arXiv subjects

Ryan Contreras

Publications and source records attributed to Ryan Contreras.

2 recordsLinked to original sources

arXiv2026

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

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.

math.AG↗

arXiv2021

Plane $\mathbb{A}^1$-curves on the complement of strange rational curves

A plane curve is called strange if its tangent line at any smooth point passes through a fixed point, called the strange point. In this paper, we study $\mathbb{A}^1$-curves on the complement of a rational strange curve of degree $p$ in characteristic $p$. We prove the connectedness of the moduli spaces of $\mathbb{A}^1$-curves with given degree, classify their irreducible components, and exhibit the inseparable $\mathbb{A}^1$-connectedness via the $\mathbb{A}^1$-curves parameterized by each irreducible component. The key to these results is the strangeness of all $\mathbb{A}^1$-curves. As an application, in every characteristic we construct explicit covering families of $\mathbb{A}^1$-curves, whose total spaces are smooth along large numbers of cusps on each general fiber.

math.AG↗