$v$-numbers of integral closure filtrations of monomial ideals

Abstract

In this article, we investigate the $v$-numbers of powers of monomial ideals and their integral closures in a polynomial ring $S$. We provide an alternative proof for determining the $v$-numbers of powers of complete intersection monomial ideals. Furthermore, we analyze the $v$-numbers associated to integral closure filtrations of irreducible monomial ideals and explore their relationship with the Castelnuovo-Mumford regularity of these ideals. Consequently, we obtain that for all $n\ge 1$, $\mathrm{reg}(S/\overline{I^n})=v(\overline{I^n})=n\alpha (I)-1$ where $I$ is an equigenerated irreducible monomial ideal. Finally, we give an upper bound for $v$-numbers associated to the integral closure filtrations of complete intersection monomial ideals and explicitly compute these $v$-numbers in certain cases. As a consequence, we show that for any integer $a\ge 1$, there exists a height two equigenerated complete intersection monomial ideal $I$ such that $\mathrm{reg}(S/\overline{I^n})-v(\overline{I^n})=a-1$ for all $n\ge 1$. Moreover, we establish that for complete intersection monomial ideals, the $v$-numbers of powers of ideals can be arbitrarily larger than the $v$-numbers of integral closures of their powers.

30 pages, Corollary 4.7 added, Comments are welcome