## 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)

Zbl 0691.14016
Full Text:

### 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
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