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Existence de faisceaux réflexifs de rang deux sur \({\mathbb{P}}^ 3\) à bonne cohomologie. (Existence of reflexive sheaves of rank two on \({\mathbb{P}}^ 3\) with good homology). (French) Zbl 0647.14004

This paper continues work of the author and Hartshorne and follows in response to Hartshorne’s problem: Characterize those Chern classes which are possible for reflexive sheaves of rank 2 on \({\mathbb{P}}^ 3\). In particular, the only main result says:
Theorem (2.3). For Chern classes \(c_ 1, c_ 2, c_ 3\) satisfying \(c_ 1c_ 2\equiv c_ 3 (mod 2)\), \(0\leq c_ 3\leq 4c_ 2-c^ 2_ 1\) and furthermore (a) if \(4c_ 2-c^ 2_ 1\) is 7 or 15 then \(c_ 3\neq 0\); (b) if \(c_ 1\) is even and \(c_ 2\) odd, then \(c_ 3\leq 4c_ 2- c^ 2_ 1-6\); there is a “suitable” rank 2 sheaf of Chern classes \(c_ 1, c_ 2, c_ 3\) admitting semi-natural cohomology.
Here a cohomology, for a sheaf \({\mathcal E}\) of rank 2 on \({\mathbb{P}}^ 3\), is semi-natural if, for \(t\geq -2-c_ 1/2\), three of the cohomology groups \(H^ i({\mathcal E}(t))\) vanish, \(0\leq i\leq 3\). The sheaves of theorem (2.3) arise as a consequence of the existence of certain curves in V(\({\mathcal O}_{{\mathbb{P}}^ 3}(-a))\) satisfying conditions relating genus, degree, the number of lines possessed and the number of points on a line meeting the curve.
Reviewer: P.Cherenack

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14H10 Families, moduli of curves (algebraic)
57R20 Characteristic classes and numbers in differential topology
14L24 Geometric invariant theory
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References:

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