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Inversion of Abelian integrals on small genus curves. (English) Zbl 0579.14023
Let M be a smooth projective curve of genus \(g>0\). Let J be the Jacobian of M and let \(M^{(i)}\) be the i-th symmetric product. Fix a base point \(p\in M\) and let \(\phi_ i: M^{(i)}\to J\) be the Abel-Jacobi map. For \(i\geq 2g-1\), Mattuck has shown that \(\phi_ i: M^{(i)}\to J\) is a \({\mathbb{P}}^{i-g}\)-bundle. The most interesting case occurs when \(i=2g-1\), since the bundles for larger i are determined by this one [see the book of R. C. Gunning on this subject entitled ”Riemann surfaces and generalized theta functions” (1976; Zbl 0341.14013)]. The inversion of abelian integrals problem asks: What is an explicit description of the transition functions of the bundle \(\phi_ i ?\) In this paper we present a solution to this question when M has genus 2 and when M is non-hyperelliptic of genus 3. For further background of this problem, see the cited book as well as the articles of Gunning, Kempf and Mattuck.

MSC:
14H05 Algebraic functions and function fields in algebraic geometry
14H40 Jacobians, Prym varieties
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References:
[1] Gunning, R.C.: Riemann surfaces and generalized theta functions. Berlin, Heidelberg, New York: Springer 1976 · Zbl 0341.14013
[2] Gunning, R.C.: On generalized theta functions. Am. J. Math.104, 183-208 (1982) · Zbl 0503.14021
[3] Kempf, G.: Toward the inversion of abelian integrals. I. Ann. Math.110, 243-273 (1979) · Zbl 0452.14011
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