Pure and Applied Mathematics Quarterly

Volume 13 (2017)

Number 1

Special Issue in Honor of Yuri Manin: Part 1

Guest Editors: Lizhen Ji, Kefeng Liu, Yuri Tschinkel, and Shing-Tung Yau

Normes de droites sur les surfaces cubiques

Pages: 123 – 130

DOI: http://dx.doi.org/10.4310/PAMQ.2017.v13.n1.a4


J.-L. Colliot-Thélène (Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, Orsay, France)

D. Loughran (School of Mathematics, University of Manchester, United Kingdom)


Let $k$ be a field and $X \subset \mathbf{P}^3_k$ a smooth cubic surface. Let $\Delta = \Delta (X) \subset \mathrm{Pic}(X)$ be the subgroup spanned by norms to $k$ of $K$-lines on $X_K = X \times \negmedspace {}_k \: K$ for $K$ running through the finite separable extensions of $k$. The quotient $\mathrm{Pic}(X) / \Delta$ is a finite, $3$-primary group. If $X$ contains a line defined over $k$, then $\Delta = \mathrm{Pic}(X)$.

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Received 19 September 2017

Published 14 September 2018