an:03889702
Zbl 0558.14015
Chang, Mei-Chu
Stable rank 2 reflexive sheaves on \({\mathbb{P}}^ 3\) with small \(c_ 2\) and applications
EN
Trans. Am. Math. Soc. 284, 57-89 (1984).
00147984
1984
j
14F05 14D20 14H10 14D22 32L10
vector bundles; reflexive sheaves; classification of coarse moduli spaces of stable rank-2 reflexive sheaves on projective 3-space; Chern classes; moduli variety of curves
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. \textit{R. Hartshorne}, Math. Ann. 238, 229-280 (1978; Zbl 0411.14002) and Math. Ann. 254, 121-176 (1980; Zbl 0437.14008); \textit{R. Hartshorne} and \textit{I. Sols}, J. Reine Angew. Math. 325, 145-152 (1981; Zbl 0448.14004), and \textit{G. Ellingsrud} and \textit{S. A. Str??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.
H.Lange
Zbl 0411.14002; Zbl 0437.14008; Zbl 0448.14004; Zbl 0448.14001