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Seminatural cohomology and stability. (English) Zbl 0598.14014

A reflexive sheaf F on \({\mathbb{P}}^ 3\) of rank 2 is said to have ”seminatural cohomology” if for every n with \(2n+c_ 1(F)+4\geq 0\), at most one group \(H^ i(F(n))\) is different from zero. R. Hartshorne and A. Hirschowitz have proved in Ann. Sci. Éc. Norm. Supér., IV. Sér. 15, 365-390 (1982; Zbl 0509.14015)] that a locally free sheaf with seminatural cohomology is necessarily stable; this is extended here to reflexive sheaves with the following exceptions. Normalise so that \(c_ 1=0\) or -1. Then the exceptional cases are \((c_ 1,c_ 2,c_ 3)=(0,1,2),\quad (0,2,6),\quad (0,3,10),\quad (-1,1,3).\) Sheaves in these cases are related to curves of degree 5, 6, 7, on quadrics or to cubic curves (but a precise classification is not given).
Reviewer: C.T.C.Wall

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14D20 Algebraic moduli problems, moduli of vector bundles
18G20 Homological dimension (category-theoretic aspects)

Citations:

Zbl 0509.14015
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