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 $\mathbb{Z}/d\mathbb{Z}$, for any $d\ge 3$. 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.

28 pages, 1 figure; final version published in Journal of Algebra

References in corpus (2)

• Abstract configurations in algebraic geometry
• On the Matrix-Element Expansion of a Circulant Determinant

Cited by in corpus (3)

• On the arithmetic Cohen-Macaulayness of varieties parameterized by Togliatti systems
• Lefschetz property and powers of linear forms in $\mathbb{K}[x,y,z]$
• Osculating behavior of Kummer surface in ${\mathbb{P}}^5$