This is a blog post on birational geometry. I will broadly be following this article for the exposition.
A birational map is a rational map such that its inverse map is also a rational map. The two (quasiprojective) varieties and are known as birational varieties. An example is , and .
Varieties are birational if and only if their function fields are isomorphic as extension fields of . What are function fields? A function field of a variety is the field of rational functions defined on . In a way, it is the rational field of the coordinate ring on . But what about the functions which are on some part of , although not all of it? They can still be inverted. In the complex domain, such functions are called meromorphic functions (isolated poles are allowed).
A variety is called rational if it is birational to affine space of some dimension. For instance, take the circle . This is birational to the affine space . Consider the map . This is a rational map, for which the inverse is .
In general, a smooth quadric hypersurface (degree 2) is rational by stereographic projection. How? Choose a point on the hypersurface, say , and consider all lines through to the various other points on the hypersurface. Each such line goes to a point in . Note that this map is not defined on the whole of the hypersurface. How do we know that the line joining and point does not pass through another point on the hypersurface? This is precisely because this is a quadric surface. A quadratic equation can only have a maximum of two distinct solutions, and one of them is already .
Now we state some well-known theorems. Chow’s Theorem states that every algebraic variety is birational to a projective variety. Hence, if one is to classify varieties up to birational isomorphism, then considering only the projective varieties is sufficient. Then Hironaka further went on to prove that every variety is birational to a smooth projective variety. Hence, we now have to classify a much smaller set of varieties. In dimension, , if two smooth projective curves are birational, then they’re isomorphic. However, this breaks down in higher dimensions due to blowing up. Due to the blowing up construction, every smooth projective variety of at least degree is birational to infinitely many “bigger” varieties with higher Betti numbers. This leads to the idea of minimal models: is there a unique “simplest variety” in each birational equivalence class? The modern definition states that a projective variety is minimal if the canonical bundle on each curve has non-negative degree. It turns out that blown up varieties are never minimal.