family of closed embeddings over such that Over any point the associated embeddings are an embedding The fiber over is the embedding of given by the zero section. This construction defines a tool analogous to differential topology where non-transverse intersections are performed in a tubular neighborhood of the intersection. Now, the intersection of with a cycle in can be given as the pushforward of an intersection of with the pullback of in . Construction One application of this is to define intersection products in the Chow ring. Suppose that X and V are closed subschemes of Y with intersection W, and we wish to define the intersection product of X and V in the Chow ring of Y. Deformation to the normal cone in this case means that we replace the embeddings of X and W in Y and V by their normal cones CY(X) and CW(V), so that we want to find the product of X and CWV in CXY. This can be much easier: for example, if X is regularly embedded in Y then its normal cone is a vector bundle, so we are reduced to the problem of finding the intersection product of a subscheme CWV of a vector bundle CXY with the zero section X. However this intersection product is just given by applying the Gysin isomorphism to CWV. Concretely, the deformation to the normal cone can be constructed by means of blowup. Precisely, let be the blow-up of along . The exceptional divisor is , the projective completion of the normal cone; for the notation used here see . The normal cone is an open subscheme of and is embedded as a zero-section into . Now, we note: The map , the followed by projection, is flat. There is an induced closed embedding that is a morphism over . M is trivial away from zero; i.e., and restricts to the trivial embedding as the divisor is the sum where is the blow-up of Y along X and is viewed as an effective Cartier divisor. As divisors and intersect at , where sits at infinity in . Item 1 is clear (check torsion-free-ness). In general, given , we have . Since is already an effective Cartier divisor on , we get yielding . Item 3 follows from the fact the blowdown map π is an isomorphism away from the center . The last two items are