## Normal Cones

Ok, so I took the weekend off to figure out where things are going and get a bit ahead.  Will probably be doing that all month.  So now, we’re going to talk about cones and normal cones, with the goal of eventually defining the intersection product itself.

We’re not going to worry about what’s coming up though, at the moment.  First, we need to define what a cone is.  Let $X$ be a scheme, and let $\mathscr{S}^*$ be a sheaf of graded algebras such that $\mathscr{O}_X\to \mathscr{S}^0$ is surjective, $\mathscr{S}^1$ is coherent, and $\mathscr{S}^*$ is generated by $\mathscr{S}^1$.  We define a cone over $X$ to be anything of the form $C=\underline{\mathrm{Spec}}(\mathscr{S}^*)$, the relative spectrum of the sheaf of algebras.  Note: every vector bundle is a cone.  This really is a good generalization of vector bundles, at least for the purposes we need.

As we go, we’ll cite the properties and definitions of cones that we need.  But right now, we’re going to start talking about the most important one: the normal cone.  Now, let $\mathscr{X}\subset\mathscr{Y}$ be a closed subscheme define by a sheaf of ideals $\mathscr{I}$.  Well, one thing we can do is look at the graded sheaf of algebras $\oplus_{n\geq 0}\mathscr{I}^n/\mathscr{I}^{n+1}$.  Then we take the cone it defines, and this is $C_X Y$, the normal cone of $X$ in $Y$.

Now, let’s take a moment to make sure that we’ve got the right definition.  What happens if $X\to Y$ is a regular embedding of codimension $d$? Well, in that case, we actually get a vector bundle of rank $d$, on $X$.  If we look back in Hartshorne, we recall that $\mathscr{I}/\mathscr{I}^2$ is the conormal sheaf, so then $\mathscr{H}om(\mathscr{I}/\mathscr{I}^2,\mathscr{O}_X)$ is the normal sheaf.  And this is then going to be locally free of rank $d$, so it gives us a vector bundle.  To see that it’s the one given by the normal cone, remember that $\underline{\mathrm{Spec}}(\mathrm{Sym}^*\mathscr{E}^\vee)$ is the total space of a vector bundle.  So we have at the least, consistency of language here.

Now, part of why the normal cone is so useful is that it is connected to the notion of a blowup.  First off, recall that $\mathrm{Bl}_X Y=\underline{\mathrm{Proj}}(\oplus_{n\geq 0}\mathscr{I}^n)$, the projectivization of a certain cone.  Let’s call the blowup $\tilde{Y}$.  Adn we’re going to look at the exceptional divisor.  That is, if we have the projection $\pi:\tilde{Y}\to Y$, we look at $\pi^{-1}(X)=E$.  This is a Cartier divisor, by the universal property of blowups.  Now, which one is it? Well, it’s actually going to be the projective cone of $(\oplus\mathscr{I}^n)\otimes_{\mathscr{O}_Y}\mathscr{O}_X$.  What is that? It’s precisely $\oplus_{n\geq 0} \mathscr{I}^n/\mathscr{I}^{n+1}$, so we get $E=P(C_X Y)$.

So then, $N_E\tilde{Y}=\mathscr{O}_{\tilde{Y}}(E)|_E=\mathscr{O}_C(-1)$, with $C=C_XY$.