An algorithm for a lifted Massey triple product of a smooth projective plane curve

Abstract

We provide an explicit algorithm to compute a lifted Massey triple product relative to a defining system for a smooth projective plane curve X defined by a homogeneous polynomial G((x) under bar) over a field. The main idea is to use the description (due to Carlson and Griffiths) of the cup product for H-1(X, C) in terms of the multiplications inside the Jacobian ring of G((x) under bar) and the Cech-deRham complex of X. Our algorithm gives a criterion whether a lifted Massey triple product vanishes or not in H-2(X) under a particular nontrivial defining system of the Massey triple product and thus can be viewed as a generalization of the vanishing criterion of the cup product in H-2(X) of Carlson and Griffiths. Based on our algorithm, we provide explicit numerical examples by running the computer program.