To understand $f^*[H] \in H^2(\tilde Y)$, where $H \subset Y = \mathbb{P}^n$ is a hyperplane, it may help to think of $H$ as the zero set of section $s$ of the anticanonical bundle $\mathcal{O}_Y(1)$. Then the zero set of $f^*s$ (essentially just the preimage of $H$) is a divisor representing $f^*[H]$. If $H$ contained the blow-up locus $X$, then the resulting divisor is the sum of the proper transform $P$ of $H$ (the blow-up of $H$ at $X$) and the exceptional set $\tilde X$. If $H$ was transverse to $X$, then the divisor is really just the proper transform of $H$ (which is exactly the pre-image of $H$, H$in this case), which is the blow-up$\tilde H$of$H$at$H \cap X$. I presume$d = n-m$, the codimension of$X$in$Y$. If$D$is a divisor in$\tilde X$representing$[D] \in H^2(\tilde X)$, then $j_*[D]$ is the class$[D] \in H^4(\tilde Y)$that you get by considering$D$as a cycle in$\tilde Y$. $g^*\xi$ can be represented by a divisor that is the preimage in$\tilde X$of a hyperplane in$X$. Writing that hyperplane as the intersection of$X$with a transverse hyperplane$H \subset Y$, we find that $j_*g^*\xi = [\tilde H \cap \tilde X] \in H^4(\tilde Y)$.$\zeta \in H^2(\tilde X)$corresponds to the normal conormal bundle of$\tilde X$in$\tilde Y$, so it is the restriction of$-[\tilde X] \in H^2(\tilde Y)$to$\tilde X$. Therefore $j_*\zeta = -[\tilde X]^2 \in H^4(\tilde Y)$. To describe $j_*\zeta$ another way, note that since$\mathcal{N} = \mathcal{O}_X(1)^d$,$\tilde X$is a trivial bundle$X \times \mathbb{P}^{d-1}$. You can get an explicit trivialisation by picking copy of$\mathbb{P}^{d-1} \cong Z \subset Y$disjoint from$X$: given points$x \in X$and$z \in Z$, the line from$x$to$z$defines an element in the projectivisation of the fibre of$\mathcal{N}$over$x$. Let$h$be the projection$\tilde X \to Z$. Then $\mathcal{O}_{\tilde X}(-1) = g^*\mathcal{O}_X(1) + h^*\mathcal{O}_{Z}(-1)$. $h^*\mathcal{O}_{Z}(1)$ corresponds to a trivial$\mathbb{P}^{d-2}$subbundle of$\tilde X$. Such a divisor is the intersection of$\tilde X$with the proper transform$P$of a hyperplane$H$containing$X$and some hyperplane in$Z$. In other words, $-\zeta = g^*\xi -[P \cap \tilde X] \in H^2(\tilde X)$, so $j_*(\zeta + g_*\xi) = [P \cap \tilde X] \in H^4(\tilde Y)$. Sanity check:$-[\tilde X] + [\tilde H] = [P]$implies$-[\tilde X]^2 + [\tilde H \cap \tilde X] = [P \cap \tilde X]$, so it adds up. 1 My understanding of this is very unsophisticated, but perhaps that means that what I can explain is precisely what you want. To understand $f^*[H] \in H^2(\tilde Y)$, where$H \subset Y = \mathbb{P}^n$is a hyperplane, it may help to think of$H$as the zero set of section$s$of the anticanonical bundle$\mathcal{O}_Y(1)$. Then the zero set of $f^*s$ (essentially just the preimage of$H$) is a divisor representing $f^*[H]$. If$H$contained the blow-up locus$X$, then the resulting divisor is the sum of the proper transform$P$of$H$(the blow-up of$H$at$X$) and the exceptional set$\tilde X$. If$H$was transverse to$X$, then the divisor is really just the pre-image of$H$, which is the blow-up$\tilde H$of$H$at$H \cap X$. I presume$d = n-m$, the codimension of$X$in$Y$. If$D$is a divisor in$\tilde X$representing$[D] \in H^2(\tilde X)$, then $j_*[D]$ is the class$[D] \in H^4(\tilde Y)$that you get by considering$D$as a cycle in$\tilde Y$. $g^*\xi$ can be represented by a divisor that is the preimage in$\tilde X$of a hyperplane in$X$. Writing that hyperplane as the intersection of$X$with a transverse hyperplane$H \subset Y$, we find that $j_*g^*\xi = [\tilde H \cap \tilde X] \in H^4(\tilde Y)$.$\zeta \in H^2(\tilde X)$corresponds to the normal bundle of$\tilde X$in$\tilde Y$, so it is the restriction of$-[\tilde X] \in H^2(\tilde Y)$to$\tilde X$. Therefore $j_*\zeta = -[\tilde X]^2 \in H^4(\tilde Y)$. To describe $j_*\zeta$ another way, note that since$\mathcal{N} = \mathcal{O}_X(1)^d$,$\tilde X$is a trivial bundle$X \times \mathbb{P}^{d-1}$. You can get an explicit trivialisation by picking copy of$\mathbb{P}^{d-1} \cong Z \subset Y$disjoint from$X$: given points$x \in X$and$z \in Z$, the line from$x$to$z$defines an element in the projectivisation of the fibre of$\mathcal{N}$over$x$. Let$h$be the projection$\tilde X \to Z$. Then $\mathcal{O}_{\tilde X}(-1) = g^*\mathcal{O}_X(1) + h^*\mathcal{O}_{Z}(-1)$. $h^*\mathcal{O}_{Z}(1)$ corresponds to a trivial$\mathbb{P}^{d-2}$subbundle of$\tilde X$. Such a divisor is the intersection of$\tilde X$with the proper transform$P$of a hyperplane$H$containing$X$and some hyperplane in$Z$. In other words, $-\zeta = g^*\xi -[P \cap \tilde X] \in H^2(\tilde X)$, so $j_*(\zeta + g_*\xi) = [P \cap \tilde X] \in H^4(\tilde Y)$. Sanity check:$-[\tilde X] + [\tilde H] = [P]$implies$-[\tilde X]^2 + [\tilde H \cap \tilde X] = [P \cap \tilde X]\$, so it adds up.