| Nontrivial Rational Polynomials In Two Variables Have Reducible Fibres (2007) | |||||||||||||
Abstract | |||||||||||||
| oof. This theorem is implicit in [7]. Suppose f is rational. As in [7], [6], [9], etc., we consider a nonsingular compactification Y = C 2 [ E of C 2 such that f extends to a holomorphic map f : Y ! P 1 . Then E is a union of smooth rational curves E 1 ; : : : ; En with normal crossings. An E i is called horizontal if f jE i is non-constant. Let ffi be the number of horizontal curves. Then we have ffi \Gamma 1 = X a2C (r a \Gamma 1); where r a is the number of irreducible components of f \Gamma1<F12. | |||||||||||||
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