Smooth components on special iterated Hilbert schemes

TL;DR

This paper uses derived categorical methods to analyze the geometry of special iterated Hilbert schemes on a smooth projective surface with specific properties, revealing a smooth component isomorphic to the surface.

Contribution

It introduces a novel approach employing derived categories to study the geometry of iterated Hilbert schemes, identifying smooth components isomorphic to the original surface.

Findings

Identification of a smooth connected component isomorphic to S

Application of derived categorical techniques to geometric problems

Advancement in understanding the structure of iterated Hilbert schemes

Abstract

Let $S$ be a smooth projective surface with $p_g=q=0$. We show how to use derived categorical methods to study the geometry of certain special iterated Hilbert schemes associated to $S$ by showing that they contain a smooth connected component isomorphic to $S$.