## Geometric collections and Castelnuovo-Mumford regularity.(English)Zbl 1130.14017

The authors extend the definition of Castelnuovo-Mumford regularity of coherent sheaves on a projective space to coherent sheaves on a smooth projective complex variety endowed with a geometric collection $$\sigma$$. Moreover they carefully study such a regularity in the case of a quadric hypersurface $$Q_n \subset\mathbb P^{n+1}$$ for $$n$$ odd.
A geometric collection $$\sigma$$ on a smooth projective variety of dimension $$n$$ is defined to be a full exceptional collection of length $$n+1$$ of coherent sheaves. We know that such collections exist in the case $$X$$ is a projective space, a smooth quadric hypersurface $$Q_n$$ for $$n$$ odd or a Fano threefold with Picard number 1 and trivial intermediate Jacobian. If $$\sigma = \{ E_0, \dots, E_n \}$$ is a geometric collection on $$X$$, then a coherent sheaf $$F$$ is $$m$$-regular with respect to $$\sigma$$ if the groups $$\text{Ext}^q(R^{(-p)}E_{-m+p},F)$$ vanish for $$q>0$$ and $$-n \leq p \leq 0$$. The regularity of a coherent sheaf $$F$$ is the smallest integer $$m$$ for which $$F$$ is $$m$$-regular. If $$X$$ is $$\mathbb P^n$$ and $$\sigma = \{ {\mathcal O}, {\mathcal O}(1), \dots, {\mathcal O}(n)\}$$, then $$F$$ is $$m$$-regular if and only if it is $$m$$-regular in the sense of Castelnuovo-Mumford. Moreover, many interesting properties of Castelnuovo-Mumford regularity remain valid for this generalization.
Finally, considering a smooth projective quadric hypersurface $$Q_n$$, for $$n$$ odd, embedded in $$\mathbb P^{n+1}$$ by $$i$$, the authors relate the regularity of a coherent sheaf $$F$$ on $$Q_n$$ to the Castelnuovo-Mumford regularity of $$i_* F$$ by giving optimal extimations. An exact calculation is performed for the elements of the geometric collection and of its dual.

### MSC:

 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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### References:

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