Sections of algebraic fibrations
Jason Starr, MIT

Given an algebraic (=holomorphic) map of complex projective varieties $f:X \rightarrow B$, what geometric condition on a general fiber of $f$ guarantees that there is an algebraic section of $f$, or even a rational section of $f$? A related question is this: given a variety $X$ defined over a non-algebraically-closed field $k$, what "geometric" condition guarantees that $X$ has a "$k$-rational point"? The answer turns out to be closely related to a geometric condition called "rational connectedness". I will discuss some older results due to Tsen, Chevalley, Lang, etc., some newer results due to Graber, Harris, de Jong, Mazur and myself for the case $\text{dim}(B)=1$, and some conjectures for the case $\text{dim}(B) > 1$ (on the way I will define rational connectedness and talk about some of its beautiful properties).