A bound on embedding dimensions of geometric generic fibers
  We limit the singularities that arise in geometric generic fibers of morphisms between smooth varieties of positive characteristic by studying changes in embedding dimension under inseparable field extensions. We then use this result in the context of the minimal model program to rule out the existence of smooth varieties fibered by certain non-normal del Pezzo surfaces over bases of small dimension. 
Zachary Maddock
math.AG