Communicated by J. Tits

Our objects of study are aftine group schemes, finitely presented and flat, over a domain A. As in 1 lo], we call such group schemes models of their generic fibers. In dimension one there are only a few possibilities for the generic fiber, and we are able to obtain a complete classification in some generality: THEOREM. Let A be an integrally closed domain with perfect fraction field. Then the smooth group schemes over A with connected afine one-dimensional fibers correspond to rank 2 projective algebras over A. The construction of these groups is straightforward; to an algebra B we assign the quotient R,,, G,/G,. The main effort comes in showing that these are the only possibilities. The key to this, and the basic technical idea in the paper, is the use of N&on blow-ups of group schemes over valuation rings. This process has been used before [ 1, 21 as a tool for resolving singularities, but in fact it also furnishes a good grasp on the structure of models in general. A further instance of this is found in the last section, where we give an analysis of all models (smooth or not) of G, over a valuation ring. It seems likely that the technique should also have other applications. 1. N~RON BLOW-UPS Let A be a discrete valuation ring with fraction field K, uniformizer 71, and residue field k. If G is a flat separated group scheme of finite type over A, and the generic fiber G, is affine, then G itself is affine [ 1, p. 301; throughout this section G will be assumed to be such a model. Our goal here is to acquire information about the blow-up process for passing from G to some * The authors ’ work was partially supported by NSF contracts