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 $\mathrm{Hilb}^{P}_n$ 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.