On ideals generated by fold products of linear forms

TL;DR

This paper proves that ideals generated by (n-2)-fold products of n linear forms in a polynomial ring have linear resolutions, providing insights into their algebraic structure and related algebras, even with proportional forms.

Contribution

It establishes linear graded free resolutions for ideals generated by fold products of linear forms, extending to cases with proportional forms and analyzing related algebraic structures.

Findings

Ideals generated by (n-2)-fold products have linear resolutions.

The symmetric ideal of these ideals can be explicitly described.

Analysis of the Orlik-Terao algebra via Sylvester forms.

Abstract

Let $\mathbb K$ be a field of characteristic 0. Given $n$ linear forms in $R=\mathbb K[x_1,\ldots,x_k]$, with no two proportional, in one of our main results we show that the ideal $I\subset R$ generated by all $(n-2)$-fold products of these linear forms has linear graded free resolution. This result helps determining a complete set of generators of the symmetric ideal of $I$. Via Sylvester forms we can analyze from a different perspective the generators of the presentation ideal of the Orlik-Terao algebra of the second order; this is the algebra generated by the reciprocals of the products of any two (distinct) of the linear forms considered. We also show that when $k=2$, and when the collection of $n$ linear forms may contain proportional linear forms, for any $1\leq a\leq n$, the ideal generated by $a$-fold products of these linear forms has linear graded free resolution.