A moduli space of stable sheaves on a cubic threefold

TL;DR

This paper establishes the existence, smoothness, and irreducibility of a specific moduli space of semistable sheaves on a cubic threefold, linking it to the space of smooth quartic rational curves.

Contribution

It proves the non-emptiness, smoothness, and irreducibility of a particular moduli space of sheaves on a cubic threefold and describes its relation to rational curves.

Findings

The moduli space is non-empty, smooth, and irreducible of dimension 8.

Provides a set-theoretic description of the moduli space.

Shows the moduli space is birational to the space of smooth quartic rational curves.

Abstract

In this paper, we prove that the moduli space $\overline{M}_{X}(\nu)$ of $H$-Gieseker semistable sheaves on a smooth cubic threefold $X$ with Chern character $\nu=(4,-H,-\frac{5}{6}H^{2},\frac{1}{6}H^{3})$ is non-empty, smooth and irreducible of dimension $8$. Moreover, we give a set-theoretic description of the moduli space $\overline{M}_{X}(\nu)$, which also yields that $\overline{M}_{X}(\nu)$ is a birational model of the moduli space of smooth quartic rational curves in $X$.