In this chapter, we introduce rational varieties through examples. After giving the fundamental definitions in the first section and settling the rationality question for curves in Section 2, we continue with the rich theory of quadric hypersurfaces in Section 3. This is essentially a special case of the theory of quadratic forms, though the questions tend to be strikingly different.

Quadrics over finite fields are discussed in Section 4. Several far–reaching methods of algebraic geometry appear here in their simplest form.

Cubic hypersurfaces are much more subtle. In Section 5, we discuss only the most basic rationality and unirationality facts for cubics. A further smattering of rational varieties is presented in Section 6, together with a more detailed look at determinantal representations for cubic surfaces.

A very general and useful nonrationality criterion, using differential forms, is discussed in Section 7.

Rational and unirational varieties

Roughly speaking, a variety is unirational if a dense open subset is parametrized by projective space, and rational if such a parametrization is one–to–one.

To be precise, fix a ground field k, and let X be a variety defined over k. It is important to bear in mind that k need not be algebraically closed and that all constructions involving the variety X are carried out over the ground field k.