In geometry, the tangent cone is a generalization of the notion of the tangent space (applied to manifolds in the usual differential geometry setting) to irregular surfaces, that may contain singularities such as angular corners.

Definitions in nonlinear analysis

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.

Clarke tangent cone

be a nonempty closed subset of the Banach space

Clarke's tangent cone is always subset of the corresponding contingent cone (and coincides with it, when the set in question is convex). It has the important property of being a closed convex cone.

Definition in convex geometry

be a closed convex subset of a real vector space be the boundary. The solid tangent cone is the closure. It is a convex cone and can also be defined as the intersection of the closed half-spaces and bounded by the supporting hyperplanes of the solid tangent cone is the tangent cone. If this is an affine subspace is called a smooth point is said to be differentiable is the ordinary tangent space.

Another intuitive definition, formulated by A. D. Aleksandrov, the tangent cone is the figure obtained when homothetically dilating.

Definition in algebraic geometry

Let X be an affine algebraic variety. For any polynomial f be the homogeneous component of f of the lowest degree, the initial term of f, and let

, the initial ideal of I. The tangent cone to X. By shifting the coordinate system 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 that is not contained in X is empty.)

For example, the nodal curve

is singular at the origin, because both partial derivatives of f(x, y) = y2 x3 x2 vanish at (0, 0). Thus the Zariski tangent space to C at the origin is the whole plane, and has higher dimension than the curve itself (two versus one). On the other hand, the tangent cone is the union of the tangent lines to the two branches of C at the origin,

Its defining ideal is the principal ideal of k[x] generated by the initial term of f, namely y2 x2 = 0.

The definition of the tangent cone can be extended to abstract algebraic varieties, and even to general Noetherian schemes. Let X be an algebraic variety, x a point of X, and (OX,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 OX,x with respect to the m-adic filtration:

If we look at our previous example, then we can see that graded pieces contain the same information. So let

then if we expand out the associated graded ring

we can see that the polynomial defining our variety

See also

• Cone

• Metric tangent cone

• Monge cone

• Normal cone

References