On the classification of Togliatti systems

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 $I\subset k[x_0,\dots ,x_n]$ 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 $\mu (I)=7$.