Cellular decompositions for nested Hilbert schemes of points
Let \(X\) be a smooth complex quasiprojective variety. The Hilbert scheme \(\text{Hilb}^nX\) parametrizes zero-dimensional subschemes of length \(n\) of \(X\). If \(X\) is projective, \(\text{Hilb}^nX\) is a natural compactification of the parameter space of \(n\) distinct points of \(X\) and when \(X\) is a surface, it is, in fact, a smooth compactification. However, when the dimension of \(X\) is greater than two, the ``Hilbert scheme of points'' of length \(n\) on \(X\) is no longer smooth unless \(n\) is very small. Just as flag varieties are a natural generalization of grassmannians, it is natural to consider ``nested'' Hilbert schemes on \(X\) parametrizing nests \(Z_1 \subset Z_2 \subset \cdots \subset Z_m\) of zero-dimensional subschemes of \(X\) of given length. By adapting the method of Ellingsrud and Strømme, we classify the nested Hilbert schemes on \(X\) which are smooth and obtain cellular decompositions for the smooth nested Hilbert schemes on affine and projective space as well as for the corresponding punctual nested Hilbert schemes. One deduces immediately that the Chow group and the (Borel-Moore) homology group of each of these spaces are isomorphic; odd homology vanishes and the \(2i\)-th homology group is the free abelian group generated by the classes of the closures of the \(i\)-cells. The number of cells of each dimension and hence the Betti numbers of the punctual nested Hilbert schemes are calculated here; the number of cells of each dimension of the smooth nested Hilbert schemes on affine and projective space can be read off formulae which are obtained in the sequels to this paper [see \textit{J. Cheah}, Math. Z. 227, No. 3, 479-504 (1998; Zbl 0890.14003) and J. Algebr. Geom. 5, No. 3, 479-511 (1996; Zbl 0889.14001)]. In fact, in the sequels, we use the cellular decompositions for the punctual nested Hilbert schemes obtained here to calculate the virtual Hodge polynomials of all the smooth nested Hilbert schemes on \(X\) (and of related varieties such as the universal families over some of these nested Hilbert schemes) in terms of the virtual Hodge polynomial of \(X\). Finally, following Göttsche, we study the Hilbert function strata of the punctual nested Hilbert schemes and cellular decompositions for these spaces.
scientific article; zbMATH DE number 1180204