Components of Gröbner strata in the Hilbert scheme of points

Abstract

We fix the lexicographic order $\prec$ on the polynomial ring $S=k[x_1,...,x_n]$ over a ring $k$. We define $\Hi^{\precΔ}_{S/k}$, the moduli space of reduced Gröbner bases with a given finite standard set $\Delta$, and its open subscheme $\Hi^{\precΔ,\et}_{S/k}$, the moduli space of families of $#Δ$ 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 $\mathrm{Spec}k$; and we determine its relative dimension over $\mathrm{Spec}k$. We show that analogous statements do not hold for the scheme $\Hi^{\precΔ}_{S/k}$. Our results prove a version of a conjecture by Bernd Sturmfels.

49 pages

References in corpus (2)

• Hilbert schemes of 8 points
• A polyhedral approach to computing border bases

Cited by in corpus (2)

• Murphy's Law for Hilbert function strata in the Hilbert scheme of points
• Gröbner scheme in the Hilbert scheme and complete intersection monomial ideals