Computing the space of differential forms of a plane curve and its Cartier-Manin matrix

Abstract

In this paper, we propose a feasible algorithm to give an explicit basis of the space of regular differential forms on the nonsingular projective model of any given plane algebraic curve. The algorithm is demonstrated for concrete examples, with our implementation over the computer algebra system Magma. As an application, we also describe the Cartier-Manin matrix of the nonsingular projective curve with respect to the basis computed by the algorithm.