Hasse principle for three classes of varieties over global function fields

TL;DR

This paper proves the Hasse principle for specific classes of varieties over global function fields, using geometric methods, and discusses related results on weak approximation and fundamental groups.

Contribution

It provides a geometric proof of the Hasse principle for certain varieties over global function fields, extending previous results and including new corollaries.

Findings

Hasse principle holds for smooth quadric hypersurfaces in odd characteristic

Hasse principle holds for smooth cubic hypersurfaces of dimension ≥4 in characteristic ≥7

Hasse principle holds for smooth complete intersections of two quadrics of dimension ≥3 in odd characteristics

Abstract

We give a geometric proof that Hasse principle holds for the following varieties defined over global function fields: smooth quadric hypersurfaces in odd characteristic, smooth cubic hypersurfaces of dimension at least $4$ in characteristic at least $7$, and smooth complete intersections of two quadrics of dimension at least $3$ in odd characteristics. In Appendix A we explain how to modify a previous argument of the author to prove weak approximation for cubic hypersurfaces defined over function fields of curves over algebraically closed fields of characteristic at least $7$. In Appendix B we prove some corollaries of Koll\'ar's results on the fundamental group of separably rationally connected varieties.