2020/05/14 by Jean-Louis Colliot-Thélène, Colliot-Thélène, Jean-Louis · 2 citations
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #History and Theory of Mathematics
paper · pdf · doi:10.48550/arxiv.2005.06876
In 1974, D. Coray showed that on a smooth cubic surface with a closed point\nof degree prime to 3 there exists such a point of degree 1, 4 or 10. We first\nshow how a combination of generisation, specialisation, Bertini theorems and\nlarge fields avoids considerations of special cases in his argument. For smooth\ncubic surfaces with a rational point, we show that any zero-cycle of degree at\nleast 10 is rationally equivalent to an effective cycle. We establish analogues\nof these results for del Pezzo surfaces of degree 2 and of degree 1. For smooth\ncubic surfaces without a rational point, we relate the question whether there\nexists a degree 3 point which is not on a line to the question whether rational\npoints are dense on a del Pezzo surface of degree 1.\n ----\n Une surface cubique lisse qui poss `ede un point ferm 'e de degr 'e premier\n `a 3 poss `ede un tel point de degr 'e 1, 4 ou 10 (Coray, 1974). Un m 'elange\nde g 'en 'erisation, de sp 'ecialisation, de th 'eor `emes de Bertini et\nd'utilisation des corps fertiles donne de la souplesse `a sa m 'ethode. Pour\nles surfaces cubiques avec un point rationnel, on montre que tout z 'ero-cycle\nde degr 'e au moins 10 est rationnellement 'equivalent `a un z 'ero-cycle\neffectif. On 'etablit l'analogue de ces r 'esultats pour les surfaces de del\nPezzo de degr 'e 2 et de degr 'e 1. On discute l'existence de points ferm 'es\nde degr 'e 3 non align 'es sur une surface cubique sans point rationnel. On la\nrelie `a la question de la densit 'e des points rationnels sur une surface de\ndel Pezzo de degr 'e 1.\n