Togliatti systems associated to the dihedral group and the weak Lefschetz property

TL;DR

This paper introduces a new family of non-monomial Togliatti systems derived from dihedral group invariants, analyzing their algebraic properties and minimal free resolutions of their associated surfaces.

Contribution

It presents the first examples of non-monomial Togliatti systems linked to dihedral groups and characterizes their algebraic and geometric properties.

Findings

The associated $GT$-surfaces are arithmetically Cohen-Macaulay.

Their homogeneous ideals are generated by quadrics.

A minimal free resolution of the ideals is provided.

Abstract

In this note, we study Togliatti systems generated by invariants of the dihedral group $D_{2d}$ acting on $k[x_{0},x_{1},x_{2}]$. This leads to the first family of non monomial Togliatti systems, which we call $GT-$systems with group $D_{2d}$. We study their associated varieties $S_{D_{2d}}$, called $GT-$surfaces with group $D_{2d}$. We prove that they are arithmetically Cohen-Macaulay surfaces whose homogeneous ideal, $I(S_{D_{2d}})$, is minimally generated by quadrics and we find a minimal free resolution of $I(S_{D_{2d}})$.