View source for Tangent cone

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====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-line (ray)|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 hyperplane|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 [[secant]]s 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 [[polynomial]]s 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 scheme]]s (see [[#References|[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====

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<TR><TD valign="top">[1]</TD> <TD valign="top"> J.-i. Igusa, "Normal point and tangent cone of an algebraic variety" ''Mem. Coll. Sci. Univ. Kyoto'' , '''27''' (1952) pp. 189–201 {{MR|0052155}} {{ZBL|0101.38501}} {{ZBL|0049.38504}} </TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> P. Samuel, "Méthodes d'algèbre abstraite en géométrie algébrique" , Springer (1967) {{MR|0213347}} {{ZBL|}} </TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top"> 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 {{MR|0199184}} {{ZBL|0122.38603}} </TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top"> 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 {{MR|0188486}} {{ZBL|0129.39402}} </TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR>
<TR><TD valign="top">[6]</TD> <TD valign="top"> A.V. Pogorelov, "Extrinsic geometry of convex surfaces" , Amer. Math. Soc. (1972) (Translated from Russian) {{MR|0346714}} {{MR|0244909}} {{ZBL|0311.53067}} </TD></TR>
<TR><TD valign="top">[7]</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>
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