I have a straightforward question :
Let $ Y$ be the union of the three lines $ L_1:x=0 , L_2 :y=0$ and $L_3:z=0$ in the Projective plane $\mathbb{P }^2$. What is the Class group of the Complement $U:=\mathbb{P}^2\setminus Y$ ?
Now all the main conditions required to define the class group are satisfied: Let $(X,\mathcal{O}_X)$ be a noetherian integral scheme which is regular in codimension one i.e. all local rings $\mathcal{O}_{X,p}$ have dimension one and are regular. The Class group $Cl(X)$ is defined as
$$ Div(X)/\sim $$ with $$ D\sim D' \iff D-D' \text{ is a principal divisor }$$
I tried to use the theorem that for any hyperplane $H:x_i=0$, if $D$ is a divisor of degree $d$ then $D\sim dH$.
I also know that if $W$ is a closed irreducible subset of $X$ codimension $1$, and $V=X\setminus W$, then there is a exact sequence:
$$\mathbb{Z} \to CL(X)\to CL(V)\to 0$$
Any hints are greatly appreciated.
The class group of $U$ is zero, dear Fellow Mathematician.
Indeed if you remove the line $z=0$ from $\mathbb P^2$, you are left with $\mathbb A^2$.
And if you then remove the points where $x=0$ or $y=0$ you are left with the product $U=(\mathbb A^1\setminus\{0\})\times (A^1\setminus\{0\})$.
This is an affine variety with coordinate ring
$$A=\mathcal O(Y)=k[T,T^{-1},U,U^{-1}]$$ Since that ring is factorial the corresponding affine variety $Y$ has zero class group: Hartshorne, Chapter II, Proposition 6.2 .