A moduli space of stable sheaves on a cubic threefold

Abstract

In this paper, we prove that the moduli space ${\overline{M}}_X(\hat{I}½)$ of $H$-Gieseker semistable sheaves on a smooth cubic threefold $X$ with Chern character $\hat{I}½=(4,-H,-\frac{5}{6}{H}^2,\frac{1}{6}{H}^3)$ is non-empty, smooth and irreducible of dimension $8$. Moreover, we give a set-theoretic description of the moduli space ${\overline{M}}_X(\hat{I}½)$, which also yields that ${\overline{M}}_X(\hat{I}½)$ is a birational model of the moduli space of smooth quartic rational curves in $X$.

32 pages. New result added. Comments are very welcome