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

TL;DR

This paper introduces an explicit algorithm to compute Massey triple products on smooth projective plane curves, linking algebraic and geometric methods, and provides practical examples through computational implementation.

Contribution

It presents a novel algorithm for computing Massey triple products on plane curves using Jacobian rings and Cech-deRham complexes, extending previous vanishing criteria.

Findings

Algorithm successfully computes Massey triple products.

Provides criteria for vanishing of Massey products.

Includes explicit numerical examples.

Abstract

We provide an explicit algorithm to compute a Massey triple product relative to a defining system for a smooth projective plane curve $X$ defined by a homogeneous polynomial $G(\underline x)$ over a field. The main idea is to use the description (due to Carlson and Griffiths) of the cup product for $H^1(X,\mathbb{C})$ in terms of the multiplications inside the Jacobian ring of $G(\underline x)$ and the Cech-deRham complex of $X$. Our algorithm gives a criterion whether a Massey triple product vanishes or not in $H^2(X)$ under a particular non-trivial 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.