Subvarieties of $\mathcal {SU}_C(2)$ and $2\theta$-divisors in the Jacobian
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- by W. M. Oxbury, C. Pauly and E. Previato
- Trans. Amer. Math. Soc. 350 (1998), 3587-3614
- DOI: https://doi.org/10.1090/S0002-9947-98-02148-5
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Abstract:
We explore some of the interplay between Brill-Noether subvarieties of the moduli space ${\mathcal {SU}}_C(2,K)$ of rank 2 bundles with canonical determinant on a smooth projective curve and $2\theta$-divisors, via the inclusion of the moduli space into $|2\Theta |$, singular along the Kummer variety. In particular we show that the moduli space contains all the trisecants of the Kummer and deduce that there are quadrisecant lines only if the curve is hyperelliptic; we show that for generic curves of genus $<6$, though no higher, bundles with $>2$ sections are cut out by $\Gamma _{00}$; and that for genus 4 this locus is precisely the Donagi-Izadi nodal cubic threefold associated to the curve.References
- 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
- Arnaud Beauville, Fibrés de rang deux sur une courbe, fibré déterminant et fonctions thêta. II, Bull. Soc. Math. France 119 (1991), no. 3, 259–291 (French, with English summary). MR 1125667, DOI 10.24033/bsmf.2167
- Arnaud Beauville and Olivier Debarre, Sur les fonctions thêta du second ordre, Arithmetic of complex manifolds (Erlangen, 1988) Lecture Notes in Math., vol. 1399, Springer, Berlin, 1989, pp. 27–39 (French). MR 1034254, DOI 10.1007/BFb0095966
- Arnaud Beauville, M. S. Narasimhan, and S. Ramanan, Spectral curves and the generalised theta divisor, J. Reine Angew. Math. 398 (1989), 169–179. MR 998478, DOI 10.1515/crll.1989.398.169
- Aaron Bertram, Moduli of rank-$2$ vector bundles, theta divisors, and the geometry of curves in projective space, J. Differential Geom. 35 (1992), no. 2, 429–469. MR 1158344
- A. Bertram, B. Feinberg, On stable rank two bundles with canonical determinant and many sections, Algebraic Geometry: the Europroj Conferences in Catania and Barcelona, Marcel Dekker (1997);
- Arthur B. Coble, Algebraic geometry and theta functions, American Mathematical Society Colloquium Publications, Vol. 10, American Mathematical Society, Providence, R.I., 1982. Reprint of the 1929 edition. MR 733252
- O. Debarre, Sur la démonstration de A. Weil du théorème de Torelli pour les courbes, Compositio Math. 58 (1986), no. 1, 3–11 (French, with English summary). MR 834045
- Ron Donagi, The fibers of the Prym map, Curves, Jacobians, and abelian varieties (Amherst, MA, 1990) Contemp. Math., vol. 136, Amer. Math. Soc., Providence, RI, 1992, pp. 55–125. MR 1188194, DOI 10.1090/conm/136/1188194
- Bert van Geemen and Gerard van der Geer, Kummer varieties and the moduli spaces of abelian varieties, Amer. J. Math. 108 (1986), no. 3, 615–641. MR 844633, DOI 10.2307/2374657
- M. L. Green, Quadrics of rank four in the ideal of a canonical curve, Invent. Math. 75 (1984), no. 1, 85–104. MR 728141, DOI 10.1007/BF01403092
- Mark Green and Robert Lazarsfeld, On the projective normality of complete linear series on an algebraic curve, Invent. Math. 83 (1986), no. 1, 73–90. MR 813583, DOI 10.1007/BF01388754
- E. Izadi, The geometric structure of ${\scr A}_4$, the structure of the Prym map, double solids and $\Gamma _{00}$-divisors, J. Reine Angew. Math. 462 (1995), 93–158. MR 1329904, DOI 10.1515/crll.1995.462.93
- 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
- Herbert Lange, Higher secant varieties of curves and the theorem of Nagata on ruled surfaces, Manuscripta Math. 47 (1984), no. 1-3, 263–269. MR 744323, DOI 10.1007/BF01174597
- Herbert Lange, Höhere Sekantenvarietäten und Vektorbündel auf Kurven, Manuscripta Math. 52 (1985), no. 1-3, 63–80 (German, with English summary). MR 790793, DOI 10.1007/BF01171486
- Yves Laszlo, Un théorème de Riemann pour les diviseurs thêta sur les espaces de modules de fibrés stables sur une courbe, Duke Math. J. 64 (1991), no. 2, 333–347 (French). MR 1136379, DOI 10.1215/S0012-7094-91-06416-1
- Shigeru Mukai, Curves and Grassmannians, Algebraic geometry and related topics (Inchon, 1992) Conf. Proc. Lecture Notes Algebraic Geom., I, Int. Press, Cambridge, MA, 1993, pp. 19–40. MR 1285374
- David Mumford, Curves and their Jacobians, University of Michigan Press, Ann Arbor, Mich., 1975. MR 0419430
- M. S. Narasimhan and S. Ramanan, $2\theta$-linear systems on abelian varieties, Vector bundles on algebraic varieties (Bombay, 1984) Tata Inst. Fund. Res. Stud. Math., vol. 11, Tata Inst. Fund. Res., Bombay, 1987, pp. 415–427. MR 893605
- Kapil Paranjape and S. Ramanan, On the canonical ring of a curve, Algebraic geometry and commutative algebra, Vol. II, Kinokuniya, Tokyo, 1988, pp. 503–516. MR 977775
- Gerald E. Welters, The surface $C-C$ on Jacobi varieties and 2nd order theta functions, Acta Math. 157 (1986), no. 1-2, 1–22. MR 857677, DOI 10.1007/BF02392589
Bibliographic Information
- W. M. Oxbury
- Affiliation: Department of Mathematical Sciences, Science Laboratories, South Road, Durham DH1 3LE, U.K.
- Email: w.m.oxbury@durham.ac.uk
- C. Pauly
- Affiliation: Laboratoire J. A. Dieudonné, Université de Nice-Sophia-Antipolis, Parc Valrose, F-06108 Nice Cedex 02, France
- Email: pauly@math.unice.fr
- E. Previato
- Affiliation: Department of Mathematics, Boston University, Boston, Massachusetts 02215
- MR Author ID: 142015
- Email: ep@math.bu.edu
- Received by editor(s): September 26, 1996
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 350 (1998), 3587-3614
- MSC (1991): Primary 14D20, 14H42, 14H60, 14K25
- DOI: https://doi.org/10.1090/S0002-9947-98-02148-5
- MathSciNet review: 1467474