Journal of Algebraic Geometry

Author(s): Angela Ortega
Journal: J. Algebraic Geom. 14 (2005), 327-356.
Posted: November 18, 2004
Retrieve article in: PDF DVI PostScript

Abstract: We prove a conjecture about the moduli space $\mathcal{SU}_C(3)$ of semi-stable rank 3 vector bundles with trivial determinant on a genus 2 curve

, due to I. Dolgachev. Given

a smooth projective curve of genus 2, and the embedding of the Jacobian

into $\vert 3\Theta\vert$ , A. Coble proved, at the beginning of the 20th century, that there exists a unique cubic hypersurface $\mathcal{C}$ in $\vert 3\Theta\vert^* \simeq \mathbb{P} ^8$ ,

-invariant and singular along

. On the other hand, we have a map of degree 2 from $\mathcal{SU}_C(3)$ over $\vert 3\Theta\vert \simeq \mathbb{P} ^{8*}$ , ramified along a sextic hypersurface $\mathcal{B}$ . Dolgachev's conjecture affirms that the sextic $\mathcal{B}$ is the dual variety of Coble's cubic $\mathcal{C}$ .

1.: W. Barth, C. Peters, A. Van-de-Ven: Compact complex surfaces. Berlin, New York: Springer-Verlag, 1984. MR 0749574 (86c:32026)
2.: W. Barth: Quadratic equations for level- abelian surfaces. Abelian varieties (Egloffstein, 1993), 1-18, de Gruyter, Berlin, 1995. MR 1336597 (96h:14064)
3.: A. Beauville: Fibrés de rang 2 sur les courbes, fibré déterminant et functions thêta. Bull. Soc. Math. France 116 (1988), 431-438. MR 1005388 (91b:14038)
4.: A. Beauville:Vector bundles on curves and generalized theta fonctions: recent results and open problems. Current topics in complex algebraic geometry, MSRI Publications 28, Cambridge University Press, 1995, 17-33. MR 1397056 (97h:14015)
5.: A. Beauville, M.S. Narasimhan, S. Ramanan: Spectral curves and the generalised theta divisor. J. Reine Angew. Math. 398 (1989), 169-179. MR 0998478 (91c:14040)
6.: N. Bourbaki: Éléments de mathématiques: groupes et algèbres de Lie. Chapitres 4. Paris: Masson, 1981. MR 0647314 (83g:17001)
7.: A. Coble: Point sets and allied Cremona groups III. Trans. Amer. Math. Soc. 18 (1917), 331-372. MR 1501073
8.: A. Coble: Algebraic geometry and theta functions. AMS Coll. Publi. 10, Providence, 1929 (3ème. edition, 1969).
9.: J.-M. Drezet, M.S. Narasimhan: Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques. Invent. Math. 97 (1989), 53-94. MR 0999313 (90d:14008)
10.: W. Fulton: Intersection theory, Springer Verlag, 1984. MR 0732620 (85k:14004)
11.: W. Fulton, J. Harris: Representation theory: a first course. Springer Verlag, 1991. MR 1153249 (93a:20069)
12.: Y. Laszlo: Local structure of the moduli space of vector bundles over curves. Comment. Math. Helvetici 71 (1996), 373-401. MR 1418944 (97j:14012)
13.: H. Lange, Ch. Birkenhake: Complex Abelian Varieties. Grundlehren 302, Springer Verlag, 1982. MR 1217487 (94j:14001)
14.: H. Lange, Ch. Birkenhake: Moduli Spaces of Abelian Surfaces with Isogeny. Geometry and analysis, Tata Inst. Fund. Res., Bombay, 1995, 225-243. MR 1351509 (96i:14035)
15.: H. Lange, Ch. Birkenhake: A Family of Abelian Surfaces and Curves of Genus Four. Manuscripta Math. 85 (1994), 393-407. MR 1305750 (95k:14064)
16.: D. Mumford: On equations defining Abelian varieties. Invent. Math. 1 (1966), 287-354. MR 0204427 (34:4269)
17.: M.S. Narasimhan, S. Ramanan: 2 $\theta$ linear systems on abelian varieties. Vector bundles on algebraic varieties, Oxford University Press, 1987, 415-427. MR 0893605 (88j:14014)
18.: P.E. Newstead: Stable bundles of rank 2 and odd degree over a curve of genus 2. Topology 7 (1968), 205-215. MR 0237500 (38:5782)
19.: A. Ortega: Variétés de Prym associées aux revêtements -cycliques d'une courbe hyperelliptique. Math. Z. 245 (2003), 97-103. MR 2023955
20.: W.M. Oxbury: Prym varieties and the moduli of spin bundles. Lect. Notes Pure Appl. Math. 200, 1998, 351-376. MR 1651104 (2000a:14032)
21.: M. Raynaud: Sections des fibrés vectoriels sur une courbe. Bull. Soc. Math. France 110 (1982), 103-125. MR 0662131 (84a:14009)

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