In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the tangent cone is a generalization of the notion of the tangent space

Tangent space

In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....

 to a manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

 to the case of certain spaces with singularities.

Definition in convex geometry

Let K be a closed

Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...

 convex subset of a real vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 V and ∂K be the boundary

Boundary (topology)

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of 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

Convex cone

In linear algebra, a convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients.-Definition:...

 in V and can also be defined as the intersection of the closed half-space

Half-space

In geometry, a half-space is either of the two parts into which a plane divides the three-dimensional euclidean space. More generally, a half-space is either of the two parts into which a hyperplane divides an affine space...

s of V containing K and bounded by the supporting hyperplane

Supporting hyperplane

Supporting hyperplane is a concept in geometry. A hyperplane divides a space into two half-spaces. A hyperplane is said to support a set S in Euclidean space \mathbb R^n if it meets both of the following:...

s of K at x. The boundary T_K 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 T_K is the ordinary tangent space

Tangent space

In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....

 to ∂K at x.

Definition in algebraic geometry

Let X be an affine algebraic variety embedded into the affine space kⁿ, with the defining ideal I ⊂ k[x₁,…,x_n]. For any polynomial f, let in(f) be the homogeneous component of f of the lowest degree, the initial term of f, and let in(I) ⊂ k[x₁,…,x_n] be the homogeneous ideal which is formed by the initial terms in(f) for all f ∈ I, the initial ideal of I. The tangent cone to X at the origin is the Zariski closed subset of kⁿ defined by the ideal in(I). By shifting the coordinate system, this definition extends to an arbitrary point of kⁿ 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

Differentiable manifold

A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...

, to all of X. (The tangent cone at a point of kⁿ that is not contained in X is empty.)

For example, the nodal curve

is singular at the origin, because both partial derivative

Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...

s of f(x, y) = y² − x³ − x² vanish at (0, 0). Thus the Zariski tangent space

Zariski tangent space

In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point P on an algebraic variety V...

 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

Noetherian

In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects; in particular,* Noetherian group, a group that satisfies the ascending chain condition on subgroups...

 schemes

Scheme (mathematics)

In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...

. Let X be an algebraic variety

Algebraic variety

In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...

, x a point of X, and (O_X,x ,m) be the local ring

Local ring

In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...

 of X at x. Then the tangent cone to X at x is the spectrum

Spectrum of a ring

In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...

 of the associated graded ring of O_X,x with respect to the m-adic filtration: