## 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

### MSC:

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

### Citations:

Zbl 0653.00006; Zbl 0186.549; Zbl 0666.14015