Let $V$ be a projective variety in $\mathbb P^n$ and $X=C(V)\subset \mathbb A^{n+1}$ its affine cone. Let $\bar X$ be the projective closure of $X=C(V)$ and let $P$ be the vertex of the cone.
Hartshorne says $Cl(\overline X) ≈ Cl(\overline X-P)$ where Cl is the Weil divisor class group. (Page 146, Ex. 6.3 (a))
Why is this true?
I know for any open set $U\subset \overline X$ we have a surjection from $Cl(\overline X) \to Cl(U)$.
Here $U$ is the open set $\overline X\setminus P$.
Why is the map an isomorphism?
Given a scheme $Y$ satisfying condition $(*)$ on page 130 of Hartshorne and a closed subscheme $Z\subset Y$ of codimension at least $2$, the restriction group morphism $Cl Y\to Cl (Y\setminus Z)$ is an isomorphism.
This is Proposition 6.5 (b), page133, which has a two line proof!
Apply to $Y=\overline X$ and $Z=\{P\}$.