Singularity theory

For non-mathematical singularity theories, see singularity.

Algebraic curve singularities

Historically singularities were first noticed in the study of algebraic curves. The double point at (0,0) of the curve

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:y2 = x3 − x2

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and the cusp there of

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:y2 = x3

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are qualitatively different, as is seen just by sketching. Isaac Newton carried out a detailed study of all cubic curves, 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 (2 for a double point, 3 for a cusp), in accounting for intersections of curves.

Related Topics:
Isaac Newton - Cubic curve - Bézout's theorem - Multiplicity

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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.

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