  Let f : X -> S be any elliptic fibration. If X has dimension 3 and is not
uniruled, then X has a minimal model (with terminal singularities) [Mori]. In
earlier work we have shown that there exists a birationally equivalent elliptic
fibration p: Y -> T such that Y is minimal and a multiple of K_Y can be
expressed as the pullback of a divisor from T. Moreover T has at worst quotient
singularities; it is not difficult to find examples where T is actually
singular. In this paper we describe the singularities of this parameter surface
T in the case of no multiple fibers. Although T is not uniquely determined by
the birational equivalence class of the fibration, any two such T are related
by a particular kind of birational map.
