TITLE: Sheaves without global sections
QUESTION [15 upvotes]: The line bundle $O(-1)$ on a projective space or $O(-\rho)$ on a flag variety has a property that all its cohomology vanish. Is there a story behind such sheaves?
Here are more precise questions. Let $X$ be a smooth complex projective surface (say, a nice one like Del Pezzo or K3). Does there always exist a coherent locally free sheaf $M$ whose derived global sections vanish? Can one describe all such sheaves? Is there a coarse moduli space of such sheaves?

REPLY [6 votes]: On a K3-surface if a line bundle $\mathcal L$ has vanishing cohomology, then
$\mathcal L^2=-4$ by the Riemann-Roch formula. Conversely, if $\mathcal L^2=-4$
and neither $\mathcal L$ nor its inverse has global sections, then also the
first cohomology vanishes.
If $X$ is a projective K3-surface with $\mathrm{Pic}(X)=\mathbb Z$, then
$\mathcal L^2\geq0$ for all line bundles and hence no line bundle has all its
cohomology vanishing. On the other hand there are (plenty of) K3-surfaces with
an ample line bundle $\mathcal M$ and a line bundle $\mathcal L$ with $\mathcal
M\cdot\mathcal L=0$ and $\mathcal L^2=-4$. Then neither $\mathcal L$ nor its
inverse has global sections (as otherwise we would get a strictly effective
divisor with zero intersection with $\mathcal M$). Thus for line bundles at
least the problem restricted to K3-surfaces depends very much on the surface.
As for the idea of consider the cokernel $B$ of $\mathcal O_X\rightarrow
F_\ast\mathcal O_X$. We have that the cohomology of $B$ vanishes if $X$ is
ordinary. The general complete intersection is ordinary (by a theorem of
Deligne) so there are plenty of those. Raynaud has also shown that if $X$ is a
curve then $B\bigotimes \mathcal L$ has vanishing cohomology for the general
line bundle $\mathcal L$ of degree $0$.