Cartan's Conjecture for Moving Hypersurfaces

Abstract

Let $f$ be a holomorphic curve in ${\mathbb{P}}^n(\mathbb{C})$ and let $\mathcal{D}=\{D_1,\dots ,D_q\}$ be a family of moving hypersurfaces defined by a set of homogeneous polynomials $\mathcal{Q}=\{Q_1,\dots ,Q_q\}$. For $j=1,\dots ,q$, denote by $Q_j=\sum _{i_0+\cdots +i_n=d_j}{a}_{j,I}(z)x_0^{i_0}\cdots x_n^{i_n}$, where $I=(i_0,\dots ,i_n)\in {\mathbb{Z}}_{\ge 0}^{n+1}$ and $a_{j,I}(z)$ are entire functions on $\mathbb{C}$ without common zeros. Let ${\mathcal{K}}_{\mathcal{Q}}$ be the smallest subfield of meromorphic function field $\mathcal{M}$ which contains $\mathbb{C}$ and all $\frac{a_{j,I^{\prime }}(z)}{a_{j,I^{\prime \prime }}(z)}$ with $a_{j,I^{\prime \prime }}(z)\not\equiv 0$, $1\le j\le q$. In previous known second main theorems for $f$ and $\mathcal{D}$, $f$ is usually assumed to be algebraically nondegenerate over ${\mathcal{K}}_{\mathcal{Q}}$. In this paper, we prove a second main theorem in which $f$ is only assumed to be nonconstant. This result can be regarded as a generalization of Cartan's conjecture for moving hypersurfaces.

arXiv commentWe add a section of preliminaries on algebraic geometry over general fields, updated the information of the references, and we also add a Lemma 3.4 which was proved by Dethloff-Tran for the sake of completeness