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 $$v$$-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 $$v$$-axis and not only the pinch point.