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On the generalized theta divisor. (English) Zbl 0890.14018

Let \(\text{SU}_X(n)\) denote the moduli space of semistable rank \(n\) bundles with trivial determinant on a Riemann surface \(X\) of genus \(>1\). Let \(L\) be the ample generator of the Picard group of \(\text{SU}_X(n)\) [cf. J.-M. Drezet and M. S. Narasimhan, Invent. Math. 97, No. 1, 53-94 (1989; Zbl 0689.14012)]. The main result of this paper is the base point freeness of the linear system \(|L^{\otimes k}|\), for any integer \(k>(n-1)^2(g-1)\).
This was known when \(n=2\) [see M. Raynaud, Bull. Soc. Math. Fr. 110, 103-125 (1982; Zbl 0505.14011)] and a weaker bound was found by J. Le Potier [in: “Moduli of vector bundles”. Papers of the 35th Taniguchi symposium, Sanda, Japan, and a symposium held in Kyoto, Japan, 1994, Lect. Notes Pure Appl. Math. 179, 83-101 (1996; see the preceding review)].
This author uses Le Potiers’ construction of canonical sections of \(|L^{\otimes k}|\), loc. cit.

MSC:

14H60 Vector bundles on curves and their moduli
14C20 Divisors, linear systems, invertible sheaves
14K25 Theta functions and abelian varieties
14H52 Elliptic curves
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