Rational connectivity and sections of families over curves
  We show that if a family of complex varieties over a base B admits a section when restricted to a very general curve in B, then the family must contain a subfamily of rationally connected varieties dominating B. As an application, we deduce the existence of one parameter families of Enriques surfaces with no section. 
T. Graber, J. Harris, B. Mazur, J. Starr