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The Bialynicki-Birula strata on the Hilbert scheme $H^n(\mathbb{A}^d)$ aresmooth in dimension $d=2$. We prove that there is a schematic structure inhigher dimensions, the Bialynicki-Birula scheme, which is natural in the sensethat it represents a functor. Let $\rho_i:H^n(\mathbb{A}^d)\rightarrow {\rmSym}^n(\mathbb{A}^1)$ be the Hilbert-Chow morphism of the ${i}^{th}$coordinate. We prove that a Bialynicki-Birula scheme associated with an actionof a torus $T$ is schematically included in the fiber $\rho_i^{-1}(0)$ if the${i}^{th}$ weight of $T$ is non-positive. We prove that the monic functorsparametrizing families of ideals with a prescribed initial ideal arerepresentable.

Bialynicki-Birula schemes in higher dimensional Hilbert schemes of points and monic functors