On non-Archimedean curves omitting few components and their arithmetic analogues

Let k be an algebraically closed field complete with respect to a non-Archimedean absolute value of arbitrary characteristic. Let D_1,...,D_n be effective nef divisors intersecting transversally in an n-dimensional nonsingular projective variety X. We study the degeneracy of non-Archimedean analytic maps from k into $X\setminus \cup_{i=1}^nD_i$ under various geometric conditions. When X is a rational ruled surface and D_1 and D_2 are ample, we obtain a necessary and sufficient condition such that there is no non-Archimedean analytic map from k into $X\setminus D_1 \cup D_2$. Using a dictionary between non-Archimedean Nevanlinna theory and Diophantine approximation, we also study arithmetic analogues of these problems, establishing results on integral points on these varieties over the integers or the ring of integers of an imaginary quadratic field.

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