A Geometric Interpretation of Reduction in the Jacobians of C ab Curves

Abstract

In this paper, we show that the reduction of divisors in the Jacobian of a curve $C$ can be performed by considering the intersections of a suitable projective model of $C$ with quadrics in projective space. We apply this idea to certain projective model of elliptic and hyperelliptic curves on one hand, and to the canonical model of $C_{ab}$ curves on the other hand, and we generalize (and recover) some well known algorithms.