Morphisms from a Surface to a Curve. Elliptic and Quasielliptic Fibrations
Let f: X → Y be a dominant morphism from an irreducible, nonsingular algebraic variety X to an algebraic variety Y, such that the field extension f*: k(Y) → k(X) between the rational function fields is separable and k(Y) is algebraically closed in k(X). Then there exists a nonempty open subset V ⊂ Y such that for every y E V the fiber f −1(y) is geometrically integral.
KeywordsExact Sequence Effective Divisor Irreducible Curve Invertible Sheaf Smooth Morphism
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- Theorem 7.1, Lemma 7.2, and Corollary 7.3 are presented after [Sha2], and Proposition 7.4 after [Hari].Google Scholar
- The concept of a curve of canonical type was introduced by Mumford in [Mum4], from which we took Theorem 7.8, Corollaries 7.9 and 7.10. and Theorems 7.11 and 7.12.Google Scholar
- The notion of an exceptional fiber of an elliptic or quasielliptic fibration appeared for the first time in [BM1], from which we took Theorem 7.15 and Corollary 7.17.Google Scholar
- Theorem 7.18 is presented after [Mum4] and [3M2].Google Scholar