Asymptotics of random Betti tables

TL;DR

This paper conjectures that syzygy module ranks of smooth projective varieties tend to be normally distributed as the embedding line bundle becomes very positive, and provides probabilistic evidence supporting this behavior through asymptotic analysis of random Betti tables.

Contribution

It introduces a conjecture about the normal distribution of syzygy ranks and proves asymptotic normality for entries of random Betti tables as their size increases.

Findings

Betti table entries become normally distributed asymptotically

Probabilistic model supports the conjecture about syzygy ranks

Asymptotic analysis applies to large Betti tables

Abstract

The purpose of this paper is twofold. First, we present a conjecture to the effect that the ranks of the syzygy modules of a smooth projective variety become normally distributed as the positivity of the embedding line bundle grows. Then, in an attempt to render the conjecture plausible, we prove a result suggesting that this is in any event the typical behavior from a probabilistic point of view. Specifically, we consider a "random" Betti table with a fixed number of rows, sampled according to a uniform choice of Boij-Soderberg coefficients. We compute the asymptotics of the entries as the length of the table goes to infinity, and show that they become normally distributed with high probability.