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
- [math]\displaystyle{ f(u,v,w) = u^2 - vw^2 + [4] \, }[/math]
where [4] denotes terms of degree 4 or more and [math]\displaystyle{ v }[/math] is not a square in the ring of functions.
For example the surface [math]\displaystyle{ 1-2x+x^2-yz^2=0 }[/math] near the point [math]\displaystyle{ (1,0,0) }[/math], meaning in coordinates vanishing at that point, has the form above. In fact, if [math]\displaystyle{ u=1-x, v=y }[/math] and [math]\displaystyle{ w=z }[/math] then {[math]\displaystyle{ u, v, w }[/math]} is a system of coordinates vanishing at [math]\displaystyle{ (1,0,0) }[/math] then [math]\displaystyle{ 1-2x+x^2-yz^2=(1-x)^2-yz^2=u^2-vw^2 }[/math] is written in the canonical form.
The simplest example of a pinch point is the hypersurface defined by the equation [math]\displaystyle{ u^2-vw^2=0 }[/math] called Whitney umbrella.
The pinch point (in this case the origin) is a limit of normal crossings singular points (the [math]\displaystyle{ v }[/math]-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 [math]\displaystyle{ v }[/math]-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. 23-25. ISBN 0-471-05059-8.
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