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 − v w 2 + [ 4 ] {\displaystyle f(u,v,w)=u^{2}-vw^{2}+[4]\,}

where [4] denotes terms of degree 4 or more and v {\displaystyle v} is not a square in the ring of functions.

For example the surface 1 − 2 x + x 2 − y z 2 = 0 {\displaystyle 1-2x+x^{2}-yz^{2}=0} near the point ( 1 , 0 , 0 ) {\displaystyle (1,0,0)}, meaning in coordinates vanishing at that point, has the form above. In fact, if u = 1 − x , v = y {\displaystyle u=1-x,v=y} and w = z {\displaystyle w=z} then {u , v , w {\displaystyle u,v,w}} is a system of coordinates vanishing at ( 1 , 0 , 0 ) {\displaystyle (1,0,0)} then 1 − 2 x + x 2 − y z 2 = ( 1 − x ) 2 − y z 2 = u 2 − v w 2 {\displaystyle 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 − v w 2 = 0 {\displaystyle 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 {\displaystyle 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 {\displaystyle v}-axis and not only the pinch point.

Pinch point (mathematics)

See also