New examples of geometrically special varieties: K3 surfaces, Enriques surfaces, and algebraic groups
Finn Bartsch
TL;DR
The paper advances the program linking geometric specialness with Campana-speciality by proving that all connected algebraic groups are geometrically special, and that this property persists after removing codimension at least two subsets under suitable hypotheses. It then shows that elliptic K3 surfaces and Enriques surfaces are geometrically special, leveraging non-torsion multisections and a method to transfer non-Jacobian cases to Jacobian ones. A density-criterion framework for sections into products is employed to generate covering sets whose graphs are dense, establishing potential density-type results in these contexts. The punctured versions demonstrate that, for general position finite sets, the geometric specialness persists, reinforcing the robustness of the property and its alignment with Campana's conjectures. Overall, the work provides concrete, technically grounded evidence for the conjectured equivalence between geometrically special and Campana-special varieties in these important geometric classes.
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.