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

Given any diagonal cyclic subgroup Λ ⊂ GL (n+ 1 , k) of order d, let Id⊂ k[x, … , xn] be the ideal generated by all monomials { m1, … , mr} of degree d which are invariants of Λ. Id is a monomial Togliatti system, provided r≤(d+n-1n-1), and in this case the projective toric variety Xd parameterized by (m1, … , mr) is called a GT-variety with group Λ. 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 Xd. 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.