In algebraic geometry, the normal cone of a subscheme of a scheme is a scheme analogous to the normal bundle or tubular neighborhood in differential geometry. Definition The normal cone or of an embedding , defined by some sheaf of ideals I is defined as the relative Spec When the embedding i is regular the normal cone is the normal bundle, the vector bundle on X corresponding to the dual of the sheaf . If X is a point, then the normal cone and the normal bundle to it are also called the tangent cone and the tangent space (Zariski tangent space) to the point. When Y = Spec R is affine, the definition means that the normal cone to X = Spec R/I is the Spec of the associated graded ring of R with respect to I. If Y is the product X × X and the embedding i is the diagonal embedding, then the normal bundle to X in Y is the tangent bundle to X. The normal cone (or rather its projective cousin) appears as a result of blow-up. Precisely, let be the blow-up of Y along X. Then, by definition, the exceptional divisor is the pre-image ; which is the projective cone of . Thus, The global sections of the normal bundle classify embedded infinitesimal deformations of Y in X; there is a natural bijection between the set of closed subschemes of , flat over the ring D of dual numbers and having X as the special fiber, and H0(X, NX Y). Properties Compositions of regular embeddings If are regular embeddings, then is a regular embedding and there is a natural exact sequence of vector bundles on X: If are regular embeddings of codimensions and if is a regular embedding of codimension then In particular, if is a smooth morphism, then the normal bundle to the diagonal embedding (r-fold) is the direct sum of copies of the relative tangent bundle . If is a closed immersion and if is a flat morphism such that , then If is a smooth morphism and is a regular embedding, then there is a natural exact sequence of vector bundles on X: (which is a special case of an exact sequence for cotangent sheaves.) Cartesian square For a Cartesian square of schemes with the vertical map, there is a closed embedding of normal cones. Dimension of components Let be a