Characteristic dependence of syzygies of random monomial ideals

TL;DR

This paper investigates when the syzygies of random monomial ideals depend on the field's characteristic, revealing that for certain random ideals, this dependence is almost always present asymptotically.

Contribution

It proves that Betti numbers of Stanley--Reisner ideals of random flag complexes typically depend on the characteristic, and introduces a heuristic for characteristic dependence in algebraic varieties.

Findings

Betti numbers depend on characteristic in random flag complexes

Asymptotic dependence is almost always present

Heuristic for characteristic dependence in algebraic varieties

Abstract

When do syzygies depend on the characteristic of the field? Even for well-studied families of examples, very little is known. For a family of random monomial ideals, namely the Stanley--Reisner ideals of random flag complexes, we prove that the Betti numbers asymptotically almost always depend on the characteristic. Using this result, we also develop a heuristic for characteristic dependence of asymptotic syzygies of algebraic varieties.