Canonical syzygies of smooth curves on toric surfaces

TL;DR

This paper investigates the syzygies of smooth curves on toric surfaces, proving the constancy of their Betti tables and verifying Green's conjecture for certain genus and Clifford index bounds, revealing new algebraic geometric insights.

Contribution

It establishes the invariance of canonical graded Betti tables for curves on Gorenstein weak Fano toric surfaces and confirms Green's conjecture for curves with genus up to 32 or Clifford index up to 6.

Findings

Betti tables are constant among smooth curves on these surfaces

Green's conjecture holds for curves with genus ≤ 32 or Clifford index ≤ 6

New relations between Betti tables and toric surface embeddings

Abstract

In a first part of this paper, we prove constancy of the canonical graded Betti table among the smooth curves in linear systems on Gorenstein weak Fano toric surfaces. In a second part, we show that Green's canonical syzygy conjecture holds for all smooth curves of genus at most 32 or Clifford index at most 6 on arbitrary toric surfaces. Conversely we use known results on Green's conjecture (due to Lelli-Chiesa) to obtain new facts about graded Betti tables of projectively embedded toric surfaces.