On the classification of Togliatti systems

Abstract

In [MeMR], Mezzetti and Miró-Roig proved that the minimal number of generators $\mu (I)$ of a minimal (smooth) monomial Togliatti system $I\subset k[x_0,\dots ,x_n]$ satisfies $2n+1\le \mu (I)\le \left(\genfrac{}{}{0ex}{}{n+d-1}{n-1}\right)$ and they classify all smooth minimal monomial Togliatti systems $I\subset k[x_0,\dots ,x_n]$ 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 $I\subset k[x_0,\dots ,x_n]$ 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 $\mu (I)=7$.

arXiv commentTo appear in Communications in Algebra. arXiv admin note: text overlap with arXiv:1506.05914