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Dimension of the space of sections of the generalized theta divisor. (La dimension de l’espace des sections du diviseur thêta généralisé.) (French) Zbl 0748.14011

Let \(C\) be a non-singular complex curve of genus \(g>1\). For a fixed line bundle \(\xi\) over \(C\) of given degree \(d\), denote by \({\mathcal M}(2,\xi)\) the (coarse) moduli space of isomorphism classes of rank-2 stable vector bundles over \(C\) with determinant line bundle isomorphic to \(\xi\). By a result by J.-M. Drezet and M. S. Narasimhan [Invent. Math. 97, No 1, 53-94 (1989; Zbl 0689.14012)], the Picard groups \(\text{Pic}{\mathcal M}(2,\xi))\) are freely generated by the line bundle \({\mathcal O_ M}(\Theta)\) associated with an ample divisor \(\Theta\) in \({\mathcal M}(2,\xi)\). This divisor \(\Theta\) is called the generalized theta divisor of \({\mathcal M}(2,\xi)\), and its global sections \(f\in H^ 0({\mathcal M}(2,\xi),{\mathcal O_ M}(\Theta))\) are called the generalized theta functions on \({\mathcal M}(2,\xi)\).
Recent developments in conformal quantum field theory gave rise to conjectures about the dimension of the spaces \(H^ 0({\mathcal M}(2,\xi),{\mathcal O_ M}(\Theta)^ k)\) of generalized theta functions of order \(k\) [cf. E. Verlinde and H. Verlinde, “Conformal field theory and geometric quantization”, Prepr. PUPT-89/1149 (1989)].
In the present paper, the author provides a partial verification of these conjectures, i.e., of the so-called Verlinde formulae. More precisely, he proves that in the special case of \(k=1\) and \(d=\deg\xi\) an odd integer, the conjectured formula \[ \dim_ \mathbb{C} H^ 0({\mathcal M}(2,\xi),{\mathcal O_ M}(\Theta))=2^{g-1}\centerdot (2^ g-1) \] indeed holds true. Moreover, using previous results of A. Beauville [Bull. Soc. Math. Fr. 119, No. 3, 259-291 (1991)], the author succeeds in exhibiting an explicit base for the vector space \(H^ 0({\mathcal M}(2,\xi),{\mathcal O_ M}(\Theta))\). The method of proof, in the particular case under investigation, is based upon the handy description of the moduli space \(M(2,\xi)\) for hyperelliptic ground curves, which is due to U. V. Desale and S. Ramanan [Invent. Math. 38, 161-185 (1976; Zbl 0323.14012)], and on a comparison argument with respect to the number of odd theta characteristics on the base curve \(C\).
A general verification of the conjectured Verlinde formulae, i.e., for \(\dim_ \mathbb{C} H^ 0({\mathcal M}(2,\xi),{\mathcal O_ M}(\Theta)^ k)\) with \(k\) and \(d=\deg \xi\) arbitrarily chosen, has recently been obtained by A. Szenes and A. Bertram [cf. “Hilbert polynomials of moduli spaces of rank-2 vector bundles”, I, II (Harvard-University, September 1991 and November 1991)]. An announcement and sketch of their general results was published by A. Szenes, after the appearance of the present article [cf. A. Szenes, Int. Math. Res. Not. 1991, No. 7, 93-98 (1991)].

MSC:

14H42 Theta functions and curves; Schottky problem
14H60 Vector bundles on curves and their moduli
14D22 Fine and coarse moduli spaces
14K25 Theta functions and abelian varieties
14H10 Families, moduli of curves (algebraic)
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References:

[1] BOURBAKI (N.) . - Groupe et algèbre de Lie , Chapitre 7. - Hermann, Paris, 1985 . · Zbl 0547.13001
[2] BEAUVILLE (A.) . - Fibrés de rang deux sur une courbe, fibré déterminant et fonctions thêta II , Bull. Soc. Math. France, t. 119, 1991 , p. 259-291. Numdam | MR 92m:14041 | Zbl 0756.14017 · Zbl 0756.14017
[3] BEAUVILLE (A.) , NARASIMHAN (M.S.) , et RAMANAN (S.) . - Spectral curves and the generalised theta divisor , J. Reine Angew. Math, t. 398, 1989 , p. 169-179. Article | MR 91c:14040 | Zbl 0666.14015 · Zbl 0666.14015
[4] DREZET (J.M.) et NARSIMHAN (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
[5] DESALE (U.V.) et RAMANAN (S.) . - Classification of vector bundles of rank 2 on hyperelliptic curves , Invent. Math., t. 38, 1976 , p. 161-185. MR 55 #2906 | Zbl 0323.14012 · Zbl 0323.14012
[6] HARTSORNE (R.) . - Principles of algebraic geometry , Graduate Text in Math. 51. - Springer Verlag, Berlin Heidelberg New-York, 1977 .
[7] HUMPHREYS (J.E.) . - Linear Algebraic groups , Graduate Text in Math. 21. - Springer Verlag, Berlin Heidelberg New-York, 1971 . MR 53 #633 | Zbl 0325.20039 · Zbl 0325.20039
[8] MACDONALD (I.G.) . - Symmetric functions and Hall polynomials . - Oxford Mathematical Monographs, Clarendon Press, 1979 . MR 84g:05003 | Zbl 0487.20007 · Zbl 0487.20007
[9] MUMFORD (D.) . - Tata lectures on theta II , Prog. in Math. 43. - Birkhäuser, Boston Basel Stuttgart, 1984 . MR 86b:14017 | Zbl 0549.14014 · Zbl 0549.14014
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