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

TL;DR

This paper proves that for certain smooth surfaces over a global function field, the Brauer-Manin obstruction is the only barrier to the existence of zero-cycles of degree one, and for cubic surfaces, to rational points.

Contribution

It establishes the sufficiency of the Brauer-Manin obstruction for zero-cycles and rational points on specific surfaces over a global function field, extending previous results.

Findings

Brauer-Manin obstruction is the only obstruction for zero-cycles of degree one.

For cubic surfaces, the same holds for rational points.

Results apply to surfaces defined by f+tg=0 with d prime to p.

Abstract

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.