Pinch point (mathematics)

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

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

where [4] denotes terms of degree 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 u,v,w 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.

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