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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 ...
Let f (x, y) be a smooth function of x and y and assume, for simplicity, that f (0, 0) = 0. Then a type A 2-singularity of f at (0, 0) can be characterised by: Having a degenerate quadratic part, i.e. the quadratic terms in the Taylor series of f form a perfect square, say L(x, y) 2, where L(x, y) is linear in x and y, and
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas).It is a hypersurface (of dimension D) in a (D + 1)-dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in D + 1 variables; for example, D = 1 in the case of conic sections.
The two families of lines on a smooth (split) quadric surface. In mathematics, a quadric or quadric hypersurface is the subspace of N-dimensional space defined by a polynomial equation of degree 2 over a field. Quadrics are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine ...
The singular set of x 2 = y 2 z 2 is the pair of lines given by the y and z axes. The only reasonable varieties to blow up are the origin, one of these two axes, or the whole singular set (both axes). However the whole singular set cannot be used since it is not smooth, and choosing one of the two axes breaks the symmetry between them so is not ...
Branch points are generally the result of a multi-valued function, such as or (), which are defined within a certain limited domain so that the function can be made single-valued within the domain. The cut is a line or curve excluded from the domain to introduce a technical separation between discontinuous values of the function.
A plane curve defined by an implicit equation (,) =,where F is a smooth function is said to be singular at a point if the Taylor series of F has order at least 2 at this point.. The reason for this is that, in differential calculus, the tangent at the point (x 0, y 0) of such a curve is defined by the equation
A singular point of an implicit surface (in ) is a point of the surface where the implicit equation holds and the three partial derivatives of its defining function are all zero. Therefore, the singular points are the solutions of a system of four equations in three indeterminates. As most such systems have no solution, many surfaces do not ...