Well, really, for intersection theory, it’s true. We start with a closed subscheme, with normal cone . We’re going to construct a family of embeddings that deforms to the zero section of . Then, because intersections should vary nicely in families, we’ll have essentially reduced the problem of doing intersections to the case of normal cones.

Now, look at . Set to be the blowup along . Now, the normal cone to this will be , so the exceptional divisor, form the connections we talked about earlier, will be .

Now, also, the blowup of along embeds as a closed subscheme of . By the universal property of the blowup, the image of is a Cartier divisor. But then, it already was, and so the blowup is an isomorphism. So now we have a map . We can also embed the blowup of along (call it , by using .

So now, there are two properties that we want for to have, in addition to flatness.

- The fibers of over finite points of should just by , with the usual embedding of .
- Over , we should get the sum of two divisors, , with embedded by the zero section of followed by the inclusion of into , and such that is the exceptional divisor of .

For the first, we know that we have maps which are all flat, so flatness follows. To get the isomorphism away from , we note that is a blowup along a subvariety contained in , and so is an isomorphism there, as desired. The second property is the one that requires some work.

For the second, we have embeddings of the two divisors, so we can just look locally on . So we reduce to , and given by some ideal . We identify with , and so get an indeterminate . The blowup, , will then just be ), with .

Now, this is covered by open sets that are the specs of (that is, when ) with running through a set of generators of in . Now, the exceptional divisor is given by , and is given by , and then , which vanishes precisely at infinity. Thus, decomposes as we’d like.

Now, why do we want to use the embedding of into the normal cone? Well, for one, there is a retraction to when is regularly embedded, and two, there’s a vector bundle on the normal bundle of rank the codimension of with a section vanishing precisely on . This is a lot like the tubular neighborhood construction in topology, which simplifies a lot of problems.

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## About Charles Siegel

Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.

Flatness follows? The first map $M \to Y \times \mathbb{P}^1$ is a blowup, so it is hardly flat.

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