V.30 The Resolution of Singularities
Virtually all important mathematical structures come with a notion of equivalence. For instance, we regard two GROUPS [I.3 §2.1] as equivalent if they are ISOMORPHIC [I.3 §4.1], and we regard two TOPOLOGICAL SPACES [III.90] as equivalent if there is a continuous map from one to the other with a continuous inverse (in which case we say that they are homeomorphic). In general, a notion of equivalence is useful if properties that we are interested in are unaffected when we replace an object by an equivalent one: for example, if G is a finitely generated Abelian group and H is isomorphic to G, then H is a finitely generated Abelian group.
A useful notion of equivalence for ALGEBRAIC VARIETIES [IV.4§7] is that of birational equivalence. Roughly speaking, two varieties V and W are said to be birationally equivalent if there is a rational map from V to W with a rational inverse. If V and W are presented as solution sets of equations in some coordinate system, then these rational maps are just rational functions in the coordinates that send points of V to points of W. However, it is important to understand that a rational map from V to W is not literally a function from V to W, because it is allowed to be undefined at certain points of V.
Consider, for example, how we might map the infinite cylinder {(x,y,z) : x2 + y2 = 1} to the cone {(x,y,z): x2 + y2 = z2}. An obvious map would be the function f (x, y, z) (zx, zy, z), which we could try to invert using the map g(x, y, z) (x/z, y/z, z). However, g is not defined at the point (0, 0, 0). Nevertheless, the cylinder and the cone are birationally equivalent, and algebraic geometers would say that g “blows up” the point (0, 0, 0) to the circle {(x, y, z) : x2 + y2 = 1, z = 0}.
The main property of a variety V that is preserved by birational equivalence is the so-called function field of V, which consists of all rational functions defined on V. (What precisely this means is not completely obvious: in some contexts, V is a subset of a larger space such as 
in which one can talk about ratios of polynomials, and then one possible definition of a rational function on V is that it is an equivalence class of such ratios, where two of them are counted as equivalent if they take the same values on V. See ARITHMETIC GEOMETRY [IV.5 §3.2] and QUANTUM GROUPS [III.75 §1] for further discussion of this equivalence relation.)
A famous theorem of Hironaka, proved in 1964, states that every algebraic variety (over a field of characteristic 0) is birationally equivalent to an algebraic variety without singularities, with some technical conditions on the birational equivalence that are needed for the theorem to be interesting and useful. The example given earlier is a simple illustration: the cone has a singularity at (0, 0, 0) but the cylinder is smooth everywhere. Hironaka’s proof was well over two hundred pages long, but his argument has since been substantially simplified by several authors.
For a further discussion of the resolution of singularities, see ALGEBRAIC GEOMETRY [IV.4 §9].