The fat locus of Hilbert schemes of points

Coppens, Marc

doi:10.1090/S0002-9939-1993-1145416-X

The fat locus of Hilbert schemes of points
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Proc. Amer. Math. Soc. 118 (1993), 777-783 Request permission

Abstract:

Let $X$ be a smooth projective variety over an algebraically closed field $k$. Let ${\operatorname {Hilb} ^d}(X)$ be its Hilbert scheme of $0$-dimensional subschemes of $X$ of degree $d$. Let $[{\operatorname {Hilb} ^d}(X)](k)$ be the set of $k$-rational points. We prove that the subset of points of $[{\operatorname {Hilb} ^d}(X)](k)$ corresponding to fat points on $X$ is a locally closed subset with respect to the Zariski topology.

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