Principe de Hasse pour les intersections de deux quadriques

TL;DR

Under certain hypotheses, smooth intersections of two quadrics in projective space satisfy the Hasse principle for dimensions greater than four, including specific cases of del Pezzo surfaces of degree four.

Contribution

The paper proves the Hasse principle for intersections of two quadrics under Schinzel's hypothesis and Tate-Shafarevich finiteness, extending to del Pezzo surfaces of degree four with specific conditions.

Findings

Hasse principle holds for n>4 under given hypotheses

Results apply to del Pezzo surfaces of degree 4 with specific Brauer group conditions

Extends known cases of the Hasse principle for algebraic surfaces

Abstract

Admettant l'hypoth\`ese de Schinzel et la finitude des groupes de Tate-Shafarevich des courbes elliptiques sur les corps de nombres, toute intersection lisse de deux quadriques dans l'espace projectif de dimension n satisfait au principe de Hasse si n>4. Le m\^eme r\'esultat vaut pour n=4, c'est-\`a-dire pour les surfaces de del Pezzo de degr\'e 4, lorsque le groupe de Brauer est r\'eduit aux constantes et que la surface est suffisamment g\'en\'erale. ----- Assuming Schinzel's hypothesis and the finiteness of Tate-Shafarevich groups of elliptic curves over number fields, smooth intersections of two quadrics in n-dimensional projective space satisfy the Hasse principle if n>4. The same result holds for n=4, i.e., for del Pezzo surfaces of degree 4, provided the Brauer group is reduced to constants and the surface is sufficiently general.