- Tangent cone
In

geometry , the**tangent cone**is a generalization of the notion of thetangent space to amanifold to the case of certain spaces with singularities.**Definition in convex geometry**Let "K" be a closed

convex subset of a realvector space "V" and ∂"K" be the convex surface which is 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 aconvex cone in "V" and can also be defined as the intersection of the closedhalf-space s of "V" containing "K" and bounded by thesupporting hyperplane 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 anaffine 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 ordinarytangent space to ∂"K" at "x".**Definition in algebraic geometry**Let "X" be an

affine algebraic variety embedded into the affine space "k"^{"n"}, with the defining ideal "I" ⊂ "k" ["x"_{1},…,"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" ["x"_{1},…,"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 "k"^{"n"}defined by the ideal in("I"). By shifting the coordinate system, this definition extends to an arbitrary point of "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 adifferentiable manifold , to all of "X". (The tangent cone at a point of "k"^{"n"}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 derivative s of "f"("x", "y") = "y"^{2}− "x"^{3}− "x"^{2}vanish at (0, 0). Thus theZariski 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 analgebraic variety , "x" a point of "X", and ("O"_{"X","x"},"m") be thelocal ring of "X" at "x". Then the**tangent cone**to "X" at "x" is the spectrum of the associatedgraded ring of "O"_{"X","x"}with respect to the "m"-adic filtration::$operatorname\{gr\}\_m\; O\_\{X,x\}=igoplus\_\{igeq\; 0\}\; m^i\; /\; m^\{i+1\}.$

**See also***

Monge cone **References***

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