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

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...

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Autores: Colarte Gómez, Liena, Mezzetti, Emilia, Miró-Roig, Rosa M. (Rosa Maria)
Tipo de recurso: artículo
Estado:Versión aceptada para publicación
Fecha de publicación:2021
País:España
Institución:Universidad de Barcelona
Repositorio:Dipòsit Digital de la UB
OAI Identifier:oai:diposit.ub.edu:2445/190625
Acceso en línea:https://hdl.handle.net/2445/190625
Access Level:acceso abierto
Palabra clave:Varietats algebraiques
Anells commutatius
Mòduls de Cohen-Macaulay
Grups algebraics diferencials
Algebraic varieties
Commutative rings
Cohen-Macaulay modules
Differential algebraic groups
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spelling On the arithmetic Cohen-Macaulayness of varieties parameterized by Togliatti systemsColarte Gómez, LienaMezzetti, EmiliaMiró-Roig, Rosa M. (Rosa Maria)Varietats algebraiquesAnells commutatiusMòduls de Cohen-MacaulayGrups algebraics diferencialsAlgebraic varietiesCommutative ringsCohen-Macaulay modulesDifferential algebraic groupsGiven 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.Springer Verlag2021info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersionapplication/pdfhttps://hdl.handle.net/2445/190625Articles publicats en revistes (Matemàtiques i Informàtica)reponame:Dipòsit Digital de la UBinstname:Universidad de BarcelonaInglésVersió postprint del document publicat a: https://doi.org/10.1007/s10231-020-01058-2Annali di Matematica Pura ed Applicata, 2021, vol. 200, p. 1757-1780https://doi.org/10.1007/s10231-020-01058-2(c) Springer Verlag, 2021info:eu-repo/semantics/openAccessoai:diposit.ub.edu:2445/1906252026-05-27T06:46:51Z
dc.title.none.fl_str_mv On the arithmetic Cohen-Macaulayness of varieties parameterized by Togliatti systems
title On the arithmetic Cohen-Macaulayness of varieties parameterized by Togliatti systems
spellingShingle On the arithmetic Cohen-Macaulayness of varieties parameterized by Togliatti systems
Colarte Gómez, Liena
Varietats algebraiques
Anells commutatius
Mòduls de Cohen-Macaulay
Grups algebraics diferencials
Algebraic varieties
Commutative rings
Cohen-Macaulay modules
Differential algebraic groups
title_short On the arithmetic Cohen-Macaulayness of varieties parameterized by Togliatti systems
title_full On the arithmetic Cohen-Macaulayness of varieties parameterized by Togliatti systems
title_fullStr On the arithmetic Cohen-Macaulayness of varieties parameterized by Togliatti systems
title_full_unstemmed On the arithmetic Cohen-Macaulayness of varieties parameterized by Togliatti systems
title_sort On the arithmetic Cohen-Macaulayness of varieties parameterized by Togliatti systems
dc.creator.none.fl_str_mv Colarte Gómez, Liena
Mezzetti, Emilia
Miró-Roig, Rosa M. (Rosa Maria)
author Colarte Gómez, Liena
author_facet Colarte Gómez, Liena
Mezzetti, Emilia
Miró-Roig, Rosa M. (Rosa Maria)
author_role author
author2 Mezzetti, Emilia
Miró-Roig, Rosa M. (Rosa Maria)
author2_role author
author
dc.subject.none.fl_str_mv Varietats algebraiques
Anells commutatius
Mòduls de Cohen-Macaulay
Grups algebraics diferencials
Algebraic varieties
Commutative rings
Cohen-Macaulay modules
Differential algebraic groups
topic Varietats algebraiques
Anells commutatius
Mòduls de Cohen-Macaulay
Grups algebraics diferencials
Algebraic varieties
Commutative rings
Cohen-Macaulay modules
Differential algebraic groups
description 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.
publishDate 2021
dc.date.none.fl_str_mv 2021
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/acceptedVersion
format article
status_str acceptedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/2445/190625
url https://hdl.handle.net/2445/190625
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Versió postprint del document publicat a: https://doi.org/10.1007/s10231-020-01058-2
Annali di Matematica Pura ed Applicata, 2021, vol. 200, p. 1757-1780
https://doi.org/10.1007/s10231-020-01058-2
dc.rights.none.fl_str_mv (c) Springer Verlag, 2021
info:eu-repo/semantics/openAccess
rights_invalid_str_mv (c) Springer Verlag, 2021
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Springer Verlag
publisher.none.fl_str_mv Springer Verlag
dc.source.none.fl_str_mv Articles publicats en revistes (Matemàtiques i Informàtica)
reponame:Dipòsit Digital de la UB
instname:Universidad de Barcelona
instname_str Universidad de Barcelona
reponame_str Dipòsit Digital de la UB
collection Dipòsit Digital de la UB
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,301603

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