eo m / 9 70 30 26 v 1 2 1 M ar 1 99 7 Heisenberg invariant quartics and SU C ( 2 ) for a curve of genus four
نویسنده
چکیده
The projective moduli variety SUC(2) of semistable rank 2 vector bundles with trivial determinant on a smooth projective curve C comes with a natural morphism φ to the linear series |2Θ| where Θ is the theta divisor on the Jacobian of C. Well-known results of Narasimhan and Ramanan say that φ is an isomorphism to P if C has genus 2 [16], and when C is nonhyperelliptic of genus 3 it is an isomorphism to a special Heisenberg-invariant quartic QC ⊂ P 7 [18]. The present paper is an attempt to extend these results to higher genus. In the nonhyperelliptic genus 3 case the so-called Coble quartic QC ⊂ |2Θ| = P is characterised by either of two properties: (i) QC is the unique Heisenberg-invariant quartic containing the Kummer variety, i.e. the image of Kum : JC → |2Θ|, x 7→ Θx + Θ−x, in its singular locus; and (ii) QC is precisely the set of 2Θ-divisors containing some translate of the curve W1 ⊂ J C .
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