Non-Archimedean analytic curves in the complements of hypersurface divisors

TL;DR

This paper investigates the behavior of non-archimedean analytic maps into complements of hypersurface divisors, establishing non-existence results under certain degree and position conditions in projective varieties.

Contribution

It provides new non-existence theorems for non-archimedean analytic maps into complements of hypersurfaces with specific degree and intersection properties.

Findings

No non-archimedean maps into complements of hypersurfaces of degree ≥ 2 in general position.

No maps into complements of two generic plane curves with degrees summing to ≥ 4.

Degeneration dimension of such maps is explicitly studied.

Abstract

We study the degeneration dimension of non-archimedean analytic maps into the complement of hypersurface divisors of smooth projective varieties. We also show that there exist no non-archimedean analytic maps into $P^n\setminus\cup_{i= 1}^n D_i$ where $D_i, 1\le i\le n$, are hypersurfaces of degree at least 2 in general position and intersecting transversally. Moreover, we prove that there exist no non-archimedean analytic maps into $P^2\setminus\cup_{i=1}^2 D_i$ when $D_1, D_2$ are generic plane curves with deg$D_1+$deg$D_2\ge 4$.