On the generalized theta divisor
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. \textit{J.-M. Drezet} and \textit{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 \textit{M. Raynaud}, Bull. Soc. Math. Fr. 110, 103-125 (1982; Zbl 0505.14011)] and a weaker bound was found by \textit{J. Le Potier} [Lect. Notes Pure Appl. Math. 179, 83--101 (1996; Zbl 0890.14017)]. This author uses Le Potiers' construction of canonical sections of \(|L^{\otimes k}|\), loc. cit.
scientific article; zbMATH DE number 1018699