Editing Tangent cone

{{Short description|Generalization of the tangent space to a manifold to the case of certain spaces}}
{{refimprove|date=November 2009}}
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 [[Singularity theory|singularities]].

== Definitions in nonlinear analysis ==
In nonlinear analysis, there are many definitions for a tangent cone, including the [[adjacent cone]], [[Georges Bouligand|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 ===
Let <math>A</math> be a nonempty closed subset of the [[Banach space]] <math>X</math>. The Clarke's tangent cone to <math>A</math> at <math>x_0\in A</math>, denoted by <math>\widehat{T}_A(x_0)</math> consists of all vectors <math>v\in X</math>, such that for any sequence <math>\{t_n\}_{n\ge 1}\subset\mathbb{R}</math> tending to zero, and any sequence <math>\{x_n\}_{n\ge 1}\subset A</math> tending to <math>x_0</math>, there exists a sequence <math>\{v_n\}_{n\ge 1}\subset X</math> tending to <math>v</math>, such that for all <math>n\ge 1</math> holds <math>x_n+t_nv_n\in A</math>

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 ==

Let <math>K</math> be a [[closed set|closed]] [[convex subset]] of a real [[vector space]] <math>V</math> and <math>\partial K</math> be the [[boundary (topology)|boundary]] of <math>K</math>. The '''solid tangent cone''' to <math>K</math> at a point <math>x\in\partial K</math> is the [[closure (mathematics)|closure]] of the cone formed by all half-lines (or rays)  emanating from <math>x</math> and intersecting <math>K</math> in at least one point <math>y</math> distinct from <math>x</math>. It is a [[convex cone]] in <math>V</math> and can also be defined as the intersection of the closed [[Half-space (geometry)|half-space]]s of <math>V</math> containing <math>K</math> and bounded by the [[supporting hyperplane]]s of <math>K</math> at <math>x</math>. The boundary <math>T_K</math> of the solid tangent cone is the '''tangent cone''' to <math>K</math> and <math>\partial K</math> at <math>x</math>. If this is an [[affine subspace]] of <math>V</math> then the point <math>x</math> is called a '''smooth point''' of <math>\partial K</math> and <math>\partial K</math> is said to be '''differentiable''' at <math>x</math> and <math>T_K</math> is the ordinary [[tangent space]] to <math>\partial K</math> at <math>x</math>.
<!--- Incorporate this formula, but think about the best way of doing it
:<math>
T_K(x) = \overline{\bigcup_{\epsilon>0} \epsilon(K-x)}. </math>
--->

== Definition in algebraic geometry ==

[[File:Node (algebraic geometry).png|thumb|right|200px|y<sup>2</sup> = x<sup>3</sup> + x<sup>2</sup> (red) with tangent cone (blue)]]

Let ''X'' be an [[affine algebraic variety]] embedded into the affine space <math>k^n</math>, with defining ideal <math>I\subset  k[x_1,\ldots ,x_n]</math>. For any polynomial ''f'', let <math>\operatorname{in}(f)</math> be the homogeneous component of ''f'' of the lowest degree, the ''initial term'' of ''f'', and let 

: <math>\operatorname{in}(I)\subset  k[x_1,\ldots ,x_n]</math> 

be the homogeneous ideal which is formed by the initial terms <math>\operatorname{in}(f)</math> for all <math>f \in I</math>, the ''initial ideal'' of ''I''. The '''tangent cone''' to ''X'' at the origin is the Zariski closed subset of <math>k^n</math> defined by the ideal <math>\operatorname{in}(I)</math>. By shifting the [[coordinate system]], this definition extends to an arbitrary point of <math>k^n</math> 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 <math>k^n</math> that is not contained in ''X'' is empty.)

For example, the nodal curve

: <math>C: y^2=x^3+x^2 </math>

is singular at the origin, because both [[partial derivative]]s of ''f''(''x'', ''y'') = ''y''<sup>2</sup> &minus; ''x''<sup>3</sup> &minus; ''x''<sup>2</sup> 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,

: <math> x=y,\quad x=-y. </math>

Its defining ideal is the [[principal ideal]] of ''k''[''x''] generated by the initial term of ''f'', namely ''y''<sup>2</sup> &minus; ''x''<sup>2</sup> = 0.

The definition of the tangent cone can be extended to abstract algebraic varieties, and even to general [[Noetherian scheme|Noetherian]] [[scheme (mathematics)|schemes]]. Let ''X'' be an [[algebraic variety]], ''x'' a point of ''X'', and (''O''<sub>''X'',''x''</sub>, ''m'') be the [[local ring]] of ''X'' at ''x''. Then the '''tangent cone''' to ''X'' at ''x'' is the [[spectrum of a ring|spectrum]] of the [[associated graded ring]] of ''O''<sub>''X'',''x''</sub> with respect to the [[Completion (ring theory)#I-adic topology|''m''-adic filtration]]:
:<math>\operatorname{gr}_m \mathcal{O}_{X,x}=\bigoplus_{i\geq 0} m^i / m^{i+1}.</math>
If we look at our previous example, then we can see that graded pieces contain the same information. So let
:<math>
(\mathcal{O}_{X,x},\mathfrak{m}) = \left(\left(\frac{k[x,y]}{(y^2 - x^3 - x^2)}\right)_{(x,y)}, (x,y)\right)
</math>
then if we expand out the associated graded ring
:<math>
\begin{align}
\operatorname{gr}_m \mathcal{O}_{X,x} &= \frac{\mathcal{O}_{X,x}}{(x,y)} \oplus \frac{(x,y)}{(x,y)^2} \oplus \frac{(x,y)^2}{(x,y)^3} \oplus \cdots \\
&= k \oplus \frac{(x,y)}{(x,y)^2} \oplus \frac{(x,y)^2}{(x,y)^3} \oplus \cdots
\end{align}
</math>
we can see that the polynomial defining our variety
:<math>
y^2 - x^3 - x^2 \equiv y^2 - x^2
</math> in <math>\frac{(x,y)^2}{(x,y)^3}</math>

== See also ==
* [[Cone]]
* [[Monge cone]]
* [[Convex cone#Examples|Normal cone]]

== References ==

* {{Springer|title=Tangent cone|id=T/t092120|author=M. I. Voitsekhovskii}}
* {{cite book | vauthors=((Aubin, J.-P.)), ((Frankowska, H.)) | date= 2009 | chapter= Tangent Cones | title=Set-Valued Analysis | publisher=Birkhäuser | series=Modern Birkhäuser Classics | pages=117–177 | doi=10.1007/978-0-8176-4848-0_4 | isbn=978-0-8176-4848-0}}

{{DEFAULTSORT:Tangent Cone}}
[[Category:Convex geometry]]
[[Category:Algebraic geometry]]
[[Category:Variational analysis]]