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Dear HNuer, a fundamental theorem on Chow groups describes the relation between the Chow ring $CH^\ast (\mathbb P(\mathcal E))$ of $X=\mathbb P(\mathcal E)$ and that $CH^\ast (Y)$ of $Y$ when $Y$ is regular.

If we call $p:\mathbb P(\mathcal E) \to Y $ the projection and $\xi$ the relative hyperplane bundle $\mathcal O_{\mathbb P(\mathcal E)}(1)$, we have

$$ CH^\ast(\mathbb P(\mathcal E) )= CH^\ast (Y)[\xi]/ < \xi^n +c_1 (p^\ast \mathcal E)\xi^{n-1} +\cdots+c_n (p^\ast \mathcal E)> $$ In particular $CH^1(\mathbb P(\mathcal E) )=p^\ast CH^1(Y)\oplus \mathbb Z \xi. $ (This is true even if $Y$ is not regular)

If you remember that locally factorial varieties (for example regular or smooth varieties) satisfy $Pic(P)=CH^1 (P)$ , your formula is proved under this hypothesis of local factoriality.

If $Y$ has bad singularities, I wouldn't bet on the correctness of the formula since there exist (non semi-normal) varieties $Y$ such that the canonical morphism $\pi^\ast: Pic (Y) \to Pic (Y) \times \mathbb A^1 $ is not surjective.

Dear HNuer, a fundamental theorem on Chow groups describes the relation between the Chow ring $CH^\ast (\mathbb P(\mathcal E))$ of $X=\mathbb P(\mathcal E)$ and that $CH^\ast (Y)$ of $Y$ when $Y$ is a regular variety over a not necessarily algebraically closed field.

If we call $p:\mathbb P(\mathcal E) \to Y $ the projection and $\xi$ the relative hyperplane bundle $\mathcal O_{\mathbb P(\mathcal E)}(1)$, we have

$$ CH^\ast(\mathbb P(\mathcal E) )= CH^\ast (Y)[\xi]/ < \xi^n +c_1 (p^\ast \mathcal E)\xi^{n-1} +\cdots+c_n (p^\ast \mathcal E)> $$ In particular $CH^1(\mathbb P(\mathcal E) )=p^\ast CH^1(Y)\oplus \mathbb Z \xi. $ (This is true even if $Y$ is not regular)

If you remember that locally factorial varieties (for example regular or smooth varieties) satisfy $Pic(P)=CH^1 (P)$ , your formula is proved under this hypothesis of local factoriality.

Edit: As the OP remarks in his comments below, the formula $Pic(\mathbb P(\mathcal E) )=p^\ast Pic(Y)\oplus \mathbb Z \xi $ is also true for any integral variety $Y$ over an algebraically closed field, locally factorial or not. The tool is then Grauert's semi-continuity theorem (cf. Hartshorne Chapter III, §12) rather than Chow groups.

Dear HNuer, a fundamental theorem on Chow groups describes the relation between the Chow ring $CH^\ast (\mathbb P(\mathcal E))$ of $X=\mathbb P(\mathcal E)$ and that $CH^\ast (Y)$ of $Y$ when $Y$ is regular.

If we call $p:\mathbb P(\mathcal E) \to Y $ the projection and $\xi$ the relative hyperplane bundle $\mathcal O_{\mathbb P(\mathcal E)}(1)$, we have

$$ CH^\ast(\mathbb P(\mathcal E) )= CH^\ast (Y)[\xi]/ < \xi^n +c_1 (p^\ast \mathcal E)\xi^{n-1} +\cdots+c_n (p^\ast \mathcal E)> $$ In particular $CH^1(\mathbb P(\mathcal E) )=p^\ast CH^1(Y)\oplus \mathbb Z \xi. $ (This is true even if $Y$ is not regular)

If you remember that for locally factorial varieties you have $Pic(P)=CH^1 (P)$ , your formula is proved under this hypothesis .

