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Points having the same residue field as their image under a morphism. (English) Zbl 1077.14505

From the text: Our result, loosely speaking, is that in a nontrivial family of varieties \(f:X\to Y\) over a perfect field \(k\), some fiber \(X_t= f^{-1}(t)\) has a point rational over the field of definition of \(t\). More precisely, denote by \(\overline{f(X)}\) the scheme-theoretic image of a morphism \(f:X\to Y\) between noetherian schemes, and by \(\kappa(x)\) the residue field of a point \(x\) of a scheme. The main results is
Theorem 1. Let \(X\) and \(Y\) be schemes of finite type over a field \(k\). Let \(f:X\to Y\) be a \(k\)-morphism such that \(\dim\overline {f(X)}\geq 1\). Then there exists a closed point \(x\in X\) such that the extension \(\kappa (x)\) of \(\kappa(f(x))\) is purely inseparable.

MSC:

14A15 Schemes and morphisms
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References:

[1] Colliot-Thélène, J.-L.; Poonen, B., Algebraic families of nonzero elements of Shafarevich-Tate groups, J. Amer. Math. Soc., 13, 83-99 (2000) · Zbl 0951.11022
[2] Grothendieck, A., Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math., 32 (1967) · Zbl 0153.22301
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