Rational connectivity and sections of families over curves

A "pseudosection" of the total space X of a family of varieties over a base variety B is a subvariety of X whose general fiber over B is rationally connected. We prove a theorem which is a converse, in some sense, of the main result of [T. Graber, J. Harris, J. Starr, Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003) 69-90]: a family of varieties over B has a "pseudosection" if its restriction to each one-parameter subfamily has a "pseudosection" (which, due to [T. Graber, J. Harris, J. Starr, Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003) 69-90], holds if and only if each one-parameter subfamily has a section). This is used to give a negative answer to a question posed by Serre to Grothendieck: There exists a family of O-acyclic varieties (a family of Enriques surfaces) parametrized by P¹ with no section.