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Let $X$ be a smooth projective variety over $\mathbb{C}$, and $D_1, \ldots, D_n$ be a collection of simple normal crossing divisors. The divisors induce a stratification $\mathcal{T}_X$ of $X$.

Let $T_1, \ldots, T_k$ be strata and $f \colon Y\to X$ be the iterated ordered blowup of $X$ at $T_1$, then at $T_2$, ... and finally at $T_k$, and let $E_1, \ldots, E_k$ be the exceptional divisors.

The exceptional divisors induce a stratification $\mathcal{T}_Y$ of $Y$.

Questions:

  1. Is there a way to relate the stratification $\mathcal{T}_X$ and the stratification $\mathcal{T}_Y$? For example, can the poset of $\mathcal{T}_Y$ be explicitly described from the poset of $\mathcal{T}_X$?

  2. Let $a_1, \ldots, a_k \in \mathbb{N}$. Is there an explicit way to express the cohomological pushforward $f_*(E_1^{a_1}\cdot \cdots \cdot E_k^{a_k})$ in terms of the strata in $\mathcal{T}_X$ and the Chern classes of their normal bundles?

(For the questions to be interesting, I think one may assume that the pairwise intersection of the strata $T_1, \ldots, T_k$ is non-transversal).

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  • $\begingroup$ It would be more natural to consider the stratification of $Y$ induced by both $E_i$ and the proper preimages of $D_j$. Anyway, the whole set up is basically smooth toroidal geometry. Whatever formulas you get under the assumption that everything is toric should also hold in the toroidal setting. $\endgroup$ Apr 3, 2022 at 23:05
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    $\begingroup$ Perhaps, Theorem 4.8 of arxiv.org/pdf/math/0206241.pdf could be useful. $\endgroup$ Apr 4, 2022 at 0:00
  • $\begingroup$ Thank you, this is very helpful! It seems to me that it answers my question 2 completely (feel free to post as an answer). $\endgroup$
    – calc
    Apr 6, 2022 at 14:48
  • $\begingroup$ I am glad it helps. Chances are, this already exists in some form elsewhere in the literature, maybe better suited to the birational map case (as opposed to generically finite case in the paper). So maybe someone else will post a better answer/reference. $\endgroup$ Apr 7, 2022 at 0:28

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