Geometry and arithmetic of certain log K3 surfaces

Harpaz, Yonatan

Annales de l'Institut Fourier, Tome 67 (2017) no. 5, pp. 2167-2200.

Dans cet article, on décrit une classification de surfaces log K3 lisses dont le groupe de Picard géométrique s’annule, et qui peuvent être réalisées comme compléments de diviseurs anti-canoniques à croisements normaux simples dans les surfaces de del Pezzo. On montre qu’une telle surface log K3 admet une compactification en une surface de del Pezzo de degree 5, avec un lacet de cinq $(-1)$-courbes comme complément, et qu’elle est déterminée à isomorphisme près par l’action de Galois sur le graphe dual du lacet. Quand le corps de base est le corps de nombres rationnels et l’action de Galois est triviale, on montre que l’ensemble des points entiers n’est pas Zariski dense sur n’importe quel modèle entier. On montre également que l’obstruction de Brauer–Manin n’est pas la seule obstruction au principe de Hasse entier pour de telles surfaces log K3, même quand ils admettent une compactification « scindée ».

In this paper we describe a classification of smooth log K3 surfaces whose geometric Picard group is trivial and which can be embedded as complements of simple normal crossing anti-canonical divisors in del Pezzo surfaces. We show that such a log K3 surface can be compactified into a del Pezzo surface of degree 5, with a compactifying divisor a cycle of five $(-1)$-curves, and is in fact determined up to isomorphism by the Galois action on the dual graph of the compactifying divisor. When the ground field is the field of rational numbers and the Galois action is trivial, we prove that the set of integral points is not Zariski dense on any integral model. We also show that the Brauer Manin obstruction is not the only obstruction for the integral Hasse principle on such log K3 surfaces, even when their compactification is “split”.

Classification : 14J10, 14J28, 14G99
Keywords: log K3 surfaces, integral points
Mot clés : surfaces log K3, points entiers

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