Let {X} be a variety over an algebraically closed field {k}. {X} is said to be rational if {X} is birational to {\mathbb{P}_k^n}. 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 {X} be a smooth projective curve. Then {X} is rational if and only if its genus is zero.
  • Let {X} be a smooth projective surface. Then {X} is rational if and only if there are no global 1-forms on {X} (i.e., {H^0(X, \Omega_{X/k}) = 0}) and the second plurigenus {H^0(X, \omega_{X/k}^{\otimes 2}) } 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 {\mathbb{P}^2_k}. The result is a criterion of Castelnuovo, extended by Zariski to characteristic p.

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

\displaystyle \mathbb{P}_k^n \dashrightarrow X;

or, equivalently, when the function field {k(X)} 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 {d} hypersurface in {\mathbb{P}^N}, {N \gg 0} 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 {\mathbb{C}}. The fundamental group of a smooth projective variety is a birational invariant, and so any rational variety has trivial {\pi_1}.

Theorem 1 (Serre) A unirational (smooth, projective) variety over {\mathbb{C}} has trivial {\pi_1}.

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