Minimal Free Resolutions of 0-Dimensional Schemes in P1 \times P1

TL;DR

This paper characterizes zero-dimensional schemes in P1 x P1 with minimal free resolutions, identifying a class of schemes whose resolutions depend solely on their Hilbert functions and point distributions, and provides explicit generators.

Contribution

It introduces a class of reduced schemes with resolutions determined by Hilbert functions, extending the understanding of minimal free resolutions beyond ACM schemes.

Findings

Resolutions depend only on Hilbert functions and point distributions.

A minimal set of generators is given by unions of lines.

Characterization of schemes with length 2 resolutions.

Abstract

Let X be a zero-dimensional scheme in P1 \times P1. Then X has a minimal free resolution of length 2 if and only if X is ACM. In this paper we determine a class of reduced schemes whose resolutions, similarly to the ACM case, can be obtained by their Hilbert functions and depends only on their distributions of points in a grid of lines. Moreover, a minimal set of generators of the ideal of these schemes is given by curves split into the union of lines.