Editing Birational geometry (section)

==Minimal models and resolution of singularities==
Every algebraic variety is birational to a [[projective variety]] ([[Chow's lemma]]). So, for the purposes of birational classification, it is enough to work only with projective varieties, and this is usually the most convenient setting.

Much deeper is [[Heisuke Hironaka|Hironaka]]'s 1964 theorem on [[resolution of singularities]]: over a field of characteristic 0 (such as the complex numbers), every variety is birational to a [[Singular point of an algebraic variety|smooth]] projective variety. Given that, it is enough to classify smooth projective varieties up to birational equivalence.

In dimension 1, if two smooth projective curves are birational, then they are isomorphic. But that fails in dimension at least 2, by the [[blowing up]] construction. By blowing up, every smooth projective variety of dimension at least 2 is birational to infinitely many "bigger" varieties, for example with bigger [[Betti number]]s.

This leads to the idea of [[minimal model program|minimal models]]: is there a unique simplest variety in each birational equivalence
class? The modern definition is that a projective variety ''X'' is '''minimal''' if the [[canonical bundle|canonical line bundle]] ''K<sub>X</sub>'' has nonnegative degree on every curve in ''X''; in other words, ''K<sub>X</sub>'' is [[nef line bundle|nef]]. It is easy to check that blown-up varieties are never minimal.

This notion works perfectly for algebraic surfaces (varieties of dimension 2). In modern terms, one central result of the [[Italian school of algebraic geometry]] from 1890–1910, part of the [[Enriques–Kodaira classification|classification of surfaces]], is that every surface ''X'' is birational either to a product <math>\mathbb{P}^1\times C</math> for some curve ''C'' or to a minimal surface ''Y''.{{sfn|Kollár|Mori|1998|loc=Theorem 1.29.}} The two cases are mutually exclusive, and ''Y'' is unique if it exists. When ''Y'' exists, it is called the [[minimal model program|minimal model]] of&nbsp;''X''.