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Hence, it is technically more correct to discuss singular points of a smooth mapping here rather than a singular point of a curve. The above definitions can be extended to cover implicit curves which are defined as the zero set of a smooth function, and it is not necessary just to consider algebraic varieties. The definitions can be ...
A cusp at (0, 1/2) In mathematics, a cusp, sometimes called spinode in old texts, is a point on a curve where a moving point must reverse direction. A typical example is given in the figure. A cusp is thus a type of singular point of a curve. For a plane curve defined by an analytic, parametric equation
The simplest example of singularities are curves that cross themselves. But there are other types of singularities, like cusps. For example, the equation y 2 − x 3 = 0 defines a curve that has a cusp at the origin x = y = 0. One could define the x-axis as a tangent at this point, but this definition can not be the same as the definition at ...
Point a is an ordinary point when functions p 1 (x) and p 0 (x) are analytic at x = a. Point a is a regular singular point if p 1 (x) has a pole up to order 1 at x = a and p 0 has a pole of order up to 2 at x = a. Otherwise point a is an irregular singular point.
It was noticed in the formulation of Bézout's theorem that such singular points must be counted with multiplicity (2 for a double point, 3 for a cusp), in accounting for intersections of curves. It was then a short step to define the general notion of a singular point of an algebraic variety; that is, to allow higher dimensions.
In classical algebraic geometry, a tacnode (also called a point of osculation or double cusp) [1] is a kind of singular point of a curve. It is defined as a point where two (or more) osculating circles to the curve at that point are tangent. This means that two branches of the curve have ordinary tangency at the double point. [1] The canonical ...
A second order Fuchsian equation with n singular points has a group of symmetries acting (projectively) on its solutions, isomorphic to the Coxeter group W(D n) of order 2 n−1 n!. The hypergeometric equation is the case n = 3, with group of order 24 isomorphic to the symmetric group on 4 points, as first described by Kummer.
The Whitney umbrella x 2 = y 2 z has singular set the z axis, most of whose point are ordinary double points, but there is a more complicated pinch point singularity at the origin, so blowing up the worst singular points suggests that one should start by blowing up the origin. However blowing up the origin reproduces the same singularity on one ...