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

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