Resolution of ideals associated to subspace arrangements

TL;DR

This paper provides a minimal free resolution for products of ideals generated by linear forms linked to subspace arrangements, using polymatroid theory to describe algebraic invariants and prime structures.

Contribution

It introduces a novel resolution supported on a polymatroid derived via Dilworth truncation, extending understanding of ideal resolutions in subspace arrangements.

Findings

Minimal free resolution supported on a polymatroid

Formulas for projective dimension and Betti numbers

Characterization of associated primes

Abstract

Let $I_1,\dots,I_n$ be ideals generated by linear forms in a polynomial ring over an infinite field and let $J = I_1 \cdots I_n$. We describe a minimal free resolution of $J$ and show that it is supported on a polymatroid obtained from the underlying representable polymatroid by means of the so-called Dilworth truncation. Formulas for the projective dimension and Betti numbers are given in terms of the polymatroid as well as a characterization of the associated primes. Along the way we show that $J$ has linear quotients. In fact, we do this for a large class of ideals $J_P$, where $P$ is a certain poset ideal associated to the underlying subspace arrangement.