Second order theta functions and vector bundles over Jacobi varieties

Yuen, David S.

doi:10.1090/S0002-9947-1990-1012508-7

Second order theta functions and vector bundles over Jacobi varieties
HTML articles powered by AMS MathViewer

by David S. Yuen

Trans. Amer. Math. Soc. 320 (1990), 457-492

DOI: https://doi.org/10.1090/S0002-9947-1990-1012508-7

PDF | Request permission

Abstract:

We consider the Picard vector bundles defined over Jacobi varieties. The rank $g + 1$ Picard bundle imbeds in the rank ${2^g}$ Clifford bundle, so the second order theta functions, viewed appropriately, span the dual of the Picard bundle over each fiber. We prove a result on the minimum number of such second order theta functions required to span the whole bundle at each point. We give an application of using these functions to describe subvarieties of the Jacobian. There follow comments on which functions we could use, and generalizations to higher order theta functions.

References

Enrico Arbarello, Fay’s trisecant formula and a characterization of Jacobian varieties, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 49–61. MR 927948, DOI 10.1090/pspum/046.1/927948
E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932, DOI 10.1007/978-1-4757-5323-3
M. F. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 (1957), 414–452. MR 131423, DOI 10.1112/plms/s3-7.1.414
William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
Alexander Grothendieck, Sur quelques points d’algèbre homologique, Tohoku Math. J. (2) 9 (1957), 119–221 (French). MR 102537, DOI 10.2748/tmj/1178244839
Robert C. Gunning, On generalized theta functions, Amer. J. Math. 104 (1982), no. 1, 183–208. MR 648486, DOI 10.2307/2374073
Robert C. Gunning, Riemann surfaces and generalized theta functions, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 91, Springer-Verlag, Berlin-New York, 1976. MR 0457787
R. C. Gunning, Some curves in abelian varieties, Invent. Math. 66 (1982), no. 3, 377–389. MR 662597, DOI 10.1007/BF01389218
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
F. Hirzebruch, Topological methods in algebraic geometry, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 131, Springer-Verlag New York, Inc., New York, 1966. New appendix and translation from the second German edition by R. L. E. Schwarzenberger, with an additional section by A. Borel. MR 0202713
Dale Husemoller, Fibre bundles, McGraw-Hill Book Co., New York-London-Sydney, 1966. MR 0229247
George Kempf, Inversion of abelian integrals, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 1, 25–32. MR 634432, DOI 10.1090/S0273-0979-1982-14962-X
George Kempf, On the geometry of a theorem of Riemann, Ann. of Math. (2) 98 (1973), 178–185. MR 349687, DOI 10.2307/1970910
J.-L. Koszul, Multiplicateurs et classes caractéristiques, Trans. Amer. Math. Soc. 89 (1958), 256–266 (French). MR 99660, DOI 10.1090/S0002-9947-1958-0099660-7
Henrik H. Martens, On the varieties of special divisors on a curve, J. Reine Angew. Math. 227 (1967), 111–120. MR 215847, DOI 10.1515/crll.1967.227.111
Arthur Mattuck, Picard bundles, Illinois J. Math. 5 (1961), 550–564. MR 150141
Arthur Mattuck, Symmetric products and Jacobians, Amer. J. Math. 83 (1961), 189–206. MR 142553, DOI 10.2307/2372727
Jean-Pierre Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier (Grenoble) 6 (1955/56), 1–42 (French). MR 82175