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Title: On the arithmetic Cohen-Macaulayness of varieties parameterized by Togliatti systems
Author: Colarte Gómez, Liena
Mezzetti, Emilia
Miró-Roig, Rosa M. (Rosa Maria)
Keywords: Varietats algebraiques
Anells commutatius
Mòduls de Cohen-Macaulay
Grups algebraics diferencials
Algebraic varieties
Commutative rings
Cohen-Macaulay modules
Differential algebraic groups
Issue Date: 6-Jan-2021
Publisher: Springer Verlag
Abstract: Given any diagonal cyclic subgroup $\Lambda \subset G L(n+1, k)$ of order $d$, let $I_d \subset k\left[x_0, \ldots, x_n\right]$ be the ideal generated by all monomials $\left\{m_1, \ldots, m_r\right\}$ of degree $d$ which are invariants of $\Lambda . I_d$ is a monomial Togliatti system, provided $r \leq\left(\begin{array}{c}d+n-1 \\ n-1\end{array}\right)$, and in this case the projective toric variety $X_d$ parameterized by $\left(m_1, \ldots, m_r\right)$ is called a $G T$-variety with group $\Lambda$. We prove that all these $G T$-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.
Note: Versió postprint del document publicat a:
It is part of: Annali di Matematica Pura ed Applicata, 2021, vol. 200, p. 1757-1780
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ISSN: 0373-3114
Appears in Collections:Articles publicats en revistes (Matemàtiques i Informàtica)

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