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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 ̲) 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 ̲) 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. [ABSTRACT FROM AUTHOR]