The Picard group of a curve contained in $\mathbb{P}_F^1$ Let $F$ be an algebraically closed field and $X$ be a curve obtained by removing at least two points on the projective line $\mathbb{P}^1_F$, i.e., $\mathbb{P}^1_F-X$ is a reduced separated divisor on $\mathbb{P}^1_F$ of degree $\geq 2$.
We know that the Picard group of $\mathbb{P}^1_F$ is $\mathbb{Z}$. How does one deduce that the Picard group of $X$ is $0$?
In this case we can view $X$ as $\mathbb{A}^1_F-\{p_1,...,p_r\}$, where $r \geq 1$, since the projective line minus a point is the affine line as seen in Is the projective line minus one point always isomorphic to the affine space?. Since $\mathbb{A}^1_F = \mathrm{Spec}\,F[x]$, we know that $\mathrm{Pic}(\mathbb{A}_F^1) = 0$. Does the removal of points have any relation to the localization of the UFD $F[x]$?
 A: All the material you need to attack this problem will be located in the chapter on divisors in your favorite algebraic geometry book. For instance, here's the relevant material from Hartshorne chapter II section 6:
Corollary 6.16: If $X$ is a noetherian, integral, separated locally factorial scheme, then there is a natural isomorphism $\operatorname{Cl} X\cong\operatorname{Pic} X$.
Proposition 6.5(c): Suppose $X$ is a noetherian integral separated scheme which is regular in codimension one. Let $Z$ be an irreducible closed subset of codimension one, and let $U$ be it's open complement. Then there is an exact sequence $$\Bbb Z\to\operatorname{Cl} X\to \operatorname{Cl} U\to 0$$ where $1\in\Bbb Z$ maps to $[Z]\in \operatorname{Cl} X$.
Combining these statements with the result that $\operatorname{Cl} \Bbb P^1_F=\Bbb Z$ generated by a closed $F$-rational point, we see that $\operatorname{Cl} \Bbb A^1_F=0$, and therefore every open subscheme also has vanishing class (and therefore Picard) group.
Alternately, one may use the homotopy invariance of the class/Picard group - see proposition 6.6 which states that if $X$ satisfies the assumptions of proposition 6.5 above, then $\operatorname{Cl} X\cong \operatorname{Cl} X\times \Bbb A^1$; or here for the statement that if $X$ is normal, then $\operatorname{Pic} X\cong \operatorname{Pic} X\times\Bbb A^1$. As the class/Picard group of $\operatorname{Spec} F$ is trivial, the class/Picard group of $\Bbb A^1_F$ is also trivial, and then we may continue on with proposition 6.5(c) as above.
