Zero-cycles and rational points on some surfaces over a global function field

Details Zero-cycles and rational points on some surfaces over a global function field Zero-cycles and rational points on some surfaces over a global function field

2010-04-16

arxiv.org/abs/1004.2797v2

Mathematics

Algebraic Geometry

The title has changed and the text has been split into sections. In the statement of Theorems 1 and 3, "degree prime to char(F

Scientific paper

Let F be a finite field of characteristic p. We consider smooth surfaces over F(t) defined by an equation f+tg=0, where f and g are forms of degree d in 4 variables with coefficients in F, with d prime to p. We prove : For such surfaces over F(t), the Brauer-Manin obstruction to the existence of a zero-cycle of degree one is the only obstruction. For d=3 (cubic surfaces), this leads to the same result for rational points. -- Soit F un corps fini de caract\'eristique p. Pour une surface lisse sur F(t) d\'efinie par une \'equation f+tg=0, o\`u f et g sont deux formes de degr\'e d sur F en 4 variables, avec d premier \`a p, nous montrons que l'obstruction de Brauer-Manin au principe de Hasse pour les z\'ero-cycles de degr\'e 1 est la seule obstruction. Pour d=3 (surfaces cubiques), on en d\'eduit le m\^{e}me \'enonc\'e pour les points rationnels.

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