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 $$Q_j=\sum \nolimits _{i_0+\cdots +i_n=d_j}a_{j,I}(z)x_0^{i_0}\cdots x_n^{i_n}$$ , 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.