On Stably Pointed Varieties and Generically Stable Groups in ACVF

We give a geometric description of the pair (V,p), where V is an algebraic variety over a non-trivially valued algebraically closed field K with valuation ring 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 O_K with residual dominant morphisms over O_K. Under this functor, the pair (an algebraic group, a generically stable generic type of a subgroup) gets sent to a group scheme over O_K. This returns a geometric description of the subgroup as the set of O_K-points of the group scheme, generalizing a previous result in the affine case. We also study a maximum modulus principle on schemes over O_K and show that the schemes obtained by this functor enjoy it.