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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 [4] Kempf, G.: Toward the inversion of abelian integrals. II. Am. J. Math.101, 184-202 (1979) · Zbl 0452.14012 [5] Kempf, G.: Inversion of abelian integrals. Bull. Am. Math. Soc. (New Ser.)6, 25-32 (1982) · Zbl 0481.14008 [6] Mattuck, A.: Picard bundles. Ill. J. Math.5, 550-564 (1961) · Zbl 0107.14702 [7] Mattuck, A.: Symmetric products and Jacobians. Am. J. Math.83, 189-206 (1961) · Zbl 0225.14020 [8] Mumford, D.: Abelian varieties. Oxford: Oxford University Press 1970 · Zbl 0223.14022
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