Components of Gr\"obner strata in the Hilbert scheme of points

TL;DR

This paper studies the structure of certain moduli spaces related to Gr"obner bases and standard sets in polynomial rings, proving irreducibility, equidimensionality, and confirming a conjecture by Sturmfels.

Contribution

It characterizes the irreducible components, connectedness, and dimensions of the moduli space of point configurations with fixed Gr"obner bases, confirming a conjecture by Sturmfels.

Findings

The moduli space of point configurations is irreducible and equidimensional.

The space has a well-defined relative dimension over the base ring.

Analogous properties do not hold for the larger scheme of all Gr"obner bases.

Abstract

We fix the lexicographic order $\prec$ on the polynomial ring $S=k[x_{1},...,x_{n}]$ over a ring $k$. We define $\Hi^{\prec\Delta}_{S/k}$, the moduli space of reduced Gr\"obner bases with a given finite standard set $\Delta$, and its open subscheme $\Hi^{\prec\Delta,\et}_{S/k}$, the moduli space of families of $#\Delta$ points whose attached ideal has the standard set $\Delta$. We determine the number of irreducible and connected components of the latter scheme; we show that it is equidimensional over ${\rm Spec}\,k$; and we determine its relative dimension over ${\rm Spec} k$. We show that analogous statements do not hold for the scheme $\Hi^{\prec\Delta}_{S/k}$. Our results prove a version of a conjecture by Bernd Sturmfels.