Pinch point (mathematics)

In geometry, a pinch point or cuspidal point is a type of singular point on an algebraic surface.

Section of the Whitney umbrella, an example of pinch point singularity.

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

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

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

For example the surface ${\displaystyle 1-2x+x^{2}-yz^{2}=0}$ near the point ${\displaystyle (1,0,0)}$, meaning in coordinates vanishing at that point, has the form above. In fact, if ${\displaystyle u=1-x,v=y}$ and ${\displaystyle w=z}$ then {${\displaystyle u,v,w}$} is a system of coordinates vanishing at ${\displaystyle (1,0,0)}$ then ${\displaystyle 1-2x+x^{2}-yz^{2}=(1-x)^{2}-yz^{2}=u^{2}-vw^{2}}$ is written in the canonical form.

The simplest example of a pinch point is the hypersurface defined by the equation ${\displaystyle u^{2}-vw^{2}=0}$ called Whitney umbrella.

The pinch point (in this case the origin) is a limit of normal crossings singular points (the ${\displaystyle v}$-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 ${\displaystyle v}$-axis and not only the pinch point.