A moduli space of stable sheaves on a cubic threefold

In this paper, we prove that the moduli space ${\bar{M}}_X(\nu )$ 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 ${\bar{M}}_X(\nu )$, which also yields that ${\bar{M}}_X(\nu )$ is a birational model of the moduli space of smooth quartic rational curves in $X$.