On the arithmetic Cohen-Macaulayness of varieties parameterized by Togliatti systems

Abstract

Given any diagonal cyclic subgroup $\Lambda \subset GL(n+1,k)$ of order $d$, let $I_d\subset k[x_0,\dots ,x_n]$ be the ideal generated by all monomials $\{m_1,\dots ,m_r\}$ of degree $d$ which are invariants of $\Lambda$. $I_d$ is a monomial Togliatti system, provided $r\le \left(\genfrac{}{}{0ex}{}{d+n-1}{n-1}\right)$, and in this case the projective toric variety $X_d$ parameterized by $(m_1,\dots ,m_r)$ is called a $GT$-variety with group $\Lambda$. We prove that all these $GT$-varieties are arithmetically Cohen-Macaulay and we give a combinatorial expression of their Hilbert functions. In the case $n=2$, we compute explicitly the Hilbert function, polynomial and series of $X_d$. We determine a minimal free resolution of its homogeneous ideal and we show that it is a binomial prime ideal generated by quadrics and cubics. We also provide the exact number of both types of generators. Finally, we pose the problem of determining whether a surface parameterized by a Togliatti system is aCM. We construct examples that are aCM and examples that are not.

To appear in Annali di Matematica Pura ed Applicata. Minor correction in the Introduction