Hilbert schemes with two Borel-fixed points in arbitrary characteristic

Borel Open Covering of Hilbert Schemes

Let p(t) be an admissible Hilbert polynomial in P of degree d and Gotzmann number r. It is well known that Hilb p(t) can be seen as a closed subscheme of the Grasmannian G(N, s), where N = ` n+r n ́ and s = N − p(r), hence, by Plücker embedding, it becomes a closed subset of a suitable projective space. Let us denote by B the finite set of the Borel fixed ideals in k[X0, . . . ,Xn] generated by ...

Hilbert schemes of 8 points

The Hilbert scheme H n of n points in A contains an irreducible component R n which generically represents n distinct points in A. We show that when n is at most 8, the Hilbert scheme H n is reducible if and only if n = 8 and d ≥ 4. In the simplest case of reducibility, the component R 8 ⊂ H 8 is defined by a single explicit equation which serves as a criterion for deciding whether a given idea...

Hilbert Schemes of 8 Points in A

The Hilbert scheme H n of n points in A d contains an irreducible component Rdn which generically represents n distinct points in A. We show that when n is at most 8, the Hilbert scheme H n is reducible if and only if n = 8 and d ≥ 4. In the simplest case of reducibility, the component R 8 ⊂ H 8 is defined by a single explicit equation which serves as a criterion for deciding whether a given id...

Hilbert Schemes of Points in Projective Spaces

Segre Classes and Hilbert Schemes of Points

We prove a closed formula for the integrals of the top Segre classes of tautological bundles over the Hilbert schemes of points of a K3 surface X. We derive relations among the Segre classes via equivariant localization of the virtual fundamental classes of Quot schemes on X. The resulting recursions are then solved explicitly. The formula proves the K-trivial case of a conjecture of M. Lehn fr...