Singularity theory

For non-mathematical singularity theories, see singularity.

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 equations 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 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, ending up with a compact manifold (for the strong topology, rather than the Zariski topology, that is).

Related Topics:
Algebraic geometry - Polynomial equation - Coordinate system - Algebraic variety - Heisuke Hironaka - Resolution of singularities - Birational geometry - Characteristic - Affine geometry - Projective geometry - Affine variety - Hyperplane at infinity - Projective space - Compactification - Zariski topology

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