On ideals generated by a-fold products of linear forms

Abstract

Let $\mathbb{𝕂}$ be any field. Given $n$ linear forms in $R=\mathbb{𝕂}\left[x_1,\dots ,x_k\right]$, some possibly proportional, in one of our main results we show that the ideal $I\subset R$ generated by all $\left(n-2\right)$-fold products of these linear forms has linear graded free resolution. When no two of the linear forms considered are proportional, 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.