Abc Estimate, Integral Points, and Geometry of P minus Hyperplanes

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