Gröbner scheme in the Hilbert scheme and complete intersection monomial ideals

Abstract

Let $k$ be a commutative ring and $S=k[x_0,\dots ,x_n]$ be a polynomial ring over $k$ with a monomial order. For any monomial ideal $J$, there exists an affine $k$-scheme of finite type, called Gröbner scheme, which parameterizes all homogeneous reduced Gröbner bases in $S$ whose initial ideal is $J$. Here we functorially show that the Gröbner scheme is a locally closed subscheme of the Hilbert scheme if $J$ is a saturated ideal. In the process, we also show that the Gröbner scheme consists of complete intersections if $J$ defines a complete intersection.

The contents are completely included in "On the functoriality of marked families"[Paolo Lella, Margherita Roggero]

References in corpus (3)

• Ideals with an assigned initial ideal
• Components of Gröbner strata in the Hilbert scheme of points
• Construction of the moduli space of reduced Groebner bases