A note on the Kuznetsov component of the Veronese double cone

TL;DR

This paper investigates the moduli spaces of complexes in the derived category of a Veronese double cone, revealing their structure and components, and connecting stable sheaves, complexes, and stable pairs.

Contribution

It characterizes the moduli spaces of stable complexes in the Kuznetsov component of a Veronese double cone, identifying their components and relations to stable pairs.

Findings

Moduli space of Gieseker stable sheaves has two components.

Moduli space of stable complexes also has two components.

One component parametrizes ideal sheaves of lines, appearing in both spaces.

Abstract

This note describes moduli spaces of complexes in the derived category of a Veronese double cone $Y$. Focusing on objects with the same class $\kappa_1$ as ideal sheaves of lines, we describe the moduli space of Gieseker stable sheaves and show that it has two components. Then, we study the moduli space of stable complexes in the Kuznetsov component of $Y$ of the same class, which also has two components. One parametrizes ideal sheaves of lines and it appears in both moduli spaces. The other components are not directly related by a wall-crossing: we show this by describing an intermediate moduli space of complexes as a space of stable pairs in the sense of Pandharipande and Thomas.