New examples of geometrically special varieties: K3 surfaces, Enriques surfaces, and algebraic groups

TL;DR

This paper demonstrates that elliptic K3 surfaces and algebraic groups possess many rational points over function fields, confirming their status as geometrically special varieties, and explores conditions under which this property persists after removing certain subsets.

Contribution

It provides new examples of geometrically special varieties, specifically elliptic K3 surfaces and algebraic groups, and analyzes the stability of their specialness under certain geometric modifications.

Findings

Elliptic K3 surfaces have many rational points over function fields.

Algebraic groups also exhibit abundant rational points in this setting.

Geometric specialness persists under removal of codimension at least two subsets under certain conditions.

Abstract

We verify that elliptic K3 surfaces and algebraic groups have many rational points over function fields, i.e., they are geometrically special in the sense of Javanpeykar-Rousseau. We also show that under additional assumptions, this geometric specialness persists under removal of closed subsets of codimension at least two.