$.

Theorems & Definitions (52)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • ...and 42 more