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A secant variety can be used to show the fact that a smooth projective curve can be embedded into the projective 3-space as follows. [2] Let be a smooth curve. Since the dimension of the secant variety S to C has dimension at most 3, if >, then there is a point p on that is not on S and so we have the projection from p to a hyperplane H, which gives the embedding :.
An algebraic manifold is an algebraic variety that is also an m-dimensional manifold, and hence every sufficiently small local patch is isomorphic to k m. Equivalently, the variety is smooth (free from singular points). When k is the real numbers, R, algebraic manifolds are called Nash manifolds. Algebraic manifolds can be defined as the zero ...
The union of the tangent and secant lines (the secant variety) of a twisted cubic C fill up P 3 and the lines are pairwise disjoint, except at points of the curve itself. In fact, the union of the tangent and secant lines of any non-planar smooth algebraic curve is three-dimensional.
The word secant comes from the Latin word secare, meaning to cut. [2] In the case of a circle, a secant intersects the circle at exactly two points. A chord is the line segment determined by the two points, that is, the interval on the secant whose ends are the two points. [3]
In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite-difference approximation of Newton's method , so it is considered a quasi-Newton method .
Let X be an algebraic variety, x a point of X, and (O X,x, m) be the local ring of X at x. Then the tangent cone to X at x is the spectrum of the associated graded ring of O X,x with respect to the m-adic filtration: , = / +.
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