Betti numbers of transversal monomial ideals

TL;DR

This paper explicitly computes the Betti numbers of transversal monomial ideals, relates them to initial ideals of generic pluri-circulant matrices, and explores their significance in algebraic geometry.

Contribution

It introduces a modified minimal free resolution for transversal monomial ideals and links Betti numbers of these ideals to those of initial ideals of generic matrices.

Findings

Betti numbers of transversal monomial ideals are explicitly computed.

Initial ideals of certain generic matrices are stable monomial ideals.

Betti numbers of initial ideals match those of transversal monomial ideals for specific cases.

Abstract

In this paper, by a modification of a previously constructed minimal free resolution for a transversal monomial ideal, the Betti numbers of this ideal is explicitly computed. For convenient characteristics of the ground field, up to a change of coordinates, the ideal of $t$-minors of a generic pluri-circulant matrix is a transversal monomial ideal . Using a Gr\"obner basis for this ideal, it is shown that the initial ideal of a generic pluri-circulant matrix is a stable monomial ideal when the matrix has two square blocks. By means of the Eliahou-Kervair resolution, the Betti numbers of this initial ideal is computed and it is proved that, for some significant values of $t$, this ideal has the same Betti numbers as the corresponding transversal monomial ideal. The ideals treated in this paper, naturally arise in the study of generic singularities of algebraic varieties.