On Stably Pointed Varieties and Generically Stable Groups in ACVF

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.