Pinch point (mathematics)

right|frame|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 - vw^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-yz^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-yz^2=(1-x)^2-yz^2=u^2-vw^2 is written in the canonical form.

The simplest example of a pinch point is the hypersurface defined by the equation u^2-vw^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.

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