Double nested Hilbert schemes and reverse plane partitions

The Hilbert scheme Hilb X of n points on a quasi-projective variety X is a geometrical object introduced by Grothendieck and it has a prominent rôle in many areas of algebraic geometry. Recently, many variants of Hilb X have been introduced. My talk will focus on the double nested Hilbert scheme of points on X defined by S. Monavari. Specifically, I will explain how, when X is a smooth irreducible curve, its geometry is influenced by the combinatorics of reverse plane partitions and exhibits several pathologies. This is a joint project with Lella, Monavari, Ricolfi, Sammartano.