On the classification of 3-dimensional ${\rm SL}\sb 2(\bold C)$-varieties

Abstract

In the present work we describe 3-dimensional complex $SL_2$-varieties where the generic $SL_2$-orbit is a surface. We apply this result to classify the minimal 3-dimensional projective varieties with Picard-number 1 where a semisimple group acts such that the generic orbits are 2-dimensional.

This is an ingredient of the classification [Keb99] of the 3-dimensional relatively minimal quasihomogeneous varieties where the automorphism group is not solvable.