Gaps in the number of generators of monomial Togliatti systems
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.
scientific article; zbMATH DE number 6994952
scientific article