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frame, Section of the Whitney umbrella, an example of pinch point singularity. In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a pinch point or cuspidal point is a type of
singular point Singularity or singular point may refer to: Science, technology, and mathematics Mathematics * Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", for example infinite or not differentiab ...
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 + \, where denotes terms of
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathemati ...
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 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 frame, Section of the surface In geometry, the Whitney umbrella (or Whitney's umbrella, named after American mathematician Hassler Whitney, and sometimes called a Cayley umbrella) is a specific self-intersecting ruled surface placed in three dim ...
. The pinch point (in this case the origin) is a limit of
normal crossings In algebraic geometry a normal crossing singularity is a singularity similar to a union of coordinate hyperplanes. The term can be confusing because normal crossing singularities are not usually normal schemes (in the sense of the local rings being ...
singular points (the v-axis in this case). These singular points are intimately related in the sense that in order to
resolve Resolve may refer to: * ''Resolve'' (Lagwagon album) * ''Resolve'' (Last Tuesday album) * "Resolve" (song), by the Foo Fighters *'' The Resolve'', a 1915 American silent short drama film * "Resolve" (''One Tree Hill'' episode) *''Resolve'', a Brit ...
the pinch point singularity one must
blow-up ''Blowup'' (sometimes styled as ''Blow-up'' or ''Blow Up'') is a 1966 mystery drama thriller film directed by Michelangelo Antonioni and produced by Carlo Ponti. It was Antonioni's first entirely English-language film, and stars David Hemmin ...
the whole v-axis and not only the pinch point.


See also

*
Whitney umbrella frame, Section of the surface In geometry, the Whitney umbrella (or Whitney's umbrella, named after American mathematician Hassler Whitney, and sometimes called a Cayley umbrella) is a specific self-intersecting ruled surface placed in three dim ...
*
Singular point of an algebraic variety In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In c ...


References

* {{cite book , author=P. Griffiths , authorlink=Phillip Griffiths , author2=J. Harris , authorlink2=Joe Harris (mathematician) , title=Principles of Algebraic Geometry , series=Wiley Classics Library , publisher=Wiley Interscience , year=1994 , isbn=0-471-05059-8 , page=23-25 Algebraic surfaces Singularity theory