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 $X$.
Categories: math.AG