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The singular points of a degenerate quadric are the points whose projective coordinates belong to the null space of the matrix A. A quadric is reducible if and only if the rank of A is one (case of a double hyperplane) or two (case of two hyperplanes).
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 singular quadric surface, the cone over a smooth conic curve. If q can be written (after some linear change of coordinates) as a polynomial in a proper subset of the variables, then X is the projective cone over a lower-dimensional quadric. It is reasonable to focus attention on the case where X is not a cone.
The study of the analytic structure of an algebraic curve in the neighborhood of a singular point provides accurate information of the topology of singularities. In fact, near a singular point, a real algebraic curve is the union of a finite number of branches that intersect only at the singular point and look either as a cusp or as a smooth curve.
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 ...
Points of V that are not singular are called non-singular or regular. It is always true that almost all points are non-singular, in the sense that the non-singular points form a set that is both open and dense in the variety (for the Zariski topology, as well as for the usual topology, in the case of varieties defined over the complex numbers). [1]
An example of a rational singularity is the singular point of the quadric cone x 2 + y 2 + z 2 = 0. {\displaystyle x^{2}+y^{2}+z^{2}=0.\,} Artin [ 2 ] showed that the rational double points of algebraic surfaces are the Du Val singularities .
Quadric surface (The union of two quadric surfaces is a special case of a quartic surface) ... Jessop, C. M. (1916), Quartic surfaces with singular points, ...