On the Geometry of $\operatorname{SL}(2)$-Equivariant Flips

In this paper, we show that any 3-dimensional normal affine quasihomogeneous $\operatorname{SL}(2)$-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 $\operatorname{SL}(2)$-variety has a unique defining equation. This allows us to construct $\operatorname{SL}(2)$-equivariant flips by different GIT-quotients of hypersurfaces. Using the theory of spherical varieties, we describe $\operatorname{SL}(2)$-flips by means of 2-dimensional colored cones.