Let $X_d \subset \mathbb{P}^{n+1}$ be a smooth hypersurface of degree $d$. How one can describe all coherent sheaves on $X_d$ with no cohomology i.e. $$ H^i(X_d, F) \cong 0, $$ for all $i \in \mathbb{Z}$. For example, is it possible to describe the corresponding graded modules? More generally, how to describe all objects with such property (replacing cohomology by hypercohomology) in the bounded derived category?
The most interesting case for me is a cubic surface in $\mathbb{P}^3$.
A cubic surface (at least over $\mathbb{C}$) has a full exceptional collection. In fact many full exceptional collections. One of them consists of sheaves $$ (O_{e_1}(-1),\dots,O_{e_6}(-1),O(-2\ell),O(-\ell),O) $$ in the standard notation. Consequently, the objects of the derived category whose hypercohomology vanish are just objects of the subcategory generated by the subcollection $$ (O_{e_1}(-1),\dots,O_{e_6}(-1),O(-2\ell),O(-\ell)). $$