Gaps in the number of generators of monomial Togliatti systems
type: Item
instance of: scholarly article
title: Gaps in the number of generators of monomial Togliatti systems
zbMATH Open document ID: 1455.13033
author: Charles Almeida
author: Aline V. Andrade
author: Rosa M. Miró-Roig
published in: Journal of Pure and Applied Algebra
publication date: 2018-12-18T00:00:00Z
review text: A Togliatti system is a homogeneous ideal \(I\) of the polynomial ring \(R=k[x_0,\ldots,x_n]\), minimally generated by \(\mu(I)\) polynomials all of the same degree \(d\), such that the quotient ring \(R/I\) is artinian and \(I\) fails the weak Lefschetz property in degree \(d-1\), i.e., there exists a linear form \(L\) such that the map of multiplication by \(L\) from \((R/I)_{d-1}\) to \((R/I)_d\) is not of maximal rank. Togliatti systems have been introduced in the article [\textit{E. Mezzetti} et al., Can. J. Math. 65, No. 3, 634--654 (2013; Zbl 1271.13036)], where it has been shown that they are connected via apolarity to projective varieties satisfying Laplace equations. A Togliatti system is minimal if there is no proper subset of its set of generators defining again a monomial Togliatti system. Aim of the article under review is to find which numbers can appear as \(\mu(I_{d,n})\) for a minimal monomial Togliatti system \(I_{d,n}\), with given \(n\) and \(d\). It is known that \(2n+1\leq\mu(I_{d,n})\leq \binom{d+n-1}{n-1}\) (see [\textit{E. Mezzetti} and \textit{R. M. Miró-Roig}, Ann. Mat. Pura Appl. (4) 195, No. 6, 2077--2098 (2016; Zbl 1357.13024)]). Here the authors prove that there are gaps in the above interval, precisely for any \(n \geq 4\) and \(d \geq 3\), the integer values between \(2n + 3\) and \(3n-1\) cannot be realized as the number of minimal generators of a minimal monomial Togliatti system. On the other hand, in the case \(n=4\), for any \(d\geq 3\), they prove that all the integer numbers in the interval \([9,\binom{d+3}{3}]\) except \(11\), can be realized. The proofs consist in an accurate analysis of the configuration of the finite subsets in \(\mathbb Z_{n+1}\) corresponding to monomials of degree \(d\) in the Macaulay inverse system, relying on the combinatorial characterization of monomial Togliatti systems. The article is completed and illustrated by many examples and counterexamples.
reviewed by: Emilia Mezzetti
Mathematics Subject Classification ID: 13E10
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 14N05
Mathematics Subject Classification ID: 14N15
Mathematics Subject Classification ID: 14N07
zbMATH DE Number: 6994952
zbMATH Keywords: monomial ideal
zbMATH Keywords: Togliatti
zbMATH Keywords: Laplace equation
zbMATH Keywords: toric variety
zbMATH Keywords: weak Lefschetz property
describes a project that uses: Macaulay2
MaRDI profile type: Publication
full work available at URL: j.jpaa.2018.07.009
OpenAlex ID: W2884666691
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Wikidata QID: Q129489368
DOI: 10.1016/J.JPAA.2018.07.009
description: scientific article; zbMATH DE number 6994952