Let $\pi_1, \pi_2$ be two planes in $\mathbb{P}^3$ intersecting in a line $L$. Let us denote its union by $X$. I would like to understand better the Picard group of $X$.
I would also like to know what is the canonical divisor of $X$. Is it a Cartier Divisor?
I have some conjectures about these questions, but I have not found a reference that allows me to prove or disprove them.
The Picard group is isomorphic to $\mathbb{Z}$ and is generated by $\mathcal{O}_{\mathbb{P}^3}(1)\vert_X$. The canonical class corresponds to the line bundle $\mathcal{O}_{\mathbb{P}^3}(-2)\vert_X$, it is a Cartier divisor.