Rank-two vector bundles on non-minimal ruled surfaces
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- by Marian Aprodu, Laura Costa and Rosa Maria Miró-Roig PDF
- Trans. Amer. Math. Soc. 370 (2018), 3913-3929 Request permission
Abstract:
We continue previous work by various authors and study the birational geometry of moduli spaces of stable rank-two vector bundles on surfaces with Kodaira dimension $-\infty$. To this end, we express vector bundles as natural extensions by using two numerical invariants associated to vector bundles, similar to the invariants defined by Brînzănescu and Stoia in the case of minimal surfaces. We compute explicitly these natural extensions on blowups of general points on a minimal surface. In the case of rational surfaces, we prove that any irreducible component of a moduli space is either rational or stably rational.References
- Marian Aprodu and Vasile Brînzǎnescu, Stable rank-2 vector bundles over ruled surfaces, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 3, 295–300 (English, with English and French summaries). MR 1464824, DOI 10.1016/S0764-4442(97)83959-6
- Marian Aprodu and Vasile Brînzǎnescu, Moduli spaces of vector bundles over ruled surfaces, Nagoya Math. J. 154 (1999), 111–122. MR 1689175, DOI 10.1017/S0027763000025332
- Arnaud Beauville, The Lüroth problem, Rationality problems in algebraic geometry, Lecture Notes in Math., vol. 2172, Springer, Cham, 2016, pp. 1–27. MR 3618664
- Vasile Brînzănescu and Manuela Stoia, Topologically trivial algebraic $2$-vector bundles on ruled surfaces. I, Rev. Roumaine Math. Pures Appl. 29 (1984), no. 8, 661–673. MR 759495
- Vasile Brînzănescu, Holomorphic vector bundles over compact complex surfaces, Lecture Notes in Mathematics, vol. 1624, Springer-Verlag, Berlin, 1996. MR 1439504, DOI 10.1007/BFb0093696
- Laura Costa and Rosa M. Miró-Roig, On the rationality of moduli spaces of vector bundles on Fano surfaces, J. Pure Appl. Algebra 137 (1999), no. 3, 199–220. MR 1685137, DOI 10.1016/S0022-4049(97)00204-1
- Laura Costa and Rosa M. Miró-Roig, Rationality of moduli spaces of vector bundles on Hirzebruch surfaces, J. Reine Angew. Math. 509 (1999), 151–166. MR 1679170, DOI 10.1515/crll.1999.036
- Laura Costa and Rosa M. Miró-Roig, Moduli spaces of vector bundles on higher-dimensional varieties, Michigan Math. J. 49 (2001), no. 3, 605–620. MR 1872759, DOI 10.1307/mmj/1012409973
- Laura Costa and Rosa M. Miro-Ŕoig, Rationality of moduli spaces of vector bundles on rational surfaces, Nagoya Math. J. 165 (2002), 43–69. MR 1892097, DOI 10.1017/S0027763000008138
- S. K. Donaldson, Polynomial invariants for smooth four-manifolds, Topology 29 (1990), no. 3, 257–315. MR 1066174, DOI 10.1016/0040-9383(90)90001-Z
- Robert Friedman, Algebraic surfaces and holomorphic vector bundles, Universitext, Springer-Verlag, New York, 1998. MR 1600388, DOI 10.1007/978-1-4612-1688-9
- David Gieseker and Jun Li, Moduli of high rank vector bundles over surfaces, J. Amer. Math. Soc. 9 (1996), no. 1, 107–151. MR 1303031, DOI 10.1090/S0894-0347-96-00171-3
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- H. Lange and M. S. Narasimhan, Maximal subbundles of rank two vector bundles on curves, Math. Ann. 266 (1983), no. 1, 55–72. MR 722927, DOI 10.1007/BF01458704
- D. Lieberman and D. Mumford, Matsusaka’s big theorem, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974) Amer. Math. Soc., Providence, R.I., 1975, pp. 513–530. MR 0379494
- Masayoshi Nagata, On self-intersection number of a section on a ruled surface, Nagoya Math. J. 37 (1970), 191–196. MR 258829
- Tohru Nakashima, Moduli of stable bundles on blown up surfaces, J. Math. Kyoto Univ. 33 (1993), no. 3, 571–581. MR 1239079, DOI 10.1215/kjm/1250519179
- Kieran G. O’Grady, Moduli of vector bundles on projective surfaces: some basic results, Invent. Math. 123 (1996), no. 1, 141–207. MR 1376250, DOI 10.1007/BF01232371
- Zhenbo Qin, Moduli spaces of stable rank-$2$ bundles on ruled surfaces, Invent. Math. 110 (1992), no. 3, 615–626. MR 1189492, DOI 10.1007/BF01231346
- Zhenbo Qin, On smooth structures of potential surfaces of general type homeomorphic to rational surfaces, Invent. Math. 113 (1993), no. 1, 163–175. MR 1223228, DOI 10.1007/BF01244306
- Zhenbo Qin, Moduli of simple rank-$2$ sheaves on $K3$-surfaces, Manuscripta Math. 79 (1993), no. 3-4, 253–265. MR 1223021, DOI 10.1007/BF02568344
- Zhenbo Qin, Equivalence classes of polarizations and moduli spaces of sheaves, J. Differential Geom. 37 (1993), no. 2, 397–415. MR 1205450
- Charles Walter, Irreducibility of moduli spaces of vector bundles on birationally ruled surfaces, Algebraic geometry (Catania, 1993/Barcelona, 1994) Lecture Notes in Pure and Appl. Math., vol. 200, Dekker, New York, 1998, pp. 201–211. MR 1651095
- Kang Zuo, Generic smoothness of the moduli spaces of rank two stable vector bundles over algebraic surfaces, Math. Z. 207 (1991), no. 4, 629–643. MR 1119961, DOI 10.1007/BF02571412
Additional Information
- Marian Aprodu
- Affiliation: Facultatea de Matematică şi Informatică, Universitatea din Bucureşti, Str. Academiei 14, 010014 Bucureşti, Romania – and – Institutul de Matematică “Simion Stoilow” al Academiei Române, Calea Griviţei 21, Sector 1, 010702 Bucureşti, Romania
- MR Author ID: 611558
- Email: marian.aprodu@fmi.unibuc.ro, marian.aprodu@imar.ro
- Laura Costa
- Affiliation: Facultat de Matemàtiques i Informàtica, Departament de Matemàtiques i Informàtica, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
- Email: costa@ub.edu
- Rosa Maria Miró-Roig
- Affiliation: Facultat de Matemàtiques i Informàtica, Departament de Matemàtiques i Informàtica, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
- MR Author ID: 125375
- ORCID: 0000-0003-1375-6547
- Email: miro@ub.edu
- Received by editor(s): March 30, 2016
- Received by editor(s) in revised form: September 1, 2016
- Published electronically: December 27, 2017
- Additional Notes: The first author was partially supported by UEFISCDI Grant PN-II-PCE-2011-3-0288
The second author was partially supported by MTM2016-78623-P
The third author was partially supported by MTM2016-78623-P - © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 3913-3929
- MSC (2010): Primary 14F05; Secondary 14D20
- DOI: https://doi.org/10.1090/tran/7062
- MathSciNet review: 3811514