Let $M$ be a rank-$2$ vector bundle on a $K3$ surface $S$ such that $h^0(M)\geq 2$ and $h^2(M)=0$. Is it possible that $h^2(\det M)>0$? If yes, can you give me some examples?



Determinant bundles of rank $2$ sheaves on $K3$ surfaces.
Choose a section $s \in H^0(M)$, let $Z$ be the zero locus of $s$, then we have a right exact sequence
$$
M \to \det M \to (\det M)\_{|Z} \to 0.
$$
Note that $\dim Z \le 1$, hence $H^2((\det M)\_{|Z}) = 0$.
On the other hand, the functor $H^2$ is right exact (since $S$ is a surface), hence $H^2(M)$ surjects onto $H^2(\det M)$, so $h^2(M) = 0$ implies $h^2(\det M) = 0$.


