Computable Białynicki-Birula decomposition of the Hilbert scheme

Abstract

We call the scheme parameterizing homogeneous ideals with fixed initial ideal the Gröbner scheme. We introduce a Białynicki-Birula decomposition of the Hilbert scheme ${\mathrm{Hilb}}_n^P$ for any Hilbert polynomial $P$ such that the cells are the Gröbner schemes in set-theoretically. Then we obtain a computable homology formula for smooth Hilbert schemes. As a corollary of our argument, we show that the Gröbner scheme for a monomial ideal defining a smooth point in the Hilbert scheme is smooth.

15 pages