Computing graded Betti tables of toric surfaces

TL;DR

This paper investigates the graded Betti tables of toric surfaces, providing explicit formulas, bounds, and an algorithm for computation, with applications to Veronese surfaces and conjectures on their Betti table entries.

Contribution

It introduces a new algorithm for computing Betti tables of toric surfaces and offers explicit formulas and bounds based on combinatorics, advancing understanding of their algebraic properties.

Findings

Explicit formulas for Betti table entries

A lower bound on the length of the linear strand

Successful computation of Betti tables for specific cases

Abstract

We present various facts on the graded Betti table of a projectively embedded toric surface, expressed in terms of the combinatorics of its defining lattice polygon. These facts include explicit formulas for a number of entries, as well as a lower bound on the length of the linear strand that we conjecture to be sharp (and prove to be so in several special cases). We also present an algorithm for determining the graded Betti table of a given toric surface by explicitly computing its Koszul cohomology, and report on an implementation in SageMath. This works well for ambient projective spaces of dimension up to roughly $25$, depending on the concrete combinatorics, although the current implementation runs in finite characteristic only. As a main application we obtain the graded Betti table of the Veronese surface $\nu_6(\mathbb{P}^2) \subseteq \mathbb{P}^{27}$ in characteristic $40\,009$. This allows us to formulate precise conjectures predicting what certain entries look like in the case of an arbitrary Veronese surface $\nu_d(\mathbb{P}^2)$.