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[[image:Whitney_unbrella.png|right|frame|240px|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 of an algebraic variety|singular point]] on an [[algebraic surface]].

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

:<math> f(u,v,w) = u^2 - vw^2 + [4] \, </math>

where [4] denotes [[Term (mathematics)|terms]] of [[Degree of a monomial|degree]] 4 or more and <math>v</math> is not a square in the ring of functions.

For example the surface <math>1-2x+x^2-yz^2=0</math> near the point <math>(1,0,0)</math>, meaning in coordinates vanishing at that point, has the form above. In fact, if <math>u=1-x, v=y</math> and <math>w=z</math> then {<math>u, v, w</math>} is a system of coordinates vanishing at <math>(1,0,0)</math> then <math>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>u^2-vw^2=0</math> called [[Whitney umbrella]].  

The pinch point (in this case the origin) is a limit of [[Normal crossing divisor|normal crossings]] singular points (the <math>v</math>-axis in this case). These singular points are intimately related in the sense that in order to [[resolution of singularities|resolve]] the pinch point singularity one must [[Blowing up|blow-up]] the whole <math>v</math>-axis and not only the pinch point. 

==See also==
*[[Whitney umbrella]]
*[[Singular point of an algebraic variety]]

==References==
{{reflist}}
* {{cite book | author=P. Griffiths | authorlink=Phillip Griffiths | coauthors=[[Joe Harris (mathematician)|J. Harris]] | title=Principles of Algebraic Geometry | series=Wiley Classics Library | publisher=Wiley Interscience | year=1994 | isbn=0-471-05059-8 | page=617 }}

[[Category:Algebraic surfaces]]
[[Category:Singularity theory]]

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