Togliatti systems and Galois coverings

Abstract

We study the homogeneous artinian ideals of the polynomial ring K[x, y, z] generated by the homogenous polynomials of degree d which are invariant under an action of the cyclic group Z/dZ, for any d ≥ 3. We prove that they are all monomial Togliatti systems, and that they are minimal if the action is defined by a diagonal matrix having on the diagonal (1, e, e a ), where e is a primitive d-th root of the unity. We get a complete description when d is prime or a power of a prime. We also establish the relation of these systems with linear Ceva configurations.Acknowledgments: The first author is member of INdAM -GNSAGA and is supported by PRIN "Geometria delle varietà algebriche" and by FRA, Fondi di Ricerca di Ateneo, Università di Trieste. The second author was partially supported by MTM2013-45075-P.

Circulant matrices and Galois-Togliatti systems

“…To determine the minimality of a GT-system is a subtle problem. In [17], in the case of three variables, the second and fourth authors proved that the ideal I d 0,a,b always satisfies the condition on the number of generators µ(I) ≤ d + 1, and conjectured the following, which we now prove using Theorem 2.3: Proof of the claim: We will prove that F d−1 generates K d−1 .…”

“…Our interest in this topic was originally motivated by its connections, exposed in [17], with a class of homogeneous ideals of a polynomial ring failing the Weak Lefschetz Property. In that context, the first question relevant to us was that of determining which monomials in the entries of a "generic" circulant matrix appear explicitly in the development of its determinant.…”

“…Note that the group generated by M a,b is the cyclic group of order d provided GCD(a, b, d) = 1, and that all actions of this group on R can be represented by a matrix of the form M a,b . In [17] it was proved that, if GCD(a, b, d) = 1, then the ideal I d a,b is a Togliatti system. This means that it fails the Weak Lefschetz Property in degree d − 1, i.e.…”

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The goal of this article is to compare the coefficients in the expansion of the permanent with those in the expansion of the determinant of a three-lines circulant matrix. As an application we solve a conjecture stated in [17] concerning the minimality of GT-systems.2010 Mathematics Subject Classification. 15B05, 15A05, 13E10, 14M25.

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Unexpected curves and Togliatti‐type surfaces

“…Note that there is a certain renaissance of interest in Togliatti systems and hypo-osculating varieties stemming partly from their connection to Lefschetz Properties and partly from the new research direction of unexpected hypersurfaces established in [2], see also [12], [4], [11].…”