Models of hypersurfaces and Bruhat-Tits buildings

Abstract

We propose a new approach to constructing semistable integral models of hypersurfaces over a discrete non-archimedian field \(K\). For each stable hypersurface over \(K\) we define a stability function on the Bruhat-Tits building of \({\rm PGL}(K)\) and show that its global minima correspond to semistable hypersurface models over some extension of \(K\). This extends work of Kollar and of Elsenhans and Stoll on minimal hypersurface models. In the case of plane curves and residue characteristic zero, our results give a practical algorithm for constructing a semistable model over a suitable extension field.