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Spectral curves and the generalized theta divisor. (English) Zbl 0666.14015
Let X be a compact Riemann surface. It is shown in this paper that a generic vector bundle on X of rank n can be obtained as the direct image of a line bundle on an n-sheeted (ramified) covering. This is used to prove the following results. Let \({\mathcal U}(n)\) (resp. \({\mathcal S}{\mathcal U}(n))\) denote the moduli space of semi-stable vector bundles of rank n and degree 0 (resp. trivial determinant) on X. For \(\xi \in J^{g-1}(X)\) and \(E\in {\mathcal U}(n)\), one defines (Cartier) divisors \(\Delta_{\xi}\) on \({\mathcal U}(n)\) and \(D_ E\) on \(J^{g-1}(X)\) by \(\Delta_{\xi}=\{E\in {\mathcal U}(n)|\) \(\Gamma(E\otimes \xi)\neq 0\}\) \(D_ E=\{\xi \in J^{g- 1}| \quad \Gamma (E\otimes \xi)\neq 0\}.\) Then:
(a) One has \(\dim\Gamma({\mathcal U}(n), {\mathcal O}(\Delta_{\xi}))=1\) for all \(\xi\) in \(J^{g-1}(X);\)
(b) The restriction h of \({\mathcal O}(\Delta_{\xi})\) to \({\mathcal S}{\mathcal U}(n)\) is independent of \(\xi\), and generates Pic(\({\mathcal S}{\mathcal U}(n));\)
(c) There is a canonical isomorphism \(\Gamma\) (\({\mathcal S}{\mathcal U}(n),h)^*\to \Gamma (J^{g-1}(X)\), \({\mathcal O}(n\Theta))\). Via this isomorphism, the rational \({\mathcal S}{\mathcal U}(n)\to | h|^*\) defined by the linear system of h is identified with the map \(E\mapsto D_ E\) from \({\mathcal S}{\mathcal U}(n)\) to \(| n\Theta |\) (in particular, dim \(\Gamma\) (\({\mathcal S}{\mathcal U}(n),h)=n^ g).\)
In the rank 2 case, these results had been previously obtained by the reviewer [Bull. Soc. Math. Fr. 116 (1988)].
Reviewer: A.Beauville

14H40 Jacobians, Prym varieties
14K25 Theta functions and abelian varieties
30F10 Compact Riemann surfaces and uniformization
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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