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Stable rank 2 reflexive sheaves on \({\mathbb{P}}^ 3\) with small \(c_ 2\) and applications. (English) Zbl 0558.14015
The paper in question investigates the coarse moduli spaces of stable rank-2 reflexive sheaves on \({\mathbb{P}}^ 2\) with Chern classes \(c_ 1=-1, 0,\quad c_ 2\leq 3,\) and \(c_ 3\), which have not been studied before. [For the remaining cases cf. R. Hartshorne, Math. Ann. 238, 229-280 (1978; Zbl 0411.14002) and Math. Ann. 254, 121-176 (1980; Zbl 0437.14008); R. Hartshorne and I. Sols, J. Reine Angew. Math. 325, 145-152 (1981; Zbl 0448.14004), and G. Ellingsrud and S. A. Strømme, Math. Ann. 255, 123-137 (1981; Zbl 0448.14001).] - For \(c_ 2\leq 2\) it is shown that the moduli spaces are nonsingular and rational varieties, whose dimension is computed. - For \(c_ 2=3\) they are irreducible and in most cases the associated reduced schemes are unirational. There are some applications to curves of low degree in \({\mathbb{P}}^ 3:\) It is shown that curves in \({\mathbb{P}}^ 3\) with certain degree and genus are of maximal rank, sometimes even projectively normal, and that the corresponding Hilbert scheme is irreducible and unirational. A consequence is the well known fact that the moduli variety of curves of genus g is unirational for \(g=5, 6, 7\), and 8.
Reviewer: H.Lange

MSC:
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14D20 Algebraic moduli problems, moduli of vector bundles
14H10 Families, moduli of curves (algebraic)
14D22 Fine and coarse moduli spaces
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
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