Editing Singularity theory (section)

==Singularities in algebraic geometry==
===Algebraic curve singularities===

[[File:Cubic with double point.svg|thumb|right|upright=0.75|A curve with double point]]
[[File:Cusp.svg|thumb|upright=0.75|right|A curve with a cusp]]

Historically, singularities were first noticed in the study of [[algebraic curve]]s. The ''double point'' at (0,&nbsp;0) of the curve

:<math>y^2 = x^2 + x^3 </math>

and the [[cusp (singularity)|cusp]] there of

:<math>y^2 = x^3\ </math>

are qualitatively different, as is seen just by sketching. [[Isaac Newton]] carried out a detailed study of all [[cubic curve]]s, the general family to which these examples belong. It was noticed in the formulation of [[Bézout's theorem]] that such ''singular points'' must be counted with [[Multiplicity (mathematics)|multiplicity]] (2 for a double point, 3 for a cusp), in accounting for intersections of curves.

It was then a short step to define the general notion of a [[singular point of an algebraic variety]]; that is, to allow higher dimensions.

===The general position of singularities in algebraic geometry===
Such singularities in [[algebraic geometry]] are the easiest in principle to study, since they are defined by [[polynomial equation]]s and therefore in terms of a [[coordinate system]]. One can say that the ''extrinsic'' meaning of a singular point isn't in question; it is just that in ''intrinsic'' terms the coordinates in the ambient space don't straightforwardly translate the geometry of the [[algebraic variety]] at the point. Intensive studies of such singularities led in the end to [[Heisuke Hironaka]]'s fundamental theorem on [[resolution of singularities]] (in [[birational geometry]] in [[characteristic (algebra)|characteristic]] 0). This means that the simple process of "lifting" a piece of string off itself, by the "obvious" use of the cross-over at a double point, is not essentially misleading: all the singularities of algebraic geometry can be recovered as some sort of very general ''collapse'' (through multiple processes). This result is often implicitly used to extend [[affine geometry]] to [[projective geometry]]: it is entirely typical for an [[affine variety]] to acquire singular points on the [[hyperplane at infinity]], when its closure in [[projective space]] is taken. Resolution says that such singularities can be handled rather as a (complicated) sort of [[compactification (mathematics)|compactification]], ending up with a ''compact'' manifold (for the strong topology, rather than the [[Zariski topology]], that is).