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Mustafin varieties are well-studied degenerations of projective spacesinduced by a choice of integral points in a Bruhat--Tits building. In recentwork, Annette Werner and the author initiated the study of degenerations ofplane curves obtained by Mustafin varieties by means of arithmetic geometry.Moreover, we applied these techniques to construct models of vector bundles onplane curves with strongly semistable reduction. In this work, we take aGroebner basis approach to the more general problem of studying degenerationsof projective varieties. Our methods include determining the behaviour ofGroebner bases under substitution over unique factorisation rings. Finally, weoutline applications to the $p-$adic Simpson correspondence, when therespective projective variety is a curve.

Mustafin models of projective varieties and vector bundles