On the classification of Togliatti systems

TL;DR

This paper classifies smooth monomial Togliatti systems with a specific number of generators for higher degrees and dimensions, extending previous classifications to new open cases.

Contribution

It provides a complete classification of smooth monomial Togliatti systems with =2n+3 for degree d4 and n2, and degree d6 for three variables with =7.

Findings

Classified smooth monomial Togliatti systems with =2n+3 for d4 and n2.

Classified monomial Togliatti systems in three variables with =7 for d6.

Abstract

In [MeMR], Mezzetti and Mir\'{o}-Roig proved that the minimal number of generators $\mu (I)$ of a minimal (smooth) monomial Togliatti system $I\subset k[x_{0},\dotsc,x_{n}]$ satisfies $2n+1\le \mu(I)\le \binom{n+d-1}{n-1}$ and they classify all smooth minimal monomial Togliatti systems $I\subset k[x_{0},\dotsc,x_{n}]$ with $2n+1\le \mu(I)\le 2n+2$. In this paper, we address the first open case. We classify all smooth monomial Togliatti systems $I\subset k[x_{0},\dotsc,x_{n}]$ of forms of degree $d\ge 4$ with $\mu(I)=2n+3$ and $n\ge 2$ and all monomial Togliatti systems $I\subset k[x_0,x_1,x_2]$ of forms of degree $d\ge 6$ with $\mu(I)=7$.