# Question regarding Chow group of a blow-up

Let $$X$$ be a smooth complex projective variety, and $$Y\hookrightarrow X$$ be a smooth projective subvariety. Let $$\pi:\tilde{X}\rightarrow X$$ be the blow-up along $$Y$$, and let $$j:E\hookrightarrow \tilde{X}$$ be the exceptional divisor.

Let $$Z$$ be a $$k$$-dimensional subvariety of $$E$$ such that $$dim\,\pi(Z)=dim\,Z$$. Now, $$[Z]\in CH_k(E)$$ is a $$k$$-cycle, and let $$Z':=j_*([Z])$$.

Question: What is the relation between $$\pi^*\pi_*(Z')$$ and $$Z'$$ ? Are they equal?

• No : Try $Z=E$.
– abx
Jul 17, 2019 at 19:53
• @abx: Thanks! I forgot to exclude the case when $\pi_*$ might become zero. I have edited the question. Jul 17, 2019 at 20:11

Say $$Y$$ is $$\mathbb P^1$$ and has codimension $$2$$, so $$E$$ is a $$\mathbb P^1$$-bundle on $$\mathbb P^1$$. Then this $$\mathbb P^1$$-bundle has many sections, which are not equivalent in the Chow group, but only equivalence class contains pullback of $$Y$$, which is the pullback of the pushforward of any section.
• @yojusmath If $Z$ is a section, then $\pi_* Z = Y$, so $\pi^* \pi_* Z = \pi^* Y$. This just comes from the definition of $\pi_*$. Jul 17, 2019 at 21:01