Tangent cone

In geometry, the tangent cone is a generalization of the notion of the tangent space to a manifold to the case of certain spaces with singularities. In geometry, the tangent cone is a generalization of the notion of the tangent space to a manifold to the case of certain spaces with singularities. In nonlinear analysis, there are many definitions for a tangent cone, including the adjacent cone, Bouligand's contingent cone, and the Clarke tangent cone. These three cones coincide for a convex set, but they can differ on more general sets. Let K be a closed convex subset of a real vector space V and ∂K be the boundary of K. The solid tangent cone to K at a point x ∈ ∂K is the closure of the cone formed by all half-lines (or rays) emanating from x and intersecting K in at least one point y distinct from x. It is a convex cone in V and can also be defined as the intersection of the closed half-spaces of V containing K and bounded by the supporting hyperplanes of K at x. The boundary TK of the solid tangent cone is the tangent cone to K and ∂K at x. If this is an affine subspace of V then the point x is called a smooth point of ∂K and ∂K is said to be differentiable at x and TK is the ordinary tangent space to ∂K at x. Let X be an affine algebraic variety embedded into the affine space k n {displaystyle k^{n}} , with defining ideal I ⊂ k [ x 1 , … , x n ] {displaystyle Isubset k} . For any polynomial f, let in ⁡ ( f ) {displaystyle operatorname {in} (f)} be the homogeneous component of f of the lowest degree, the initial term of f, and let be the homogeneous ideal which is formed by the initial terms in ⁡ ( f ) {displaystyle operatorname {in} (f)} for all f ∈ I {displaystyle fin I} , the initial ideal of I. The tangent cone to X at the origin is the Zariski closed subset of k n {displaystyle k^{n}} defined by the ideal in ⁡ ( I ) {displaystyle operatorname {in} (I)} . By shifting the coordinate system, this definition extends to an arbitrary point of k n {displaystyle k^{n}} in place of the origin. The tangent cone serves as the extension of the notion of the tangent space to X at a regular point, where X most closely resembles a differentiable manifold, to all of X. (The tangent cone at a point of k n {displaystyle k^{n}} that is not contained in X is empty.)