It is cyclic, generated by $\mathscr{O}(1)$. Indeed this is true for $X$ by the Lefschetz theorem (SGA2, Exp. XII, Cor. 3.7), and the restriction map $\operatorname{Pic}(X)\rightarrow \operatorname{Pic}(X\smallsetminus V) $ is an isomorphism, because the local rings of $X$ are parafactorial by SGA2, Exp. XI, Thm. 3.13).
abx
- 38k
- 3
- 86
- 146