On the geometry of SL(2)-equivariant flips

TL;DR

This paper characterizes 3-dimensional normal affine quasihomogeneous SL(2)-varieties as quotients of hypersurfaces, describes their Cox rings, and constructs SL(2)-equivariant flips using GIT-quotients and spherical variety theory.

Contribution

It provides a new description of SL(2)-varieties as hypersurface quotients and introduces a method to construct equivariant flips via GIT and spherical varieties.

Findings

3D SL(2)-varieties are quotients of 4D hypersurfaces.

Cox rings of these varieties have unique defining equations.

SL(2)-flips are described using 2D colored cones.

Abstract

In this paper, we show that any 3-dimensional normal affine quasihomogeneous SL(2)-variety can be described as a categorical quotient of a 4-dimensional affine hypersurface. Moreover, we show that the Cox ring of an arbitrary 3-dimensional normal affine quasihomogeneous SL(2)-variety has a unique defining equation. This allows us to construct SL(2)-equivariant flips by different GIT-quotients of hypersurfaces. Using the theory of spherical varieties, we describe SL(2)-flips by means of 2-dimensional colored cones.