Paper detail

Hyperbolic Embeddings in Toric Geometry: Effectivity and Deformation Stability

We study the deformation behavior of Kobayashi hyperbolic embeddings for complements of divisors in projective toric varieties. In the toric setting, entire curves in divisor complements propagate along algebraic subtori, allowing hyperbolicity questions to be translated into combinatorial conditions on lattice-point configurations of Newton polytopes. Building on a theorem of Tiba, which guarantees hyperbolic embedding for a general divisor under suitable facewise lattice conditions, we develop an effective refinement of his argument. We construct an explicit Zariski closed exceptional locus in the coefficient parameter space, characterized by the presence of translated subtori in the support or complement of the divisor. This description makes the exceptional set amenable to explicit computation. Using this effectivity, we prove a deformation stability result: along any algebraic one--parameter family of divisors whose initial member avoids the exceptional locus, hyperbolic embedding fails for at most finitely many parameters. Under strengthened lattice-point hypotheses, we further exhibit distinguished one--parameter families for which hyperbolic embedding persists without any exceptional parameters. We also analyze deformations arising from diagonal torus reparametrizations, showing that the number of exceptional parameters in such similarity families is controlled by a simple multiplicative rank invariant of the scaling vector. Finally, we illustrate the theory through explicit examples in projective spaces and Hirzebruch surfaces, including a benchmark computation in $\mathbb{P}^3$ where the exceptional locus contains a hypersurface component of degree $126$.