scheme of finite type over a field and a closed subscheme. If is of ; i.e., every irreducible component has dimension r, then is also of pure dimension r. (This can be seen as a consequence of #Deformation to the normal cone.) This property is a key to an application in intersection theory: given a pair of closed subschemes in some ambient space, while the scheme-theoretic intersection has irreducible components of various dimensions, depending delicately on the positions of , the normal cone to is of pure dimension. Examples Let be an effective Cartier divisor. Then the normal bundle to it (or equivalently the normal cone to it) is Non-regular Embedding Consider the non-regular embedding then, we can compute the normal cone by first observing If we make the auxiliary variables and we get the relation We can use this to give a presentation of the normal cone as the relative spectrum Since is affine, we can just write out the relative spectrum as the affine scheme giving us the normal cone. Geometry of this normal cone The normal cone's geometry can be further explored by looking at the fibers for various closed points of . Note that geometrically is the union of the -plane with the -axis , so the points of interest are smooth points on the plane, smooth points on the axis, and the point on their intersection. Any smooth point on the plane is given by a map for and either or . Since it's arbitrary which point we take, for convenience let's assume . Hence the fiber of at the point is isomorphic to giving the normal cone as a one dimensional line, as expected. For a point on the axis, this is given by a map hence the fiber at the point is which gives a plane. At the origin , the normal cone over that point is again isomorphic to . Nodal cubic For the nodal cubic curve given by the polynomial over , and the point at the node, the cone has the isomorphism showing the normal cone has more components than the scheme it lies over. Deformation to the normal cone Suppose is an embedding. This can be deformed to the embedding of inside the normal cone (as the zero section) in the following sense: there is a flat family with generic fiber and special fiber such that there exists a