Morphisms of varieties over ample fields
Bull. Korean Math. Soc. 2018 Vol. 55, No. 4, 1023-1035
Published online July 1, 2018
Lior Bary-Soroker, Wulf-Dieter Geyer, Moshe Jarden
Tel Aviv University, Universitat Erlangen, Tel Aviv University
Abstract : We strengthen a result of Michiel Kosters by proving the following theorems: $(*)$ Let $\phi\colon W\to V$ be a finite surjective morphism of algebraic varieties over an ample field $K$. Suppose $V$ has a simple $K$-rational point ${\bf a}$ such that ${\bf a}\notin\phi(W(K_{ins}))$. Then, ${card}(V(K)\setminus \phi(W(K))={\rm card}(K)$. $(**)$ Let $K$ be an infinite field of positive characteristic and let $f\in K[X]$ be a non-constant monic polynomial. Suppose all zeros of $f$ in $\tilde K$ belong to $K_{\rm ins}\setminus K$. Then, ${\rm card}(K\setminus f(K))={\rm card}(K)$.
Keywords : ample fields, morphisms of varieties
MSC numbers : 12E30
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