Grothendieck–Plücker Images of Hilbert Schemes are Degenerate

We study the decompositions of Hilbert schemes induced by the Schubert cell decomposition of the Grassmannian variety and show that Hilbert schemes admit a stratification into locally closed subschemes along which the generic initial ideals remain the same. We give two applications. First, we give completely geometric proofs of the existence of the generic initial ideals and of their Borel fixed properties. Second, we prove that when a Hilbert scheme of non-constant Hilbert polynomial is embedded by the Grothendieck–Plücker embedding of a high enough degree, it must be degenerate.

Universal Gröbner Bases and Cartwright–Sturmfels Ideals

The main theoretical contribution of the paper is the description of two classes of multigraded ideals named after Cartwright and Sturmfels and the study of their surprising properties. Among other things we prove that these classes of ideals have very special multigraded generic initial ideals and are closed under several operations including arbitrary multigraded hyperplane sections. As a main application we describe the universal Gröbner basis of the ideal of maximal minors and the ideal of 2-minors of a multigraded matrix of linear forms generalizing earlier results of various authors including Bernstein, Sturmfels, Zelevinsky, and Boocher.