2-step ideals and the reducibility of Hilbert schemes of points

Abstract: Inspired by two famous examples by Iarrobino showing that the Hilbert scheme of d points in the affine 3-space is reducible for d sufficiently large, we study homogenous ideals in a polynomial ring with n variables with a small difference between the initial degree and the regularity. First, we describe how to construct families of such ideals with a given Hilbert function. Second, we use these families to certify that several Hilbert schemes and nested Hilbert schemes are reducible. For n greater than 3, we discover several new reduced irreducible components and some new generically non-reduced irreducible component. This is a joint project with Franco Giovenzana, Luca Giovenzana and Michele Graffeo.

algebraic geometry

Audience: researchers in the topic

SISSA algebraic geometry seminar