Tangent cone

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

References

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

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

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

V.I. Danilov

References

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