Constructing stable vector bundles from curves with torsion normal bundle
DOI:
https://doi.org/10.7146/math.scand.a-136533Abstract
Given a smooth irreducible curve $S$ with torsion normal bundle on a projective surface $X$, we provide a criterion for the non-emptiness of the moduli of slope stable vector bundles with prescribed Chern classes. The criterion is given in terms of the topology of the pair $(X,S)$.
References
Ballico, E., On vector bundles on algebraic surfaces and $0$-cycles, Nagoya Math. J. 130 (1993), 19–23. https://doi.org/10.1017/S0027763000004402
Barth, W., Hulek, K., Peters, C., and Van de Ven, A., Compact complex surfaces, Second edition, Springer-Verlag Berlin 2004. https://doi.org/10.1007/978-3-642-57739-0
Beltrametti, M., Francia, P., and Sommese, A. J., On Reiders's method and higher order embeddings, Duke math. 58 (1989), no. 2, 425–439. https://doi.org/10.1215/S0012-7094-89-05819-
Beltrametti, M., and Sommese, A. J., Zero cycles and $k$th order embeddings of smooth projective surfaces., Sympos. Math., XXXII, Problems in the theory of surfaces and their classification (Cortona, 1988), 33–48, Academic Press, London, 1991.
Beltrametti, M., and Sommese, A. J., On the preservation of $k$-very ampleness under adjunction, Math. Z. 212 (1993), no. 2, 257–283. https://doi.org/10.1007/BF02571657
Brussee, R., Stable bundles on blown up surfaces, Math. Z. 205 (1990), no. 4, 551–565. https://doi.org/10.1007/BF02571262
Dold, A., Lectures on algebraic topology, second ed., Grundlehren der Mathematischen Wissenschaften 200, Springer-Verlag, Berlin-New York, 1980.
Donaldson, S. K., Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. (3) 50 (1985), no. 1, 1–26. https://doi.org/10.1112/plms/s3-50.1.1
Eisenbud, D., Green, M., and Harris, J., Cayley-Bacharach theorems and conjectures, Bull. Amer. Math. Soc. (N.S.) 33 (1996), no. 3, 295–324. https://doi.org/10.1090/S0273-0979-96-00666-0
Friedman, R., Algebraic surfaces and holomorphic vector bundles, Universitext. Springer-Verlag, New York, 1998. https://doi.org/10.1007/978-1-4612-1688-9
Gieseker, D., A construction of stable bundles on an algebraic surface, J. Differential Geom. 27 (1988), no. 1, 137–154. http://projecteuclid.org.zorac.aub.aau.dk/euclid.jdg/1214441654
Hamm, H. A., and Lê, D. T., On the Picard group for non-complete algebraic varieties, Singularités Franco-Japonaises, 71–86, Sémin. Congr., 10, Soc. Math. France, Paris, 2005.
Hartshorne, R., Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics, Vol. 156 Springer-Verlag, Berlin-New York 1970.
Hartshorne, R., Algebraic geometry, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.
Hirschowitz, A., and Laszlo, Y., À propos de l'existence de fibrés stables sur les surfaces, J. Reine Angew. Math. 460 (1995), 55–68.
Iitaka, S., Logarithmic forms of algebraic varieties, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), no. 3, 525–544.
Iitaka, S., Algebraic geometry. An introduction to birational geometry of varieties, Graduate Texts in Mathematics, 76. Springer-Verlag, New York-Berlin, 1982.
Kim, J. S., Stability of symmetric powers of vector bundles of rank two with even degree on a curve, Internat. J. Math. 33 (2022), no. 1, Paper No. 2250001, 36 pp. https://doi.org/10.1142/S0129167X2250001X
Knapp, A. W., Representation theory of semisimple groups. An overview based on examples, Princeton Mathematical Series, 36. Princeton University Press, Princeton, NJ, 1986. https://doi.org/10.1515/9781400883974
Li, W. P., and Qin, Z., Stable vector bundles on algebraic surfaces, Trans. Amer. Math. Soc. 345 (1994), no. 2, 833–852. https://doi.org/10.2307/2155001
Licanic, S., On boundedness of families of holomorphic foliations, Internat. J. Math. 20 (2009), no. 1, 15–43. https://doi.org/10.1142/S0129167X09005157
Maruyama, M., The theorem of Grauert-Mulich-Spindler, Math. Ann. 255 (1981), no. 3, 317–333. https://doi.org/10.1007/BF01450706
Mehta, V. B., and Ramanathan, A., Semistable sheaves on projective varieties and their restriction to curves, Math. Ann. 258 (1981/82), no. 3, 213–224. https://doi.org/10.1007/BF01450677
Mistretta, E. C., Stable vector bundles as generators of the Chow ring, Geom. Dedicata 117 (2006), 203–213. https://doi.org/10.1007/s10711-005-9024-1
Nakashima, T., Moduli of stable bundles on blown up surfaces, J. Math. Kyoto Univ. 33 (1993), no. 3, 571–581. https://doi.org/10.1215/kjm/1250519179
Narasimhan, M. S., and Seshadri, C. S., Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2) 82 (1965), 540–567. https://doi.org/10.2307/1970710
Peternell, T., Le Potier, J., and M. Schneider, M., Vanishing theorems, linear and quadratic normality, Invent. Math. 87 (1987), no. 3, 573–586. https://doi.org/10.1007/BF01389243
Tan, S. L., and Viehweg, E., A note on the Cayley-Bacharach property for vector bundles, Complex analysis and algebraic geometry, 361–373, de Gruyter, Berlin, 2000.
Totaro, B., The topology of smooth divisors and the arithmetic of abelian varieties, Michigan Math. J. 48 (2000), 611–624, https://doi.org/10.1307/mmj/1030132736
