English

Entire holomorphic curves into $\mathbb{P}^n(\mathbb{C})$ intersecting $n+1$ general hypersurfaces

Complex Variables 2023-11-30 v2 Algebraic Geometry

Abstract

Let {Di}i=1n+1\{D_i\}_{i=1}^{n+1} be n+1n+1 hypersurfaces in Pn(C)\mathbb{P}^n(\mathbb{C}) with total degrees i=1n+1degDin+2\sum_{i=1}^{n+1} \deg D_i\geqslant n+2, in general position and satisfying a generic geometric condition: every nn hypersurfaces intersect only at smooth points and the intersection is transversal. Then, for every algebraically nondegenerate entire holomorphic curve f ⁣:CPn(C)f\colon\mathbb{C}\rightarrow\mathbb{P}^n(\mathbb{C}), we show a Second Main Theorem: i=1n+1δf(Di)<n+1 \sum_{i=1}^{n+1} \delta_f(D_i) < n+1 in terms of defect inequality in Nevanlinna theory. This is the first result in the literature on Second Main Theorem for n+1n+1 general hypersurfaces in Pn(C)\mathbb{P}^n(\mathbb{C}) with optimal total degrees.

Keywords

Cite

@article{arxiv.2310.05433,
  title  = {Entire holomorphic curves into $\mathbb{P}^n(\mathbb{C})$ intersecting $n+1$ general hypersurfaces},
  author = {Zhangchi Chen and Dinh Tuan Huynh and Ruiran Sun and Song-Yan Xie},
  journal= {arXiv preprint arXiv:2310.05433},
  year   = {2023}
}

Comments

20 pages, 6 figures, comments are welcome

R2 v1 2026-06-28T12:44:16.169Z