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 ź k x 0 , ź , x n generated by cubics, failing the weak Lefschetz property and whose apolar cubic system I - 1 defines a smooth toric variety. Equivalently, we classify all minimal monomial artinian ideals I ź k x 0 , ź , x n generated by cubics whose apolar cubic system I - 1 defines a smooth toric variety satisfying at least a Laplace equation of order 2. Our methods rely on combinatorial properties of monomial ideals.