A general defect relation and height inequality for divisors in subgeneral position
The paper deals with a defect relation for holomorphic curves. Let \(X\) be an \(n\)-dimensional complex projective variety. Let \(D_1,\ldots,D_q\) be effective Cartier divisors on \(X\). We say that \(D_1,\ldots,D_q\) are in \(l\)-subgeneral position if for any subset \(I\subseteq \{1,\ldots,q\}\) with \(\sharp I\leq l+1\), \[ \dim \bigcap_{i\in I} \text{Supp}~D_i \leq l-\sharp I, \] where \(\dim\emptyset =-1\). Let \(\mathscr{L}\) be a line sheaf on \(X\), we set \(h^0(\mathscr{L})=H^0(X, \mathscr{L})\) and \(\mathscr{L}(-D)=\mathscr{L}\otimes \mathscr{O}(-D)\) for a divisor \(D\) on \(X\). For a nonzero effective Cartier divisor \(D\) on \(X\), we define \[ \gamma(\mathscr{L}, D)=\limsup_{N\to +\infty} \frac{Nh^0(\mathscr{L}^N)}{\sum_{m\geq 1}h^0(\mathscr{L}^N(-mD))}, \] where \(N\) passes over all positive integer such that \(h^0(\mathscr{L}^N(-D))\neq 0\). In the case where \(h^0(\mathscr{L}^N(-D))\neq 0\) for all \(N\), we set \(\gamma(\mathscr{L}, D)=+\infty\). Then the main result in this paper can be stated as follows: Let \(X\) and \(D_j\) be as above. Suppose that \(D_1,\ldots,D_q\) are in \(l\)-subgeneral position with \(l+n-2\). Let \(\mathscr{L}\) be a line sheaf with \(h^0(\mathscr{L}^N(-D))\neq 0\) for sufficiently big \(N\). Let \(f:{\mathbb C}\to X\) be a holomorphic curve with Zariski dense image. Then for \(\varepsilon>0\), \[ \sum_{j=1}^q m_f(r, D_j)\leq_{\mathrm{exc}}\frac{l(l-1)}{(l+n-2)} \left(\max_{1\leq j \leq q}\gamma(\mathscr{L}, D_j)+\varepsilon\right) T_{f, \mathscr{L}}. \] Here \(T_{f, \mathscr{L}}:=T_{f, D}\) for \(D \in |H^0(X, \mathscr{L})|\). The authors also give a general height inequality for rational points approximating the given divisors on algebraic varieties.
scientific article; zbMATH DE number 6915168