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 where [4] denotes terms of degree 4 or more and

is not a square in the ring of functions.

, meaning in coordinates vanishing at that point, has the form above.

In fact, if

} is a system of coordinates vanishing at

is written in the canonical form.

The simplest example of a pinch point is the hypersurface defined by the equation

called Whitney umbrella.

The pinch point (in this case the origin) is a limit of normal crossings singular points (the

-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

-axis and not only the pinch point.