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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|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 $S$.
 
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|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 $S$.
  
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Schneider, "Boundary structure and curvature of convex bodies" J. Tölke (ed.) J.M. Wills (ed.) , ''Contributions to geometry'' , Birkhäuser (1979) pp. 13–59 {{MR|0568493}} {{ZBL|0427.52003}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Schneider, "Boundary structure and curvature of convex bodies" J. Tölke (ed.) J.M. Wills (ed.) , ''Contributions to geometry'' , Birkhäuser (1979) pp. 13–59 {{MR|0568493}} {{ZBL|0427.52003}} </TD></TR></table>
  
The tangent cone to an algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212019.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212020.png" /> is the set of limiting positions of the secants passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212021.png" />. More precisely, if the algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212022.png" /> is imbedded in an affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212023.png" /> and if it is defined by an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212024.png" /> of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212025.png" /> so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212026.png" /> has coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212027.png" />, then the tangent cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212028.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212029.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212030.png" /> is given by the ideal of initial forms of the polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212031.png" />. (If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212032.png" /> is the expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212033.png" /> in homogeneous polynomials and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212034.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212035.png" /> is called the initial form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212036.png" />.) There is another definition, suitable for Noetherian schemes (see [[#References|[1]]]): Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212037.png" /> be the [[Local ring|local ring]] of a [[Scheme|scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212038.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212039.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212040.png" /> be its maximal ideal. Then the spectrum of the graded ring
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212041.png" /></td> </tr></table>
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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 [[#References|[1]]]): Let $O_{X,x}$ be the [[Local ring|local ring]] of a [[Scheme|scheme]] $X$ at the point $x$, and let $\mathcal{M}$ be its maximal ideal. Then the spectrum of the graded ring
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$$ \bigoplus_{n\ge 0} (\mathfrak{M}^n / \mathfrak{M}^{n+1}) $$
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is called the tangent cone to $X$ at the point $x$.
  
is called the tangent cone to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212042.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212043.png" />.
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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|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.
  
In a neighbourhood of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212044.png" /> the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212045.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212046.png" />. The dimension and multiplicity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212047.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212048.png" /> 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|Zariski tangent space]] if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212049.png" /> is a non-singular point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092120/t09212050.png" />. A morphism of varieties induces a mapping of the tangent cones.
 
  
 
====References====
 
====References====

Revision as of 11:46, 27 August 2013

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 $S$.


References

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


Comments

References

[a1] 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


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 $\mathcal{M}$ 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

V.I. Danilov

Comments

References

[a1] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
How to Cite This Entry:
Tangent cone. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tangent_cone&oldid=30255
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article