Let $\pi : \tilde{\mathbb{P}^{3}} \longrightarrow \mathbb{P}^{3}$ be the blowup of $\mathbb{P}^{3}$ along a smooth algebraic curve $C$ with exceptional divisor $E$ We know that $\text{Pic}(\mathbb{P}^{3}) \simeq \mathbb{Z}$. How do I determine the Picard Group of $\tilde{\mathbb{P}^{3}}$?
I thank you for your help.
Best Answer
Whenever $\pi: Y \rightarrow X$ is a blowup of a smooth variety along a smooth subvariety, we have
$$\operatorname{Pic}(Y) = \pi^* \operatorname{Pic}(X) \oplus \mathbf Z[E] $$
where $E$ is the class of the exceptional divisor.
So in your case, $\operatorname{Pic} \tilde{\mathbf P}^3 = \mathbf Z^2$.