On the Tate constant
It is well known that versal deformations of nonsimple singularities depend on moduli. However they can be topologically trivial along some or all of them. The first step in the investigation of this phenomenon is to determine the versal discriminant (unstable locus), which roughly speaking is the obstacle to analytic triviality. The next one is to construct continuous liftable vector fields smooth far from the versal discriminant and to integrate them. In this paper we extend the results of J....
We explicitly determine the elliptic surfaces with section and maximal singular fibre. If the characteristic of the ground field is different from , for each of the two possible maximal fibre types, and , the surface is unique. In characteristic the maximal fibre types are and , and there exist two (resp. one) one-parameter families of such surfaces.