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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.

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.
Definition in convex geometry

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.
Definition in algebraic geometry

y² = x³ + x² (red) with tangent cone (blue)

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, 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

\( C: y^2=x^3+x^2 \)

is singular at the origin, because both partial derivatives of \( f(x, y) = y^2=x^3+x^2 \) 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,

\( x=y,\quad x=-y. \)

Its defining ideal is the principal ideal of k[x] generated by the initial term of f, namely \( y^2 − x^2 = 0 \).

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:

\( \operatorname{gr}_m O_{X,x}=\bigoplus_{i\geq 0} m^i / m^{i+1}. \)

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Tangent cone