Computable Bia{\l}ynicki-Birula decomposition of the Hilbert scheme

We call the scheme parameterizing homogeneous ideals with fixed initial ideal the Gr"obner scheme. We introduce a Bia{\l}ynicki-Birula decomposition of the Hilbert scheme ${\mathrm{H}\mathrm{i}\mathrm{l}\mathrm{b}}_n^P$ for any Hilbert polynomial $P$ such that the cells are the Gr"obner 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 Gr"obner scheme for a monomial ideal defining a smooth point in the Hilbert scheme is smooth.