Abstract
The birational geometry of projective threefolds on which $\mathrm {SL}(2)$ acts with 2-dimensional general orbits is studied from the viewpoint of the minimal model theory of projective threefolds. These threefolds are closely related to the minimal rational threefolds classified by Enriques, Fano and Umemura. The main results are (i) the $\mathrm {SL}(2)$-birational classification of such threefolds and (ii) the classification of relatively minimal models in the fixed point free cases.
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