Simple description of a Chow ring of blow-ups.

2011-01-13T22:52:35Z

Is there a simple description of a Chow ring of a blow-up of a point on a smooth projective variety? Or at least of successive blow-ups of $\mathbb{P}^n$?
Maybe something like $A(\tilde{X})=f^*(A(X))\oplus\mathbb{Z}(E)$, where $f\colon\tilde{X}\to{}X$ is a blow-up, E is an exceptional divisor, with multiplication given by $E\cdot{}E_k=-E_{k-1}$, $E_0=f^*(P)$, where $E_k{}$ is a k-dimensional linear subspace of an exceptional divisor $E(=E_{n-1})$, and $P$ is a point we are blowing up. What I'm suggesting is true for surfaces (exercise 6.5 in appendix A of Hartshorne), and seems geometrically plausible in the case $X=\mathbb{P}^n$.
Also, it'd be great to know what cycles are effective.
I'm afraid all this is really trivial for someone understanding Fulton's book, but I'm not at that level yet.

Answer by Passenger for Simple description of a Chow ring of blow-ups.

2011-01-14T09:48:44Z

The general formula about the intersection ring of blow-ups is discussed in Fulton's book. In your case you want to study the intersection ring of a smooth algebraic variety $V$ blown up at a point $Z$. There is a simple formula for this situation by Keel. You can find it in his paper: Intersection Theory of Moduli Space of Stable N-Pointed Curves of Genus Zero.

Another nice reference is the paper "A compactification of configuration spaces" by Fulton-MacPherson. In section 5 of this paper they mention the Keel's formula and state the facts needed in the computation of the Chow ring. I summarize it below.

The key fact is that the restriction map from the Chow ring of the variety $V$ to the Chow ring of the point $Z$ is surjective. The intersection ring of the blow-up $\widetilde{V}$ is generated over $A(V)$ by the class of the exceptional divisor $E$ with the ideal $I$ of relations described bellow:

1) Let $J_{Z/V}$ be the kernel of the restriction map from $A(V)$ to $A(Z)$. It contains all elements in $A(V)$ of positive degree, for example.

2) Assume that you can write $Z$ as a transversal intersection $\cap_{i=1}^r D_i$ of the divisor classes $D_i$. Define the polynomial $P_{Z/Y} \in A(V)[t]$ by the rule $P(t)=\prod_{i=1}^r(t+D_i)$. This polynomial is called a Chern polynomial of $Z$. It depends on the choice of the divisor classed $D_i$. It means that it is not unique and is determined up-to an element in $J_{Z/V}$.

The ideal $I$ is generated by $J_{Z/V}\cdot E$ and $P_{Z/V}(-E)$. The Chow ring of $\widetilde{V}$ is therefore equal to $\frac{A(V)[E]}{I}$.

Simple description of a Chow ring of blow-ups. - MathOverflow

Simple description of a Chow ring of blow-ups.

Answer by Passenger for Simple description of a Chow ring of blow-ups.