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PreprintCC BY 4.0arXiv

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

  • Ryan Contreras
  • Joseph Cummings
  • Eric Riedl
  • Jaziel Torres
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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 ℙ2 × ℙ2, 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.

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Cite

BibTeXRIS
BibTeX
@misc{contreras2026computing,
  title = {{Computing Jet Differentials and the Green-Griffiths-Lang Conjecture for Complements of Smooth Plane Curves}},
  author = {Ryan Contreras and Joseph Cummings and Eric Riedl and Jaziel Torres},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.20479},
  eprint = {2608.20479},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.20479},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Computing Jet Differentials and the Green-Griffiths-Lang Conjecture for Complements of Smooth Plane Curves
AU  - Ryan Contreras
AU  - Joseph Cummings
AU  - Eric Riedl
AU  - Jaziel Torres
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2608.20479
UR  - https://arxiv.org/abs/2608.20479
AB  - 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 ℙ^2 × 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.
ER  -