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
−
v
w
2
+
[
4
]
{\displaystyle f(u,v,w)=u^{2}-vw^{2}+[4]\,}
where [4] denotes terms of degree 4 or more and
v
{\displaystyle v}
is not a square in the ring of functions.
For example the surface
1
−
2
x
+
x
2
−
y
z
2
=
0
{\displaystyle 1-2x+x^{2}-yz^{2}=0}
near the point
(
1
,
0
,
0
)
{\displaystyle (1,0,0)}
, meaning in coordinates vanishing at that point, has the form above. In fact, if
u
=
1
−
x
,
v
=
y
{\displaystyle u=1-x,v=y}
and
w
=
z
{\displaystyle w=z}
then {
u
,
v
,
w
{\displaystyle u,v,w}
} is a system of coordinates vanishing at
(
1
,
0
,
0
)
{\displaystyle (1,0,0)}
then
1
−
2
x
+
x
2
−
y
z
2
=
(
1
−
x
)
2
−
y
z
2
=
u
2
−
v
w
2
{\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
u
2
−
v
w
2
=
0
{\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
v
{\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
v
{\displaystyle v}
-axis and not only the pinch point.
