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Smooth monomial Togliatti systems of cubics

2016; Elsevier BV; Volume: 143; Linguagem: Inglês

10.1016/j.jcta.2016.05.004

ISSN

1096-0899

Autores

Mateusz Michałek, Rosa M. Miró-Roig,

Tópico(s)

Algebraic Geometry and Number Theory

Resumo

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] 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⊂k[x0,⋯,xn] 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.

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