Explicit local stable resolution of cusps

Abstract

This article gives an explicit local stable resolution of cusps on GIT-stable plane quartic models. More generally, we consider a smoothing of an ordinary plane cusp over a complete discretely valued field. We show that, after a finite separable extension and a suitable choice of coordinates, a single weighted blow-up with weights ((1,2,3)) gives the stable resolution. The exceptional component is an explicit semistable Weierstrass cubic. The construction is effective and is implemented in the Sage package "StabilityFunction".