Pinch point (mathematics)

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Section of the Whitney umbrella, an example of pinch point singularity.

In geometry, a pinch point or cuspidal point is a type of singular point on an algebraic surface.

The equation for the surface near a pinch point may be put in the form

where [4] denotes terms of degree 4 or more and is not a square in the ring of functions.

For example the surface near the point , meaning in coordinates vanishing at that point, has the form above. In fact, if and then {} is a system of coordinates vanishing at then is written in the canonical form.

The simplest example of a pinch point is the hypersurface defined by the equation called Whitney umbrella.

The pinch point (in this case the origin) is a limit of normal crossings singular points (the -axis in this case). These singular points are intimately related in the sense that in order to resolve the pinch point singularity one must blow-up the whole -axis and not only the pinch point.

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