2\(\theta\)-linear systems on abelian varieties. (English) Zbl 0685.14023

Vector bundles on algebraic varieties, Pap. Colloq., Bombay 1984, Stud. Math., Tata Inst. Fundam. Res. 11, 415-427 (1987).
[For the entire collection see Zbl 0653.00006.]
Let X be a projective nonsingular curve of genus g and let \(SU_ X(2)\) be the moduli space of semistable vector bundles of rank 2 having trivial determinant. Let J be the Jacobian of X and let \(\theta\) be the natural divisor on \(J^{g-1}\). In a former paper [cf. Ann. Math., II. Ser. 89, 14-51 (1969; Zbl 0186.549)], the authors showed: If \(g=2\) then \(SU_ X(2)\) is canonically isomorphic to the projective space of divisors on J, linearly equivalent to \(2\theta\) [and is smooth]; if \(g\geq 3\) then the Kummer variety \({\mathcal K}\)- which is canonically embedded in \(SU_ X(2)\)- is the singular locus of \(SU_ X(2)\). In this paper, the authors sketch a proof of the following theorem: If X is non-hyperelliptic of genus 3, then \(SU_ X(2)\) is isomorphic to a quartic hypersurface in \({\mathbb{P}}_ 7\), and the Kummer surface \({\mathcal K}\) can be defined by cubic polynomials. Details of the proof are to appear elsewhere.
[One should also compare a paper by A. Beauville and the authors, cf. J. Reine Angew. Math. 398, 169-179 (1989; Zbl 0666.14015)].
Reviewer: K.Kiyek


14K10 Algebraic moduli of abelian varieties, classification
14K25 Theta functions and abelian varieties
14H52 Elliptic curves