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

2021; Springer Science+Business Media; Volume: 200; Issue: 4 Linguagem: Inglês

10.1007/s10231-020-01058-2

1618-1891

Liena Colarte-Gómez, Emilia Mezzetti, Rosa M. Miró-Roig,

Algebraic Geometry and Number Theory

Given any diagonal cyclic subgroup $$\Lambda \subset \text {GL}(n+1,k)$$ of order d, let $$I_d\subset k[x_0,\ldots , x_n]$$ be the ideal generated by all monomials $$\{m_{1},\ldots , m_{r}\}$$ of degree d which are invariants of $$\Lambda$$ . $$I_d$$ is a monomial Togliatti system, provided $$r \le \left( {\begin{array}{c}d+n-1\\ n-1\end{array}}\right)$$ , and in this case the projective toric variety $$X_d$$ parameterized by $$(m_{1},\ldots , m_{r})$$ 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.