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Integrable systems and moduli spaces of rank two vector bundles on a non-hyperelliptic genus 3 curve. (English) Zbl 1087.14027

The paper under review is concerned with the study of the moduli space of rank 2 vector bundles with fixed determinant on a non-hyperelliptic compact Riemann surface of genus 3. Let \(M\) denote the natural compactification of this moduli space. A result of M. S. Narasimhan and S. Ramanan [in: Vector bundles on algebraic varieties, Pap. Colloq., Bombay 1984, Stud. Math., Tata Inst. Fundam. Res. 11, 415–427 (1987; Zbl 0685.14023)] claims that \(M\) is a quartic hypersurface in \({\mathbb P}^7\). Many importants facts on \(M\) are known; e.g. Wirtinger duality yields a canonical isomorphism between its singular locus and the Kummer variety of the Riemann surface and, furthermore, the quartic hypersurface coincides with the Coble’s quartic [see M. S. Narasimhan and S. Ramanan, Ann. Math. (2) 89, 14–51 (1969; Zbl 0186.54902)]; for the Coble quartic see [A. B. Coble, “Algebraic geometry and theta functions”, AMS Colloquium Publications, vol. 10 (1929; JFM 55.0808.02)].
In the paper under review, the author computes an explicit equation of this quartic hypersurface for a family of non-hyperelliptic compact Riemann surfaces of genus 3. Indeed, the coefficients of such equation are given in terms of the algebraic equation of the Riemann surface as a plane algebraic curve. The present approach should be useful when dealing with problems of deformation theory for \(M\) as the Riemann surface varies. The technique consists of thinking the Riemann surface as the spectral curve of an integrable system. Then, an explicit embedding of the Kummer variety into \({\mathbb P}^7\) is given. Eventually, the desired equation is obtained through an integration procedure.

MSC:

14H60 Vector bundles on curves and their moduli
14H70 Relationships between algebraic curves and integrable systems
35Q58 Other completely integrable PDE (MSC2000)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
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