abc estimate, integral points, and geometry of $\bold P^\bold n$ minus hyperplanes

Let $K$ be a field and $\Cal H$ be a set of hyperplanes in $P^n(K)$. When $K$ is a function field, we show that the following are equivalent. (a) $\Cal H$ is nondegenerate over $K$. (b) The height of the $(S,\Cal H)$-integral points of $P^n(K)-\Cal H$ is bounded. (c) $P^n_K-\Cal H$ is an abc variety. When $K$ is a number field and $\Cal H$ is nondegenerate over $K$, we establish an explicit bound on the number of $(S,\Cal H)$-integral points of $P^n(K)-\Cal H$. Finally, we discuss the geometric properties of holomorphic maps into $P^n(\Bbb C)$ omitting a set of hyperplanes with moving targets.

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Published 1 January 1999