I’m trying to compute the Picard group of ${\rm Spec}\left(\frac{k[x,y]}{xy(x+y+1)}\right)$ where $k$ is a field.
The question came up when I was trying to compute the Picard group of a ‘triangle’ over $k$ (I.e Three copies of $\mathbb{A}^1_k$ glued together where $0$ on the first line is identified with $1$ on the second, $0$ on the second with $1$ on the third and $0$ on the third with $1$ on the first), which I showed is isomorphic to the given scheme.
But I have no idea how to compute the Picard group.
Any help would be appreciated!
This is a hint, but too long for a comment :
Let $X=\operatorname{Spec}k[x,y]/(xy(x+y+1))$ and $L_1,L_2,L_3$ be the three lines whose union is $X$. Let $\mathcal{L}$ be a line bundle on $X$. Then $\mathcal{L}|_{L_i}$ is trivial and admit a non-vanishing section $s_i$ well defined up to a non-zero constant.
Actually, fix $s_1$, then you have a unique choice for $s_2$ with the requirement that $s_1=s_2$ on the intersection point $L_1\cap L_2$. Similarily, you have a unique choice for $s_3$ with the requirement that $s_2=s_3$ on $L_2\cap L_3$. Finally, on $L_1\cap L_3$ you don't have necessarily $s_1=s_3$ and there is a constant $u\in k^*$ such that $s_1=us_3$ on $L_1\cap L_3$.
Show that the map $\operatorname{Pic}(X)\to k^*$ such that $\mathcal{L}\mapsto u$ is well defined and an isomorphism.