Every rationally connected variety over the function field of a curve has a rational point

In a paper from 1992, Kollár, Miyaoka and Mori posed the following question: Given a proper flat morphism f: P → X with target a nonsingular curve and whose geometric generic fiber is rationally-connected, does it follow that f admits a regular section? In the case that the ground field is an algebraically-closed field of characteristic zero, this was answered affirmatively by Graber, Harris, and Starr using a topological argument. We prove that f admits a regular section when the ground field is an algebraically-closed field of arbitrary characteristic. The key ingredients in the proof are the following: (a) Proposition 2.1 which shows that after taking the reduced fiber product of f with a generically étale morphism π: Y → X, we may assume that the smooth locus of f intersects every geometric fiber, and (b) Proposition 1.1 which is a purely algebraic analogue of the "moving branch points" argument in Graber-Harris-Starr. The reader is cautioned that our definition of "separably rationally connected" differs slightly from the original definition.