Smooth monomial Togliatti systems of cubics

The goal of this paper is to prove the conjecture stated in [6], extending and correcting a previous conjecture of Ilardi [5], and classify smooth minimal monomial Togliatti systems of cubics in any dimension.More precisely, we classify all minimal monomial artinian ideals I⊂k[x0,⋯,xn]I⊂k[x0,⋯,xn] generated by cubics, failing the weak Lefschetz property and whose apolar cubic system I−1I−1 defines a smooth toric variety. Equivalently, we classify all minimal monomial artinian ideals I⊂k[x0,⋯,xn]I⊂k[x0,⋯,xn] generated by cubics whose apolar cubic system I−1I−1 defines a smooth toric variety satisfying at least a Laplace equation of order 2. Our methods rely on combinatorial properties of monomial ideals.