×

Fibre bundles of rank two on a curve, the determinant bundle and theta functions. (Fibrés de rang deux sur une courbe, fibré déterminant et fonctions thêta. II.) (French) Zbl 0756.14017

[For part I see ibid. 116, No. 4, 431-448 (1988; Zbl 0691.14016).]
Let \(C\) be a smooth and complete algebraic curve of genus \(g\) over the field \(\mathbb{C}\) of complex numbers and let \({\mathcal M}_ d\) be the moduli space of semi-stable vector bundles of rank 2, degree \(d\) and fixed determinant. Let \({\mathcal L}\) be the determinant bundle over \({\mathcal M}_ d\). In this paper the author constructs two natural homomorphisms \(\varphi^*_ 0:S^ 2V\to H^ 0({\mathcal M}_ 0,{\mathcal L}^ 2)\), \(\varphi^*_ p:\bigwedge^ 2V\to H^ 0({\mathcal M}_ 1,{\mathcal L})\) (the second depending on a point \(p\in C)\), where \(V\) is the vector space of second-order theta functions on the Jacobian of \(C\). These homomorphisms are in general isomorphisms. The author constructs in this way bases \(\{d_ k\}\) for the vector spaces \(H^ 0({\mathcal M}_ 0,{\mathcal L}^ 2)\) and \(H^ 0({\mathcal M}_ 1,{\mathcal L})\) in terms of the theta- characteristics of the curve. These results are based on the study of the Kummer variety of \(C\) and the study of the action of the Heisenberg group on \(S^ 2V\) and \(\bigwedge^ 2V\).

MSC:

14H42 Theta functions and curves; Schottky problem
14M12 Determinantal varieties
14K25 Theta functions and abelian varieties
14H10 Families, moduli of curves (algebraic)

Citations:

Zbl 0691.14016
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] BEAUVILLE (A.) . - Fibrés de rang 2 sur une courbe, fibré déterminant et fonctions thêta , Bull. Soc. Math. France, t. 116, 1988 , p. 431-448. Numdam | MR 91b:14038 | Zbl 0691.14016 · Zbl 0691.14016
[2] DREZET (J.-M.) et NARASIMHAN (M.S.) . - Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques , Invent. Math., t. 97, 1989 , p. 53-94. MR 90d:14008 | Zbl 0689.14012 · Zbl 0689.14012
[3] DESALE (R.V.) et RAMANAN (S.) . - Classification of vector bundles of rank 2 on hyperelliptic curves , Invent. Math., t. 38, 1976 , p. 181-185. MR 55 #2906 | Zbl 0323.14012 · Zbl 0323.14012
[4] KNUDSEN (F.) et MUMFORD (D.) . - The projectivity of the moduli space of stable curves , I, Math. Scand., t. 39, 1976 , p. 19-55. MR 55 #10465 | Zbl 0343.14008 · Zbl 0343.14008
[5] LASZLO (Y.) . - Dimension de l’espace des sections du diviseur thêta généralisé , Bull. Soc. Math. France, t. 119, 1991 , p. 293-306. Numdam | MR 92i:14009 | Zbl 0748.14011 · Zbl 0748.14011
[6] MUMFORD (D.) . - On the equations defining Abelian varieties , Invent. Math., t. 1, 1966 , p. 287-354. MR 34 #4269 | Zbl 0219.14024 · Zbl 0219.14024
[7] MUMFORD (D.) . - Theta characteristics of an algebraic curve , Ann. Sci. École Norm. Sup., t. 4, 1971 , p. 181-192. Numdam | MR 45 #1918 | Zbl 0216.05904 · Zbl 0216.05904
[8] NARASIMHAN (M.S.) et RAMANAN (S.) . - Moduli of vector bundles on a compact Riemann surface , Ann. of Math., t. 89, 1969 , p. 19-51. MR 39 #3518 | Zbl 0186.54902 · Zbl 0186.54902
[9] THADDEUS (M.) . - Conformal field theory and the cohomology of the moduli space of stable bundles , - Preprint, 1990 .
[10] VERLINDE (E.) . - Fusion rules and modular transformations in 2D-conformal field theory , Nuclear Phys., t. B300, 1988 , p. 360. MR 89h:81238 · Zbl 1180.81120
[11] WITTEN (E.) . - Quantum field theory and the Jones polynomial , Comm. Math. Phys., t. 121, 1989 , p. 351-399. Article | MR 90h:57009 | Zbl 0667.57005 · Zbl 0667.57005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.