Pinch point (mathematics)

Pinch point (mathematics)
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 + x2 − yz2 = 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 + x2 − yz2 = (1 − x)2 − yz2 = u2 − vw2 is written in the canonical form.

The simplest example of a pinch point is the hypersurface defined by the equation u2 − vw2 = 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

References

  • P. Griffiths; J. Harris (1994). Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. p. 617. ISBN 0-471-05059-8. 
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