Tangent cone

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

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 at a point is the set of limiting positions of the secants passing through . More precisely, if the algebraic variety is imbedded in an affine space and if it is defined by an ideal of the ring so that has coordinates , then the tangent cone to at is given by the ideal of initial forms of the polynomials in . (If is the expansion of in homogeneous polynomials and , then is called the initial form of .) There is another definition, suitable for Noetherian schemes (see [1]): Let be the local ring of a scheme at the point , and let be its maximal ideal. Then the spectrum of the graded ring

is called the tangent cone to at the point .

In a neighbourhood of a point the variety 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 . The dimension and multiplicity of at 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 is a non-singular point of . 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

References

[a1]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001