On rational chain connectedness of globally +-regular varieties

TL;DR

This paper proves that certain algebraic varieties with globally plus-regularity conditions are rationally chain connected in specific dimensions and characteristics, and introduces a stronger regularity notion with broader implications.

Contribution

It establishes rational chain connectedness for globally plus-regular varieties in dimension three and in mixed characteristic, and introduces strongly globally plus-regular varieties with wider applicability.

Findings

Globally plus-regular varieties are rationally chain connected in dimension three for residue characteristic p>5.

Strongly globally plus-regular varieties are rationally chain connected over dense open subsets of Spec(Z).

The paper introduces a new notion of strong global plus-regularity with significant geometric consequences.

Abstract

We prove that globally $+$-regular varieties are rationally chain connected in dimension three and mixed characteristic with residue field characteristic $p>5$. We also introduce a notion of strongly globally $+$-regular, and show that varieties of arbitrary dimension which are strongly globally $+$-regular over a dense open subset of $\mathrm{Spec}(\mathbb{Z})$ are rationally chain connected.