A Schmidt's subspace theorem for moving hyperplane targets over function fields (Q6597143)

The authors study a function field analogue of Schmidt's subspace theorem for moving targets [\textit{M. Ru} and \textit{P. Vojta}, Invent. Math. 127, No. 1, 51--65 (1997; Zbl 1013.11044)]. Their main theorem is as follows:\N\NTheorem (Schmidt's subspace theorem for moving targets over function fields of characteristic zero). Let \(K\) be the function field of a non-singular variety \(V\) defined over an algebraically closed field \(k\) of characteristic zero. Let \(S\) be a finite set of prime divisors of \(V\). Let \(\Lambda\) be an infinite index set and let \(H_j\) be moving hyperplanes in \(\mathbb{P}^M(K)\), \(j = 1,\dots,q\), indexed by \(\Lambda\). \N\NLet \(\mathbf{x}= [x_0 : \cdots : x_M] : \Lambda\rightarrow \mathbb{P}^M(K)\) be a sequence of points. Assume that\N\begin{itemize}\N\item[1.] \(H_1(\alpha),\dots,H_q(\alpha)\) are in general position for every \(\alpha\in \Lambda\),\N\item[2.] \(\mathbf{x}\) is linearly non-degenerate with respect to \(H = \{ H_1 , \dots, H_q \}\),\N\item[3.] \(h(H_j(\alpha)) = o(h(\mathbf{x}(\alpha)))\) for all \(j = 1,\dots,q\), where \(h(\mathbf{x})\) denotes the logarithmic height of \(\mathbf{x}\).\N\end{itemize}\NThen, for any \(\varepsilon>0\), there exists an infinite index subset \(A\subseteq \Lambda\) such that \N\[\sum_{j=1}^q\sum_{\mathfrak{p}\in S}\lambda_{\mathfrak{p},H_j(\alpha)}(\mathbf{x}(\alpha))\leq (M+1+\varepsilon)h(\mathbf{x}(\alpha)) + O(1)\] \Nfor all \(\alpha\in A\).\N\NTo prove the theorem, the authors prove that for every \(\alpha\in \Lambda\), they can reduce the problem to the case in which \(H_i(\alpha)\) is constant for every \(i\), so that they can use the results of [\textit{J. T. Y. Wang}, Math. Z. 246, No. 4, 811--844 (2004; Zbl 1051.11041)] about Schmidt's subspace theorem for fixed hyperplanes.