Stable rank 2 reflexive sheaves on \({\mathbb{P}}^ 3\) with small \(c_ 2\) and applications.

*(English)*Zbl 0558.14015The 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. StrĂ¸mme, 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 |