Geodesic
Bialynicki-Birula schemes in higher dimensional Hilbert schemes of points and monic functors
Épijournal de Géométrie Algébrique, Tome 5 (2021)

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The Bialynicki-Birula strata on the Hilbert scheme $H^n(\mathbb{A}^d)$ are smooth in dimension $d=2$. We prove that there is a schematic structure in higher dimensions, the Bialynicki-Birula scheme, which is natural in the sense that it represents a functor. Let $\rho_i:H^n(\mathbb{A}^d)\rightarrow {\rm Sym}^n(\mathbb{A}^1)$ be the Hilbert-Chow morphism of the ${i}^{th}$ coordinate. We prove that a Bialynicki-Birula scheme associated with an action of 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 functors parametrizing families of ideals with a prescribed initial ideal are representable. 
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     author = {Evain, Laurent and Lederer, Mathias},
     title = {Bialynicki-Birula schemes in higher dimensional {Hilbert} schemes of points and monic functors},
     journal = {\'Epijournal de G\'eom\'etrie Alg\'ebrique},
     publisher = {mathdoc},
     volume = {5},
     year = {2021},
     doi = {10.46298/epiga.2021.volume5.5618},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/epiga.2021.volume5.5618/}
}
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Evain, Laurent; Lederer, Mathias. Bialynicki-Birula schemes in higher dimensional Hilbert schemes of points and monic functors. Épijournal de Géométrie Algébrique, Tome 5 (2021). doi: 10.46298/epiga.2021.volume5.5618

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