Nagoya Mathematical Journal

Rationality of moduli spaces of vector bundles on rational surfaces

Laura Costa and Rosa M. Miro-Ŕoig

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Abstract

Let $X$ be a smooth rational surface. In this paper, we prove the rationality of the moduli space ${M}_{X, L}(2; c_{1}, c_{2})$ of rank two $L$-stable vector bundles $E$ on $X$ with det $(E) = c_{1} \in $ Pic$(X)$ and $c_{2}(E) = c_{2} \gg 0$.

Article information

Source
Nagoya Math. J., Volume 165 (2002), 43-69.

Dates
First available in Project Euclid: 27 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1114631697

Mathematical Reviews number (MathSciNet)
MR1892097

Zentralblatt MATH identifier
1020.14012

Subjects
Primary: 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx]
Secondary: 14J26: Rational and ruled surfaces

Citation

Costa, Laura; Miro-Ŕoig, Rosa M. Rationality of moduli spaces of vector bundles on rational surfaces. Nagoya Math. J. 165 (2002), 43--69. https://projecteuclid.org/euclid.nmj/1114631697


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References

  • I. V. Artamkin, Stable bundles with $c_1 = 0$ on rational surfaces , Math. USSR Izvestiya, 36 , 231–246 (1990).
  • L. Costa and R. M. Miró-Roig, On the rationality of moduli spaces of vector bundles on Fano surfaces. , J. Pure Applied Algebra, 137 , 199–220 (1999).
  • S. Donaldson, Polynomial invariants for smooth $4$-manifolds , Topology, 29 , 257–315 (1986).
  • G. Ellisngsrud and S. A. Stromme, On the rationality of the moduli space for stable rank-$2$ vector bundles on $\PP^2$ , LNM, 1273, 363–371 (1987).
  • D. Gieseker, On the moduli of vector bundles on an algebraic surface , Ann. Math., 106 , 45–60 (1977).
  • D. Gieseker and J. Li, Moduli of high rank vector bundles over surfaces , J. Amm. Math. Soc., 9 , 107–151 (1996).
  • R. Hartshorne, Algebraic Geometry, GTM 52, Springer-Verlag (1977).
  • A. Hirschowitz and Y. Laszlo, Fibrés génériques sur le plan projectif , Math. Annalen, 297 , 85–102 (1993).
  • T. Maeda, An elementary proof of the rationality of the moduli space of rank $2$ vector bundles on $\PP^2$ , Hirosh. Math. J., 20 , 103–107 (1990).
  • M. Maruyama, Stable vector bundles on an algebraic surface , Nagoya Math. J., 58 , 25–68 (1975).
  • M. Maruyama, On the rationality of the moduli space of vector bundles of rank $2$ on $\PP^2$, Proc. Sendai Conf. (1985).
  • RM. Miró-Roig, The moduli spaces of rank $2$ stable vector bundles over Veronesean surfaces , Manusc. Math., 72 , 391–402 (1993).
  • T. Nakashima, Moduli of stable bundles on blown up surfaces , J. Math. Kyoto Univ., 33-3 , 571–581 (1993).
  • K. O'Grady, Moduli of vector bundles on projective surfaces : some basic results , Inv. Math., 123 , 141–206 (1996).
  • Z. Qin, Equivalence classes of polarizations and moduli spaces of sheaves , J. Differential Geometry, 37 , 397–415 (1993).
  • C. Walter, Irreducibility of moduli spaces of vector bundles on birationally ruled surfaces , Algebraic Geometry (Catania, 1993/Barcelona, 1994), 201–211, Lect. Not. Pure Appl. Math., 200, Deckker, 1998.
  • K. Zuo, Generic smoothness of the moduli of rank $2$ stable bundles over an algebraic surface , Math. Z., 207 , 629–643 (1991).