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Some varieties with points only in a field extension. (English) Zbl 0532.14011
Several authors have investigated the following problem. Let \(\Gamma\) be an irreducible algebraic variety of degree \(d\), in projective \(n\)-space \(\mathbb P^ n\), defined over a field \(k\); and suppose \(K\) is a finite extension of \(k\) with \([K:k]\) prime to \(d\). If \(\Gamma\) has a point defined over \(K\), then does it necessarily have a point defined over \(k\)? For instance, the result is known to be true for quadrics in \(\mathbb P^ n\), and for cubic plane curves; it is also true for the intersection of two quadrics, whereas it fails to hold for three quadrics. The current paper comprises several further counterexamples, which are the following.
(I) Two cubics in \(\mathbb P^ 2\) having a common zero over a quadratic extension of \(\mathbb Q\), but having no common zero in \(\mathbb Q\).
(II) Two examples of quartic curves defined over \(\mathbb Q\), everywhere locally solvable, having no rational point, yet possessing points defined over a cubic extension of \(\mathbb Q\).
(III) Two examples akin to those of (II), but now over the function field \(\mathbb Q(t)\).
(IV) A quartic form in 16 variables with a point over a cubic extension of \(\mathbb Q\), but with no point in \(\mathbb Q\) itself.
Reviewer: Andrew Bremner

14G25 Global ground fields in algebraic geometry
14H25 Arithmetic ground fields for curves
11D25 Cubic and quartic Diophantine equations
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