## 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

### Citations:

Zbl 0689.14012; Zbl 0505.14011; Zbl 0890.14017
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