On the Classification of 3-dimensional SL₂(C)-varieties

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 [Keb98, math.AG/9805042] of the 3-dimensional relatively minimal quasihomogeneous varieties where the automorphism group is not solvable.