Paper detail

Components of Gröbner strata in the Hilbert scheme of points

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 $Δ$, and its open subscheme $\Hi^{\precΔ,\et}_{S/k}$, the moduli space of families of $#Δ$ points whose attached ideal has the standard set $Δ$. 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 $\Hi^{\precΔ}_{S/k}$. Our results prove a version of a conjecture by Bernd Sturmfels.