A lower bound for the number of components of the moduli schemes of stable rank 2 vector bundles on projective 3-folds

Abstract:

Fix a smooth projective 3-fold $X$, $c_1$, $H\in \mathrm {Pic}(X)$ with $H$ ample, and $d\in \mathbf {Z}$. Assume the existence of integers $a,b$ with $a\not =0$ such that $ac_1$ is numerically equivalent to $bH$. Let $M(X,2,c_1,d,H)$ be the moduli scheme of $H$-stable rank 2 vector bundles, $E$, on $X$ with $c_1(E)=c_1$ and $c_2(E)\cdot H=d$. Let $m(X,2,c_1,d,H)$ be the number of its irreducible components. Then $\limsup _{d\rightarrow \infty }m(X,2,c_1,d,H)= +\infty$.