On Stably Pointed Varieties and Generically Stable Groups in ACVF

Abstract

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 OK and p is a Zariski dense generically stable type concentrated on V, by defining a fully faithful functor to the category of schemes over OK with residual dominant morphisms over OK.Under this functor, the pair (an algebraic group, a generically stable generic type of a subgroup) gets sent to a group scheme over OK. This returns a geometric description of the subgroup as the set of OK-points of the group scheme, generalizing a previous result in the affine case.We also study a maximum modulus principle on schemes over OK and show that the schemes obtained by this functor enjoy it.