Computing Jet Differentials and the Green-Griffiths-Lang Conjecture for Complements of Smooth Plane Curves

Abstract

We study the Green-Griffiths-Lang Conjecture for complements of smooth plane curves. We develop an effective method for computing a family of negatively twisted invariant logarithmic 2-jet differentials. By realizing the first logarithmic jet space as a hypersurface in $\mathbb{P}^2 \times \mathbb{P}^2$, we encode these jet differentials in a finitely generated bigraded module that can be computed explicitly. We use this description to give a computational criterion for the Green-Griffiths-Lang Conjecture and verify it for several families of smooth plane curves. In examples with sufficiently many independent jet differentials, we determine the exceptional locus explicitly.