In light of my [previous question](https://mathoverflow.net/questions/90409/second-chern-class-of-projective-space-blown-up-in-a-linear-subvariety), I am interested in the following scenario: Let $\tilde Y$ be the blow-up of $Y=\mathbb{P}^n$ along a linear subvariety $X\subseteq Y$ of codimension $d$, i.e. $X\cong\mathbb{P}^{n-d}$. Let $E$ denote the exceptional divisor. Let $H$ be a hyperplane in $Y$ and denote by $P$ the strict transform. I am now interested in the degrees of $P^a E^b$ for $a+b=n$. The only thing related I could find was [this post](https://mathoverflow.net/questions/72710/self-intersection-of-exceptional-divisor), but I am in a more general setting.


**PS:** I am working over $\mathbb{C}$, so you may assume that, or just any algebraically closed field, or less of course.



Intersection powers of the exceptional divisor (and the transform of a hyperplane)
Assuming $H$ contains $X$, I think $P^{n-b} \cdot E^b = (-1)^{b-1+\dim X} {b-1 \choose \dim X}$, where $\dim X = n-d$. One can, if one wishes, reduce to the case $a=0$ by letting $Y'$ be the intersection of $a$ generic hyperplanes through $X$, for then $P^a\cdot E^b = (E|\_{Y'})^b$. Or, without making this reduction, let $T=P+E$ be the total transform of $H$ in $\tilde{Y}$, and note that (i) $E$ is a $\mathbb{P}^{d-1}$-bundle over $X$, (ii) $P|\_E$ is a $\mathbb{P}^{d-2}$-subbundle over $X$, meeting every fiber of $E\to X$ in a hyperplane, (iii) $T|\_E$ is the pullback to $E$ of a hyperplane section of $X$. It follows geometrically that, if $a+b=n$, then $P^{a}\cdot T^{b-1} \cdot E = (P|\_E)^{a}\cdot (T|\_E)^{b-1}$ equals 1 if $(a,b)=(d-1,n-d+1)$ and 0 otherwise. Then we can expand $P^a \cdot E^b = P^a \cdot (T-P)^{b-1} \cdot E$ using multilinearity of the intersection form.


