Inversion of abelian integrals on small genus curves

Abstract

Let M be a smooth projective curve of genus g > 0. Let J be the Jacobian of M and let M ") be the i th symmetric product. Fix a base point PeM and define a map ~bi: M ") ~ J by Oi(D) = DiP. Mattuck [6, 7] has shown that, if i > 2g-1, then 0i is a Ipi-g-bundle. The most interesting case occurs when i = 2g-1, since in [1 ] the bundles for larger i are determined by this one, The inversion of abelian integrals problem asks: What is an explicit description of the transition functions of the bundle 0~ ? For an excellent introduction to the problem, see Kempf's article [5]. Until now, the only complete answer was given in genus 1 by the Abel Inversion Theorem, the Riemann-Roch Theorem, and Riemann's approach through theta functions. Even without the transition functions, Gunning [1, 2] and Kempf [3,4] were able to extract a great deal of information about M from the bundles 0,. In this paper, we present a solution to the inversion of abelian integrals problem in the cases (i) curves of genus 2 (ii) non-hyperelliptic curves of genus 3. The techniques we use are very geometric but essentially elementary. We strongly emphasize the role played by effective divisors and by the Riemann-Roch Theorem. To build sections, we use the commutative diagram M(g) +Oo ~ M(2g-1) j +no-(g-l)e ~ j where D O is a fixed effective divisor of degree g-1. The Jacobi Inversion Theorem says that 4~ is a birational surjection; hence, it has an inverse on an open set. It is then necessary to choose enough different D0's to trivialize the bundle. The geometry first enters when we must describe the open set over which a given collection of Do, s determines a trivialization. We then use explicit knowledge of the