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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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