Stable sheave moduli of rank $2$ with Chern classes $c_1=-1$, $c_2=2$, $c_3=0$ on $Q_3$

In this paper we consider the scheme $M_Q(2;-1,2,0)$ of stable torsion free sheaves of rank $2$ with Chern classes $c_1=-1$, $c_2=2$, $c_3=0$ on a smooth $3$-dimensional projective quadric $Q$. The manifold $M_Q(-1,2)$ of moduli bundles of rank $2$ with Chern classes $c_1=-1$, $c_2=2$ on $Q$ was studied by Ottaviani and Szurek in 1994. In 2007 the author described the closure $M_Q(-1,2)$ in the scheme $M_Q(2;-1,2,0)$. In this paper we prove that in $M_Q(2;-1,2,0)$ there exists a unique irreducible component different from $\overline{M_Q(-1,2)}$ which is a rational variety of dimension $10$.