A projective algebraic variety $X$ over a field $k$ is a Fano variety if its anticanonical divisor is ample (an ample $Q$-Cartier divisor if one means to have some singular varieties under consideration). An algebraic coherent sheaf $F$ on $X$ is called rigid if a tangent space of a deformation functor is zero which means that ${\rm Ext}^i(F,F)=0$.
It happens to be that in order to study rigid sheavs on Fano varieties one whould look closely on a particular case of exceptional sheaves. A sheaf $E$ is called exceptional if ${\rm Ext}^i(F,F)=0$ for $i>0$ and ${\rm Hom}(E,E)=k$. The reason for this is that when $X$ is smooth and $\dim X=2$ then any rigid sheaf $F$ on $X$ is a direct sum of exceptional sheaves $$F=\bigopus n_iE_i.$$ (multiplicities $n_i$ are possible). This is proved by S. Kuleshov in general ([5]) and by J.-M. Drezet for $X=P^2$ ([1]). Not any set of exceptional sheaves are good here but the members $\{E_i\}$ of a sum should meet some conditions. Appropriate sets $\{E_i\}$ are related to so called exceptional systems.
The system $(E_1,\cdots, E_m)$ of exceptional sheaves is called exceptional system if $${\rm Ext}^i(E_\alpha,E_\beta)=0\ {\rm for}\ i\geq 0}\ {\rm and}\ \alpha>\beta.$$ This is quite a general definition and it is especially good for a derived category because a braid group action exists on the set of exceptional systems of a given length $m$ in a derived category ([6]).
On the other hand exceptional systems appears to be behind Beilinson type spectral sequences theorems for sheaves on projective spaces ([8]), quadrics, grassmannians, flag varieties et al. ([2]), and their algebraic generalizations ([3]).
The braid group action stated above was found first as fairly constructive way to produce exceptional sheaves and exceptional systems. It is because for many varieties among Fano ones the action provides us with transformations which operator with sheaves themselves besides objects of a derived category ([6], [8]). Those transformations were called mutations and their properties were studied under the name of a helix theory.
On the other hand it was realized that there are reach and interesting algebraic structures involved with exceptional sheaves ([3], [4], [7]) which could be interesting in a much broader context that a study of rigid sheaves on a variety and which appear to be a good source of conjectures.
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