Mazur, BarryGraber, T.Harris, J.Starr, J.2019-01-142005-09Graber, T, J. Harris, B. Mazur, and J. Starr. 2005. “Rational Connectivity and Sections of Families over Curves.” Annales Scientifiques de l’École Normale Supérieure 38 (5): 671–92. https://doi.org/10.1016/j.ansens.2005.07.003.0012-9593http://nrs.harvard.edu/urn-3:HUL.InstRepos:37989547A "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. Staff, 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. Staff, 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 P(1) with no section.en-USJournal Article2019-01-1410.1016/j.ansens.2005.07.003