Entire holomorphic curves into ℙⁿ(ℂ) intersecting n+1 general hypersurfaces

2023/10/09 by Chen, Zhangchi, Huynh, Dinh Tuan, Sun, Ruiran +1 · 1 citation
#14J99 #32H30 #32Q45 #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2310.05433

Abstract

Let \D_i\_i=1ⁿ⁺¹ be n+1 hypersurfaces in ℙⁿ(ℂ) with total degrees ∑_i=1ⁿ⁺¹ °D_i\geqslant n+2, in general position and satisfying a generic geometric condition: every n hypersurfaces intersect only at smooth points and the intersection is transversal. Then, for every algebraically nondegenerate entire holomorphic curve f\colonℂ→ℙⁿ(ℂ), we show a Second Main Theorem: ∑_i=1ⁿ⁺¹ δ_f(D_i) lt; n+1 in terms of defect inequality in Nevanlinna theory. This is the first result in the literature on Second Main Theorem for n+1 general hypersurfaces in ℙⁿ(ℂ) with optimal total degrees.

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