Invariant two-jets and effective hyperbolicity for complements of two plane curves

Abstract

Let $D=C_1+C_2\subset\mathbb{P}^2$ be a simple normal crossing union of smooth plane curves of degrees $1\leqslant d_1\leqslant d_2$. We prove an effective Second Main Theorem for a general ordered pair whenever \[ d_1,d_2\geqslant3, \qquad\text{or}\qquad d_1=2,\ d_2\geqslant5, \qquad\text{or}\qquad d_1=1,\ d_2\geqslant8. \] For each admissible degree pair, there is a nonempty Zariski-open set of ordered pairs $(C_1,C_2)$ for which every algebraically nondegenerate entire curve $f:\mathbb{C}\to\mathbb{P}^2$ whose image is not contained in $D=C_1+C_2$ satisfies \[ T_f(r)\leqslant \mathcal{A}_{d_1,d_2}N_f^{[1]}(r,D)+o(T_f(r)) \ \|. \] For two cubics one may take $\mathcal{A}_{3,3}=57$; for a conic and a quintic, $\mathcal{A}_{2,5}=45$; and for a line and an octic, $\mathcal{A}_{1,8}=69$. Intersecting the resulting Zariski-open parameter locus with Xi Chen's very-general algebraic-hyperbolicity locus yields Kobayashi hyperbolicity and hyperbolic embedding of the complement. The proof first constructs one negatively twisted invariant two-jet differential. It then obtains a second equation either from a Demailly--El Goul zero-locus argument or by differentiating with mixed $\mathcal{O}_{\mathbb{P}^2}(3)$ slanted vector fields. A finite calculation is needed only for a short list of low twists. In those cases, exact rank certificates over finite fields prove the required Key Vanishing Lemma.