If $Y$ has bad singularities, I wouldn't bet on the correctness of the formula since there exist (non semi-normal) varieties $Y$ such that the canonical morphism $\pi^\ast: Pic (Y) \to Pic (Y) \times \mathbb A^1 $ is not surjective.

Dear HNuer, a fundamental theorem on Chow groups describes the relation between the Chow ring $CH^\ast (\mathbb P(\mathcal E))$ of $X=\mathbb P(\mathcal E)$ and that $CH^\ast (Y)$ of $Y$ when $Y$ is regular.

If we call $p:\mathbb P(\mathcal E) \to Y $ the projection and $\xi$ the relative hyperplane bundle $\mathcal O_{\mathbb P(\mathcal E)}(1)$, we have

$$ CH^\ast(\mathbb P(\mathcal E) )= CH^\ast (Y)[\xi]/ < \xi^n +c_1 (p^\ast \mathcal E)\xi^{n-1} +\cdots+c_n (p^\ast \mathcal E)> $$ In particular $CH^1(\mathbb P(\mathcal E) )=p^\ast CH^1(Y)\oplus \mathbb Z \xi. $ (This is true even if $Y$ is not regular)

If you remember that locally factorial varieties (for example regular or smooth varieties) satisfy $Pic(P)=CH^1 (P)$ , your formula is proved under this hypothesis of local factoriality.

If $Y$ has bad singularities, I wouldn't bet on the correctness of the formula since there exist (non semi-normal) varieties $Y$ such that the canonical morphism $\pi^\ast: Pic (Y) \to Pic (Y) \times \mathbb A^1 $ is not surjective.

Dear HNuer, a fundamental theorem on Chow groups describes the relation between the Chow ring $CH^\ast (\mathbb P(\mathcal E))$ of $X=\mathbb P(\mathcal E)$ and that $CH^\ast (Y)$ of $Y$.

If we call $p:\mathbb P(\mathcal E) \to Y $ the projection and $\xi$ the relative hyperplane bundle $\mathcal O_{\mathbb P(\mathcal E)}(1)$, we have

$$ CH^\ast(\mathbb P(\mathcal E) )= CH^\ast (Y)[\xi]/ < \xi^n +c_1 (p^\ast \mathcal E)\xi^{n-1} +\cdots+c_n (p^\ast \mathcal E)> $$ In particular $CH^1(\mathbb P(\mathcal E) )=p^\ast CH^1(Y)\oplus \mathbb Z \xi. $

If you remember that for locally factorial varieties you have $Pic(P)=CH^1 (P)$ , your formula is proved under this hypothesis .

If $Y$ has bad singularities, say not semi-normal, I wouldn't bet on the correctness of the formula since then there exist varieties such that the canonical morphism $\pi^\ast: Pic Y \to Pic Y \times \mathbb A^1 $ is not surjective.

Dear HNuer, a fundamental theorem on Chow groups describes the relation between the Chow ring $CH^\ast (\mathbb P(\mathcal E))$ of $X=\mathbb P(\mathcal E)$ and that $CH^\ast (Y)$ of $Y$ when $Y$ is regular.

If we call $p:\mathbb P(\mathcal E) \to Y $ the projection and $\xi$ the relative hyperplane bundle $\mathcal O_{\mathbb P(\mathcal E)}(1)$, we have

$$ CH^\ast(\mathbb P(\mathcal E) )= CH^\ast (Y)[\xi]/ < \xi^n +c_1 (p^\ast \mathcal E)\xi^{n-1} +\cdots+c_n (p^\ast \mathcal E)> $$ In particular $CH^1(\mathbb P(\mathcal E) )=p^\ast CH^1(Y)\oplus \mathbb Z \xi. $ (This is true even if $Y$ is not regular)

If you remember that for locally factorial varieties you have $Pic(P)=CH^1 (P)$ , your formula is proved under this hypothesis .

If $Y$ has bad singularities, I wouldn't bet on the correctness of the formula since there exist (non semi-normal) varieties $Y$ such that the canonical morphism $\pi^\ast: Pic (Y) \to Pic (Y) \times \mathbb A^1 $ is not surjective.