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I'm trying to understand how to compute the Chow ring of a blow-up.

Let $W\subset \mathbb P^4$ be a smooth surface and let $X$ be the blow-up of $\mathbb P^4$ along $W$ with exceptional divisor $E$. By Griffiths-Harris, we have $$A^2(X)=A^2(\mathbb P^4)\oplus A^1(W),$$ so if $H$ denotes the pullback of $O(1)$ on $\mathbb P^n$, there is a basis of $A^2(X)$ given by $H^2$ and line bundles on $W$.

However I can also form the 2-cycles $H\cdot E$ and $E^2$. How can these be expressed in terms of the above basis?

Edit: It seems that $H\cdot E$ should correspond to the line bundle $O_{\mathbb P^4}(1)$ pulled back to $E=\mathbb P N_W^*$. What about $E^2$?

In other words, if $i:E\to X$ is the inclusion and $\pi:E=\mathbb P N_W^*\to W$ is the bundle, then what is $i_*O_E(-1)$ in terms of $H$ and $A^1(W)$?)

Let $W\subset \mathbb P^4$ be a smooth surface and let $X$ be the blow-up of $\mathbb P^4$ along $W$ with exceptional divisor $E$. By Griffiths-Harris, we have $$A^2(X)=A^2(\mathbb P^4)\oplus A^1(W),$$ so if $H$ denotes the pullback of $O(1)$ on $\mathbb P^n$, there is a basis of $A^2(X)$ given by $H^2$ and line bundles on $W$.

However I can also form the 2-cycles $H\cdot E$ and $E^2$. How can these be expressed in terms of the above basis?

Edit: It seems that $H\cdot E$ should correspond to the line bundle $O_{\mathbb P^4}(1)$ pulled back to $E=\mathbb P N_W^*$. What about $E^2$?

In other words, if $i:E\to X$ is the inclusion and $\pi:E=\mathbb P N_W^*\to W$ is the bundle, then what is $i_*O_E(-1)$ in terms of $H$ and $A^1(W)$?)

Let $W\subset \mathbb P^n$ be a smooth projective variety of codimension 2 and let $X$ be the blow-up of $\mathbb P^n$ along $W$ with exceptional divisor $E$. By Griffiths-Harris, we have $A^2(X)=A^2(\mathbb P^n)\oplus A^1(W)$, so if $H$ denotes the pullback of $O(1)$ on $\mathbb P^n$, there is a basis of $A^2(X)$ given by $H^2$ and line bundles on $W$.

However I can also form the 2-cycles $H\cdot E$ and $E^2$. How can these be expressed in terms of the above basis?

I'm trying to understand how to compute the Chow ring of a blow-up.

Let $W\subset \mathbb P^4$ be a smooth surface and let $X$ be the blow-up of $\mathbb P^4$ along $W$ with exceptional divisor $E$. By Griffiths-Harris, we have $$A^2(X)=A^2(\mathbb P^4)\oplus A^1(W),$$ so if $H$ denotes the pullback of $O(1)$ on $\mathbb P^n$, there is a basis of $A^2(X)$ given by $H^2$ and line bundles on $W$.

However I can also form the 2-cycles $H\cdot E$ and $E^2$. How can these be expressed in terms of the above basis?

Edit: It seems that $H\cdot E$ should correspond to the line bundle $O_{\mathbb P^4}(1)$ pulled back to $E=\mathbb P N_W^*$. What about $E^2$?

In other words, if $i:E\to X$ is the inclusion and $\pi:E=\mathbb P N_W^*\to W$ is the bundle, then what is $i_*O_E(-1)$ in terms of $H$ and $A^1(W)$?)

Let $W\subset \mathbb P^n$ be a smooth projective variety of codimension 2 and let $X$ be the blow-up of $\mathbb P^n$ along $W$ with exceptional divisor $E$. By Griffiths-Harris, we have $A^2(X)=A^2(\mathbb P^n)\oplus A^1(W)$, so if $H$ denotes the pullback of $O(1)$ on $\mathbb P^n$, there is a basis of $A^2(X)$ is given by $H^2$ and line bundles on $W$. However I can also form the 2-cycles $H\cdot E$ and $E^2$. How can these be expressed in this basis?

Let $W\subset \mathbb P^n$ be a smooth projective variety of codimension 2 and let $X$ be the blow-up of $\mathbb P^n$ along $W$ with exceptional divisor $E$. By Griffiths-Harris, we have $A^2(X)=A^2(\mathbb P^n)\oplus A^1(W)$, so if $H$ denotes the pullback of $O(1)$ on $\mathbb P^n$, there is a basis of $A^2(X)$ given by $H^2$ and line bundles on $W$. 

However I can also form the 2-cycles $H\cdot E$ and $E^2$. How can these be expressed in terms of the above basis?