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.
Flatness follows? The first map $M \to Y \times \mathbb{P}^1$ is a blowup, so it is hardly flat.
For those wondering: Flatness holds because a map to a smooth curve is flat if all of the associated points of the domain are mapped to the generic point of the curve… (this is an exercise in Ravi’s notes, in the flatness section). If we assume that Y is integral, then the blow up is integral, so has no associated points other than the generic point, and removing a closed subscheme doesn’t change this. The generic point of the blow up is mapped to the generic point of P^1 because the map is surjective.
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