Strong Rational Connectedness of Surfaces

TL;DR

This paper proves that for many smooth rationally connected surfaces, rational connectedness implies strong rational connectedness, confirming a conjecture by Hassett and Tschinkel.

Contribution

It establishes the equivalence of rational and strong rational connectedness for broad classes of smooth surfaces, including log del Pezzo surfaces.

Findings

Rational connectedness implies strong rational connectedness in many cases.

Confirms Hassett and Tschinkel's conjecture for smooth rationally connected surfaces.

Includes the smooth locus of log del Pezzo surfaces.

Abstract

We discuss the strong rational connectedness of smooth rationally connected surfaces. We prove in lots of cases, including the smooth locus of a log del Pezzo surface, the rational connectedness indeed implies the strong rational conectedness. This confirms a conjecture due to Hassett and Tschinkel in \cite{ht08}.