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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

$f(u,v,w)=u^2-v{w}^2+[4]$

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

For example the surface $1-2x+x^2-y{z}^2=0$ near the point $(1,0,0)$, meaning in coordinates vanishing at that point, has the form above. In fact, if $u=1-x,v=y$ and $w=z$ then {$u,v,w$} is a system of coordinates vanishing at $(1,0,0)$ then $1-2x+x^2-y{z}^2=(1-x{)}^2-y{z}^2=u^2-v{w}^2$ is written in the canonical form.

The simplest example of a pinch point is the hypersurface defined by the equation $u^2-v{w}^2=0$ called Whitney umbrella.

The pinch point (in this case the origin) is a limit of normal crossings singular points (the $v$-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 $v$-axis and not only the pinch point.

See also

• Whitney umbrella
• Singular point of an algebraic variety

References

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