# Regularity of powers of cover ideals of unimodular hypergraphs

## TL;DR

This paper proves that the regularity of powers of cover ideals of unimodular hypergraphs grows linearly with the power, and provides bounds for the linearity of certain algebraic invariants.

## Contribution

It establishes the linearity of the regularity and $a_i$-invariants of powers of cover ideals for unimodular hypergraphs, extending understanding of their algebraic properties.

## Key findings

- $
eg J( ext{H})^s$ is linear in $s$ for large $s$
- $a_i(R/J( ext{H})^s)$ is linear in $s$ for $s o ext{large}$
- Provides explicit bounds for the linearity in terms of hypergraph parameters

## Abstract

Let $\H$ be a unimodular hypergraph over the vertex set $[n]$ and let $J(\H)$ be the cover ideal of $\H$ in the polynomial ring $R=K[x_1,\ldots,x_n]$. We show that $\reg J(\H)^s$ is a linear function in $s$ for all $s\geqslant r\left\lceil \frac{n}{2}\right\rceil+1$ where $r$ is the rank of $\H$. Moreover for every $i$, $a_i(R/J(\H)^s)$ is also a linear function in $s$ for $s \geqslant n^2$.