Symbolic powers of polymatroidal ideals

TL;DR

This paper studies the properties of symbolic powers of polymatroidal ideals, proposing conjectures about their regularity and componentwise linearity, and verifies these for specific classes of ideals.

Contribution

It introduces conjectures on the regularity and linearity of symbolic powers of polymatroidal ideals and proves them for several important classes of ideals.

Findings

Proved that regularity of symbolic powers is at least that of ordinary powers for certain ideals.

Established criteria ensuring symbolic powers have linear quotients and are componentwise linear.

Verified conjectures for classes like squarefree Veronese, matching-matroidal, and transversal polymatroidal ideals.

Abstract

In this paper, we investigate the componentwise linearity and the Castelnuovo-Mumford regularity of symbolic powers of polymatroidal ideals. For a polymatroidal ideal $I$, we conjecture that every symbolic power $I^{(k)}$ is componentwise linear and $$ \text{reg}\,I^{(k)}=\text{reg}\,I^k $$ for all $k \ge 1$. We prove that $\text{reg}\,I^{(k)}\ge\text{reg}\,I^k$ for all $k \ge 1$ when $I$ has no embedded associated primes, for instance if $I$ is a matroidal ideal. Moreover, we establish a criterion on the symbolic Rees algebra $\mathcal{R}_s(I)$ of a monomial ideal of minimal intersection type which guarantees that every symbolic power $I^{(k)}$ has linear quotients and, hence, is componentwise linear for all $k\ge1$. By applying our criterion to squarefree Veronese ideals and certain matching-matroidal ideals, we verify both conjectures for these families. We establish the Conforti-Cornu\'ejols conjecture for any matroidal ideal, and we show that a matroidal ideal is packed if and only if it is the product of monomial prime ideals with pairwise disjoint supports. Furthermore, we identify several classes of non-squarefree polymatroidal ideals for which the ordinary and symbolic powers coincide. Hence, we confirm our conjectures for transversal polymatroidal ideals and principal Borel ideals. Finally, we verify our conjectures for all polymatroidal ideals either generated in small degrees or in a small number of variables.