Let $X$ be a smooth, projective variety of dimension $n \ge 2$, $L$ a very ample line bundle on $X$ and $\pi: \widetilde{X} \to X$ be the blow-up along a closed subvariety of codimension at least $2$. Denote by $E$ the associated exceptional divisor. Is it true that for all $1 \le i \le n-1$, we have $(\pi^*L)^{n-i} E^i=0$?
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3$\begingroup$ By projection formula this vanishing holds if $\pi_*(E^i) = 0$ in Chow groups; this must be the case when $i$ is less than codimension of the center of the blowup. On the other hand, when $n = 3$, $i = 2$ and we blow up a curve, the answer is not necessary zero (it is roughly degree of the curve). $\endgroup$– Evgeny ShinderCommented Apr 7, 2022 at 11:32
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1$\begingroup$ See What's the cohomology ring structure of a blow-up?. $\endgroup$– abxCommented Apr 7, 2022 at 16:39
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$\begingroup$ @EvgenyShinder and abx: Thanks for your comments. $\endgroup$– JanaCommented Apr 7, 2022 at 20:30
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