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 formula_2 is not a square in the ring of functions.
For example the surface formula_3 near the point formula_4, meaning in coordinates vanishing at that point, has the form above. In fact, if formula_5 and formula_6 then {formula_7} is a system of coordinates vanishing at formula_4 then formula_9 is written in the canonical form.
The simplest example of a pinch point is the hypersurface defined by the equation formula_10 called Whitney umbrella. 
The pinch point (in this case the origin) is a limit of normal crossings singular points (the formula_2-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 formula_2-axis and not only the pinch point. 