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Geometric collections and Castelnuovo-Mumford Regularity | | L. Costa
; R.M. Miró-Roig
; | | Date: |
20 Sep 2006 | | Subject: | Algebraic Geometry | | Abstract: | The paper begins by overviewing the basic facts on geometric exceptional collections. Then, we derive, for any coherent sheaf $cF$ on a smooth projective variety with a geometric collection, two spectral sequences: the first one abuts to $cF$ and the second one to its cohomology. The main goal of the paper is to generalize Castelnuovo-Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on smooth projective varieties $X$ with a geometric collection $sigma $. We define the notion of regularity of a coherent sheaf $cF$ on $X$ with respect to $sigma$. We show that the basic formal properties of the Castelnuovo-Mumford regularity of coherent sheaves over projective spaces continue to hold in this new setting and we show that in case of coherent sheaves on $PP^n$ and for a suitable geometric collection of coherent sheaves on $PP^n$ both notions of regularity coincide. Finally, we carefully study the regularity of coherent sheaves on a smooth quadric hypersurface $Q_n subset PP^{n+1}$ ($n$ odd) with respect to a suitable geometric collection and we compare it with the Castelnuovo-Mumford regularity of their extension by zero in $PP^{n+1}$. | | Source: | arXiv, math/0609561 | | Services: | Forum | Review | PDF | Favorites | |
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