Let be a variety over an algebraically closed field . is said to be **rational** if is birational to . In general, it is difficult to determine when a variety in higher dimensions is rational, although there are numerical invariants in dimensions one and two.

- Let be a smooth projective curve. Then is rational if and only if its genus is zero.
- Let be a smooth projective surface. Then is rational if and only if there are no global 1-forms on (i.e., ) and the second plurigenus vanishes. This is a statement about the negativity of the cotangent bundle (or, equivalently, of the positivity of the tangent bundle) which is a birational invariant and which holds for . The result is a criterion of Castelnuovo, extended by Zariski to characteristic .

In higher dimensions, it is harder to tell when a variety is rational. An easier problem is to determine when a variety is **unirational**: that is, when there is a dominant rational map

or, equivalently, when the function field has a finite extension which is purely transcendental. In dimensions one and two (and in characteristic zero), the above invariants imply that a unirational variety is rational. In higher dimensions, there are many more unirational varieties: for example, a theorem of Harris, Mazur, and Pandharipande states that a degree hypersurface in , is always unirational.

The purpose of this post is to describe a theorem of Serre that shows the difficulty of distinguishing rationality from unirationality. Let’s work over . The fundamental group of a smooth projective variety is a birational invariant, and so any rational variety has trivial .

Theorem 1 (Serre)A unirational (smooth, projective) variety over has trivial .

The reference is Serre’s paper “On the fundamental group of a unirational variety,” in J. London Math Soc. 1959. (more…)