Regularity of powers of cover ideals of unimodular hypergraphs

Abstract

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