A remark by Daniel Ferrand on bundles on Fano threefolds

Abstract

Let $X$ be a Fano threefold with index $i_X$ and fundamental line bundle ${\mathcal{O}}_X(h)$. We classify $\mu$-semistable rank two bundles $\mathcal{E}$ on $X$ with $c_1(\mathcal{E})=0$, $h^0(\mathcal{E})\ne 0$ and $h^1(\mathcal{E}(-\lceil \frac{i_X}{2}\rceil h))=0$.

v.2: improved main theorem by describing the multiple structure in the case $i_X=2$, thanks to suggestion of anonymous referee. Added Remarks 3.6, 5.3 and more examples. v3: minor improvements