Let $X$ be a smooth, projective variety, $Y \subset X$ a smooth, projective subvariety of codimension $3$. Denote by $\pi:\tilde{X} \to X$ the blow-up of $X$ along $Y$ and by $E$ the exceptional divisor. We know, that $\pi|_E:E \to Y$ is a $\mathbb{P}^2$-bundle. Let $\pi':F \to Y$ be a $\mathbb{P}^1$-sub-bundle of $E$ in the sense that there exists a closed immersion $i:F \hookrightarrow E$ such that $\pi|_E \circ i=\pi'$. Is it possible to say something about the kernel of the push-forward morphism $i_*:\mathrm{Pic}(F) \to A^2(E)$ i.e., about its generators, where $A^2(E)$ denotes the Chow group of codiension $2$ cycles on $E$. If necessary assume that $\mathrm{Pic}(Y)=\mathbb{Z}$.

I know that this is not the Gysin map on Chow groups. The question is motivated by Gysin map on cohomology, rather its restriction to certain Hodge structures. To simplify the original question, I have given the above formulation.

  • $\begingroup$ By a $\mathbb{P}^1$-subbundle, do you mean that the fibers of $F$ over a point of $Y$ are lines in $\mathbb{P}^2$? $\endgroup$ – abx Nov 21 '17 at 6:03

OK, let me assume that your $F$ is a projective sub-bundle. Then $i_*$ is injective.

Note that the blowing up setting is completely irrelevant to the situation: you just have two projective bundles $p:E\rightarrow Y$, $q:F\rightarrow Y$ and an embedding $i:F\hookrightarrow E$ over $Y$. Now the Chow group of such bundles are well-known. If $h$ is the class of $F$ in $\operatorname{Pic}(E) $, we have $$\operatorname{Pic}(F)=q^*\operatorname{Pic}(Y)\oplus \mathbb{Z}i^*h\quad\mbox{and}\quad A^2(E)= p^*A^2(Y)\oplus h\cdot p^*\operatorname{Pic}(Y)\oplus \mathbb{Z}h^2 .$$ For $d\in \operatorname{Pic}(Y)$ we have $i_*q^*d=i_*(i^*p^*d)=h\cdot p^*d$, so $i_*$ injects the first summand of $\operatorname{Pic}(F)$ into the second summand of $A^2(E)$. Since $i_*i^*h= h^2$, this shows that $i_*$ is injective.

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