Tangent cone

Last updated February 26, 2019

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.

Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

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 that gives the displacement of the one point from the other.

Manifold topological space that at each point resembles Euclidean space

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. In this more precise terminology, a manifold is referred to as an n-manifold.

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.

Georges Louis Bouligand was a French mathematician who introduced paratingent cones and contingent cones.

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 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 to ∂K at x.

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. In a complete metric space, a closed set is a set which is closed under the limit operation.

A vector space is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below.

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 S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set S include bd(S), fr(S), and ∂S. Some authors use the term frontier instead of boundary in an attempt to avoid confusion with the concept of boundary used in algebraic topology and manifold theory. However, frontier sometimes refers to a different set, which is the set of boundary points which are not actually in the set ; that is, S \ S known alternatively as the residue of S. Other authors use the term boundary for those points that are in S but not in its interior.

Definition in algebraic geometry

y = x + x (red) with tangent cone (blue) Node (algebraic geometry).png — y = x + x (red) with tangent cone (blue)

Let X be an affine algebraic variety embedded into the affine space

k^{n}

, with defining ideal

I\subset k[x_{1},\ldots ,x_{n}]

. For any polynomial f, let

\operatorname {in} (f)

be the homogeneous component of f of the lowest degree, the initial term of f, and let

\operatorname {in} (I)\subset k[x_{1},\ldots ,x_{n}]

be the homogeneous ideal which is formed by the initial terms $\operatorname {in} (f)$ for all $f\in I$ , 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 $\operatorname {in} (I)$ . 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 a differentiable manifold, to all of X. (The tangent cone at a point of $k^{n}$ that is not contained in X is empty.)

In mathematics, 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 each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.

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) = 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,

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. Partial derivatives are used in vector calculus and differential geometry.

In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point P on an algebraic variety V. It does not use differential calculus, being based directly on abstract algebra, and in the most concrete cases just the theory of a system of linear equations.

x=y,\quad x=-y.

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 (O_X,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 O_X,x with respect to the m-adic filtration:

\operatorname {gr} _{m}O_{X,x}=\bigoplus _{i\geq 0}m^{i}/m^{i+1}.

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

({\mathcal {O}}_{X,x},{\mathfrak {m}})=\left(\left({\frac {k[x,y]}{(y^{2}-x^{3}-x^{2})}}\right)_{(x,y)},(x,y)\right)

then if we expand out the associated graded ring

{\begin{aligned}\operatorname {gr} _{m}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{aligned}}

we can see that the polynomial defining our variety

y^{2}-x^{3}-x^{2}\equiv y^{2}-x^{2}

in

{\frac {(x,y)^{2}}{(x,y)^{3}}}

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References

M. I. Voitsekhovskii (2001) [1994], "Tangent cone", in Hazewinkel, Michiel, Encyclopedia of Mathematics , Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

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