On the Segre invariant for rank two vector bundles on ℙ2
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L. Roa-Leguizamón
Abstract
We extend the concept of the Segre invariant to vector bundles on a surface X. For X = ℙ2 we determine what numbers can appear as the Segre invariant of a rank 2 vector bundle with given Chern classes. The irreducibility of strata with fixed Segre invariant is proved and their dimensions are computed. Finally, we present applications to the Brill–Noether Theory for rank 2 vector bundles on ℙ2.
Funding: This paper was partially supported by CONACyT Grant CB-257079. The first author acknowledges the financial support of the Fondo Institucional de Fomento Regional para el Desarrollo Científico, Tecnológico y de Innovación, FORDECYT 265667.
Communicated by: I. Coskun
References
[1] W. P. Barth, K. Hulek, C. A. M. Peters, A. Van de Ven, Compact complex surfaces. Springer 2004. MR2030225 Zbl 1036.1401610.1007/978-3-642-57739-0Suche in Google Scholar
[2] L. Brambila-Paz, H. Lange, A stratification of the moduli space of vector bundles on curves. J. Reine Angew. Math. 494 (1998), 173–187. MR1604403 Zbl 0919.1401610.1515/crll.1998.005Suche in Google Scholar
[3] I. Coskun, J. Huizenga, Weak Brill–Noether for rational surfaces. In: Local and global methods in algebraic geometry, volume 712 of Contemp. Math., 81–104, Amer. Math. Soc. 2018. MR3832400 Zbl 1401.1418610.1090/conm/712/14343Suche in Google Scholar
[4] L. Costa, R. M. Miró-Roig, Brill–Noether theory for moduli spaces of sheaves on algebraic varieties. Forum Math. 22 (2010), 411–432. MR2652705 Zbl 1190.1404010.1515/forum.2010.023Suche in Google Scholar
[5] R. Friedman, Algebraic surfaces and holomorphic vector bundles. Springer 1998. MR1600388 Zbl 0902.1402910.1007/978-1-4612-1688-9Suche in Google Scholar
[6] L. Göttsche, Change of polarization and Hodge numbers of moduli spaces of torsion free sheaves on surfaces. Math. Z. 223 (1996), 247–260. MR1417430 Zbl 0884.1401410.1007/PL00004557Suche in Google Scholar
[7] L. Göttsche, A. Hirschowitz, Weak Brill–Noether for vector bundles on the projective plane. In: Algebraic geometry (Catania, 1993/Barcelona, 1994), 63–74, Dekker 1998. MR1651090 Zbl 0937.14010Suche in Google Scholar
[8] R. Hartshorne, Algebraic geometry. Springer 1977. MR0463157 Zbl 0367.1400110.1007/978-1-4757-3849-0Suche in Google Scholar
[9] R. Hartshorne, Stable reflexive sheaves. Math. Ann. 254 (1980), 121–176. MR597077 Zbl 0431.1400410.1007/BF01467074Suche in Google Scholar
[10] D. Huybrechts, M. Lehn, The geometry of moduli spaces of sheaves. Friedr. Vieweg & Sohn, Braunschweig 1997. MR1450870 Zbl 0872.1400210.1007/978-3-663-11624-0Suche in Google Scholar
[11] H. Lange, Zur Klassifikation von Regelmannigfaltigkeiten. Math. Ann. 262 (1983), 447–459. MR696517 Zbl 0492.1400310.1007/BF01456060Suche in Google Scholar
[12] H. Lange, M. S. Narasimhan, Maximal subbundles of rank two vector bundles on curves. Math. Ann. 266 (1983), 55–72. MR722927 Zbl 0507.1400510.1007/BF01458704Suche in Google Scholar
[13] J. Le Potier, Lectures on vector bundles. Cambridge Univ. Press 1997. MR1428426 Zbl 0872.14003Suche in Google Scholar
[14] M. Maruyama, On classification of ruled surfaces, volume 3 of Lectures in Mathematics, Department of Mathematics, Kyoto University. Kinokuniya Book-Store Co., Ltd., Tokyo 1970. MR0276243 Zbl 0214.20103Suche in Google Scholar
[15] M. Maruyama, Stable vector bundles on an algebraic surface. Nagoya Math. J. 58 (1975), 25–68. MR396576 Zbl 0337.1402610.1017/S0027763000016688Suche in Google Scholar
[16] M. Maruyama, Openness of a family of torsion free sheaves. J. Math. Kyoto Univ. 16 (1976), 627–637. MR429899 Zbl 0404.1400410.1215/kjm/1250522875Suche in Google Scholar
[17] C. Okonek, M. Schneider, H. Spindler, Vector bundles on complex projective spaces. Birkhäuser, Boston, Mass. 1980. MR561910 Zbl 0438.3201610.1007/978-3-0348-0151-5Suche in Google Scholar
[18] B. Russo, M. Teixidor i Bigas, On a conjecture of Lange. J. Algebraic Geom. 8 (1999), 483–496. MR1689352 Zbl 0942.14013Suche in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- An 𝔽p2-maximal Wiman sextic and its automorphisms
- Pseudo-algebraic Ricci solitons on Einstein nilradicals
- The wobbly divisors of the moduli space of rank-2 vector bundles
- Symmetries of complex analytic vector fields with an essential singularity on the Riemann sphere
- Betti numbers and pseudoeffective cones in 2-Fano varieties
- The generating rank of a polar Grassmannian
- The Beckman–Quarles theorem via the triangle inequality
- On Huisman’s conjectures about unramified real curves
- Geodesic orbit Finsler spaces with K ≥ 0 and the (FP) condition
- On the Segre invariant for rank two vector bundles on ℙ2
- Lifting coarse homotopies
- How to construct all metric f-K-contact manifolds
- An extremum problem for the power moment of a convex polygon contained in a disc
- Sharply transitive sets in PGL2(K)
Artikel in diesem Heft
- Frontmatter
- An 𝔽p2-maximal Wiman sextic and its automorphisms
- Pseudo-algebraic Ricci solitons on Einstein nilradicals
- The wobbly divisors of the moduli space of rank-2 vector bundles
- Symmetries of complex analytic vector fields with an essential singularity on the Riemann sphere
- Betti numbers and pseudoeffective cones in 2-Fano varieties
- The generating rank of a polar Grassmannian
- The Beckman–Quarles theorem via the triangle inequality
- On Huisman’s conjectures about unramified real curves
- Geodesic orbit Finsler spaces with K ≥ 0 and the (FP) condition
- On the Segre invariant for rank two vector bundles on ℙ2
- Lifting coarse homotopies
- How to construct all metric f-K-contact manifolds
- An extremum problem for the power moment of a convex polygon contained in a disc
- Sharply transitive sets in PGL2(K)
