Equality of ordinary and symbolic powers and the Conforti-Cornuéjols conjecture for $(n-2)$-uniform clutters

Abstract

Let $I$ be an equigenerated squarefree monomial ideal in the polynomial ring $\mathbb{K}[x_1,\dots ,x_n]$, and let $\mathcal{H}$ be a uniform clutter on the vertex set $\{x_1,\dots ,x_n\}$ such that $I=I(\mathcal{H})$ is its edge ideal. A central and challenging problem in combinatorial commutative algebra is to classify all clutters $\mathcal{H}$ for which $I(\mathcal{H}{)}^{(k)}=I(\mathcal{H}{)}^k$ for a fixed positive integer $k$, where $I(\mathcal{H}{)}^{(k)}$ denotes the $k^{\text{th}}$ symbolic power of $I(\mathcal{H})$. In this article, we give a complete solution to this problem for $(n-2)$-uniform clutters. Moreover, we provide a simple combinatorial classification of all $(n-2)$-uniform clutters having the packing property. As a consequence, we confirm the celebrated Conforti-Cornuéjols conjecture for $(n-2)$-uniform clutters. We also compare our results with the known families of clutters for which the conjecture is known to be true. Finally, we present an application of our results to the theory of Linear Programming duality problems.

Major revision. Among others, an application to LP duality problems is presented. Comments are welcome!