Extremal elliptic surfaces in characteristic 2 and 3

We show that extremal elliptic surfaces in characteristic 2 and 3 are unirational surfaces. Our strategy of proof is to determine explicit equations for the generic fibers. The method also applies to the classification of elliptic curves over k(T) with given places of bad reduction where k is any perfect field of characteristic 2 or 3. Received: 17 September 1999 / Revised version: 19 April 2000