On Stably Pointed Varieties and Generically Stable Groups in ACVF

TL;DR

This paper provides a geometric framework for understanding generically stable types on affine varieties over valued fields, linking model theory with algebraic geometry through schemes over valuation rings.

Contribution

It introduces a fully faithful functor from pairs of varieties and types to schemes over valuation rings, revealing geometric properties of generically stable types in ACVF.

Findings

Established a geometric description of generically stable types as schemes over valuation rings.

Proved the schemes obtained satisfy a maximum modulus principle.

Connected model-theoretic stability with algebraic geometric structures.

Abstract

We give a geometric description of the pair $(V,p)$, where $V$ is an affine algebraic variety over a non-trivially valued algebraically closed field $K$ with valuation ring $\mathcal{O}_K$ and $p$ is a Zariski dense generically stable type concentrated on $V$, by defining a fully faithful functor to the category of schemes over $\mathcal{O}_K$ with residual dominant morphisms over $\mathcal{O}_K$. We also study a maximum modulus principle on schemes over $\mathcal{O}_K$ and show that the schemes obtained by this functor enjoy it.