# Tangent cone

#### Tangent cone to a convex surface (by M.I. Voitsekhovskii)

The tangent cone to a convex surface $S$ at a point $O$ is the surface $V(O)$ of the cone formed by the half-lines emanating from $O$ and intersecting the convex body bounded by $S$ in at least one point distinct from $O$. (This cone itself is sometimes called the solid tangent cone.) In other words, $V(O)$ is the boundary of the intersection of all half-spaces containing $S$ and defined by the supporting planes to $S$ at $O$. If $V(O)$ is a plane, then $O$ is called a smooth point of $S$; if $V(O)$ is a dihedral angle, $O$ is called a ridge point; finally, if $V(O)$ is a non-degenerate convex cone, $O$ is called a conic point of $S$.

#### Tangent cone to an algebraic variety (by V.I. Danilov)

The tangent cone to an algebraic variety $X$ at a point $x$ is the set of limiting positions of the secants passing through $x$. More precisely, if the algebraic variety $X$ is imbedded in an affine space $A^n$ and if it is defined by an ideal $\mathfrak{A}$ of the ring $k[T_1,\ldots,T_n]$ so that $x\in X$ has coordinates $(0,\ldots,0)$, then the tangent cone $C(X,x)$ to $X$ at $x$ is given by the ideal of initial forms of the polynomials in $\mathfrak{A}$. (If $F = F_k + F_{k+1} + \cdots$ is the expansion of $F$ in homogeneous polynomials and $F_k \ne 0$, then $F_k$ is called the initial form of $F$.) There is another definition, suitable for Noetherian schemes (see [1]): Let $O_{X,x}$ be the local ring of a scheme $X$ at the point $x$, and let $\mathfrak{M}$ be its maximal ideal. Then the spectrum of the graded ring

$$ \bigoplus_{n\ge 0} (\mathfrak{M}^n / \mathfrak{M}^{n+1}) $$

is called the tangent cone to $X$ at the point $x$.

In a neighbourhood of a point $x$ the variety $X$ is, in a certain sense, structured in the same way as the tangent cone. For example, if the tangent cone is reduced, normal or regular, then so is the local ring $\mathcal{O}_{X,x}$. The dimension and multiplicity of $X$ at $x$ are the same as the dimension of the tangent cone and the multiplicity at its vertex. The tangent cone coincides with the Zariski tangent space if and only if $x$ is a non-singular point of $X$. A morphism of varieties induces a mapping of the tangent cones.

#### References

[1] | J.-i. Igusa, "Normal point and tangent cone of an algebraic variety" Mem. Coll. Sci. Univ. Kyoto , 27 (1952) pp. 189–201 MR0052155 Zbl 0101.38501 Zbl 0049.38504 |

[2] | P. Samuel, "Méthodes d'algèbre abstraite en géométrie algébrique" , Springer (1967) MR0213347 |

[3] | J. Hironaka, "Resolution of singularities of an algebraic variety over a field of characteristic zero I, II" Ann. of Math. , 79 (1964) pp. 109–203; 205–326 MR0199184 Zbl 0122.38603 |

[4] | H. Whitney, "Local properties of analytic varieties" S.S. Cairns (ed.) , Differential and Combinatorial Topol. (Symp. in honor of M. Morse) , Princeton Univ. Press (1965) pp. 205–244 MR0188486 Zbl 0129.39402 |

[5] | I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001 |

[6] | A.V. Pogorelov, "Extrinsic geometry of convex surfaces" , Amer. Math. Soc. (1972) (Translated from Russian) MR0346714 MR0244909 Zbl 0311.53067 |

[7] | R. Schneider, "Boundary structure and curvature of convex bodies" J. Tölke (ed.) J.M. Wills (ed.) , Contributions to geometry , Birkhäuser (1979) pp. 13–59 MR0568493 Zbl 0427.52003 |

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Tangent cone.

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