Cartan's Conjecture for Moving Hypersurfaces

Abstract

Let $f$ be a holomorphic curve in $\mathbb{P}^n({\mathbb{C}})$ and let $\mathcal{D}=\{D_1,\ldots,D_q\}$ be a family of moving hypersurfaces defined by a set of homogeneous polynomials $\mathcal{Q}=\{Q_1,\ldots,Q_q\}$. For $j=1,\ldots,q$, denote by $Q_j=\sum\limits_{i_0+\cdots+i_n=d_j}a_{j,I}(z)x_0^{i_0}\cdots x_n^{i_n}$, where $I=(i_0,\ldots,i_n)\in\mathbb{Z}_{\ge 0}^{n+1}$ 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 $\frac{a_{j,I'}(z)}{a_{j,I''}(z)}$ with $a_{j,I''}(z)\not\equiv 0$, $1\le j\le q$. 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.