Components of Gröbner strata in the Hilbert scheme of points

Details Components of Gröbner strata in the Hilbert scheme of points Components of Gröbner strata in the Hilbert scheme of points

2010-06-18

arxiv.org/abs/1006.3653v2

Mathematics

Algebraic Geometry

49 pages

Scientific paper

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 $\Hi^{\prec\Delta,0}_{S/k}$, 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 is 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.

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