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.

Added AI disclosure and corrected a few typos