Periodicity of betti numbers of monomial curves

TL;DR

This paper proves that the Betti numbers of defining ideals of shifted monomial curves become periodic with a specific period, and describes their Betti tables for large shifts.

Contribution

It confirms Herzog and Srinivasan's conjecture on the periodicity of Betti numbers for shifted monomial curves and characterizes their Betti tables asymptotically.

Findings

Betti numbers are eventually periodic with period a_n - a_1

Explicit description of Betti tables for large shifts

Validation of Herzog and Srinivasan's conjecture

Abstract

Let $K$ be an arbitrary field. Let $\a = (a_1< ... <a_n)$ be a sequence of positive integers. Let $C(\a)$ be the affine monomial curve in ${\mathbb A}^n$ parametrized by $t\to (t^{a_1}, ..., t^{a_n})$. Let $I(\a)$ be the defining ideal of $C(\a)$ in $K[x_1, ..., x_n]$. For each positive integer $j$, let $\a+j$ be the sequence $(a_1 + j, ..., a_n+j)$. In this paper, we prove the conjecture of Herzog and Srinivasan saying that the betti numbers of $I(\a + j)$ are eventually periodic in $j$ with period $a_n -a_1$. When $j$ is large enough, we describe the betti table for the closure of $C(\a+j)$ in $\PP^n$.