×

Ample divisors on fine moduli spaces on the projective plane. (English) Zbl 0533.14006

Let \(c_ 1\) and \(c_ 2\) be two integers with discriminant \(D=c_ 1\!^ 2-4c_ 2\) satisfying \(D<0\), \(D\neq -4\), and \(D\not\equiv 0(mod 8),\) and let \(M(c_ 1,c_ 2)\) be the fine moduli space for stable rank- 2 coherent sheaves on \({\mathbb{P}}^ 2\) with the given Chern classes. We prove that the Picard group of \(M(c_ 1,c_ 2)\) is free on two generators, and the geometric significance of these is given. The ample cone is described in terms of the generators. Furthermore, the divisor class of the non-locally free sheaves is computed, as well as the canonical class.

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14C20 Divisors, linear systems, invertible sheaves
14D22 Fine and coarse moduli spaces
57R20 Characteristic classes and numbers in differential topology
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Barth, W.: Some properties of stable rank-2 vector bundles on IP n . Math. Ann.226, 125-150 (1977) · Zbl 0417.32013 · doi:10.1007/BF01360864
[2] Barth, W.: Moduli of vector bundles on the projective plane. Invent. Math.42, 63-91 (1977) · Zbl 0386.14005 · doi:10.1007/BF01389784
[3] Ellingsrud, G., Strømme, S.A.: On the moduli space for stable rank-2 vector bundles on IP2. Preprint, Oslo (1979) · Zbl 0632.14013
[4] Ellingsrud, G., Strømme, S.A.: The Picard group of the moduli space for stable rank-2 vector bundles on IP2 with odd first Chern class. Preprint, Oslo (1979)
[5] Gieseker, D.: On the moduli of vector bundles on an algebraic surface. Ann. of Math.106, 45-60 (1977) · Zbl 0381.14003 · doi:10.2307/1971157
[6] Grothendieck, A.: Techniques de construction et théorèmes d’existence en géométrie algébrique IV: Les schémas de Hilbert. Sem. Bourbaki 221 (1960/61)
[7] Hartshorne, R.: Stable reflexive sheaves. Math. Ann.254, 121-176 (1980) · Zbl 0437.14008 · doi:10.1007/BF01467074
[8] Hartshorne, R.: Algebraic Geometry. Graduate Texts in Math. 52. Berlin, Heidelberg, New York: Springer 1977 · Zbl 0367.14001
[9] Hulek, K.: Stable rank-2 vector bundles on ?2 withc 1 odd. Math. Ann.242, 241-266 (1979) · Zbl 0407.32013 · doi:10.1007/BF01420729
[10] Kleiman, S.: Relative duality for quasi-coherent sheaves. Compositio Math.41, 39-60 (1980) · Zbl 0403.14003
[11] Le Potier, J.: Fibrés stables de rang 2 sur ?2(?). Math. Ann.241, 217-256 (1979) · Zbl 0405.14008 · doi:10.1007/BF01421207
[12] Le Potier, J.: Sur le groupe de Picard de l’espace de modules de fibrés stables sur IP2. Ann. scient. Éc. Norm. Sup.14, 141-155 (1981) · Zbl 0482.14006
[13] Maruyama, M.: Moduli of stable sheaves, II. J. Math. Kyoto Univ.18, 557-614 (1978) · Zbl 0395.14006
[14] Mumford, D.: Geometric invariant theory. Berlin, Heidelberg, New York: Springer 1965 · Zbl 0147.39304
[15] Schneider, M.: Holomorphic vector bundles on ? n . Sem. Bourbaki 530 (1978/79)
[16] Schwarzenberger, R.L.E.: Vector bundles on algebraic surfaces. Proc. London Math. Soc.11, 601-622 (1961) · Zbl 0212.26003 · doi:10.1112/plms/s3-11.1.601
[17] Schwarzenberger, R.L.E.: The secant bundle of a projective variety. Proc. London Math. Soc. (3)14, 369-384 (1964) · Zbl 0123.38201 · doi:10.1112/plms/s3-14.2.369
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.