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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

MSC:
 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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