Geometry of equivariant compactifications of Ga

In this paper we begin a systematic study of equivariant compactifications of Ga . The question of classifying non-equivariant compactifications was raised by F. Hirzebruch ([10]) and has attracted considerable attention since (see [5], [15], [12] and the references therein). While there are classification results for surfaces and non-singular threefolds with small Picard groups, the general perception is that a complete classification is out of reach. On the other hand, there is a rich theory of equivariant compactifications of reductive groups. The classification of normal equivariant compactifications of reductive groups is combinatorial. Essentially, the whole geometry of the compactification can be understood in terms of (colored) fans. In particular, these varieties do not admit moduli. For more details see [14], [4], [2] and the references therein. Our goal is to understand equivariant compactifications of Ga . The first step in our approach is to classify possible Ga-structures on simple varieties, like projective spaces or Hirzebruch surfaces. Then we realize general smooth Ga-varieties as appropriate (i.e., equivariant) blow-ups of simple varieties. This gives us a geometric description of the moduli space of equivariant compactifications of Ga . In section 2 we discuss general properties of equivariant compactifications of Ga (Ga-varieties). In section 3 we classify all possible Ga-structures on projective spaces P. In section 4 we study curves, paying particular attention to non-normal examples. In section 5 we carry out our program completely for surfaces. In particular, we classify all possible Ga-structures on minimal rational surfaces. In section 6 we turn to threefolds. We give a classification