Consider the following cubic hypersurface in $\mathbb{P}^5$:


$$
X = \{z\_0z\_3z\_5-z\_1^2z\_5-z\_0z\_4^2+2z\_1z\_2z\_4-z\_2^2z\_3 = 0\}\subset\mathbb{P}^5
$$


The singular locus of $X$ is the Veronese surface $V\subset X$. I would like to ask if it is known what is the Picard group of $X\setminus V$?


Thank you very much.



Picard group of a cubic hypersurface
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).


**Edit:** This is wrong, as pointed out by @F\_L in the comments (thanks!). The mistake is that parafactoriality must be checked at *all* points of $V$, and not only the closed points. The local ring $\mathscr{O}\_{X,v}$ at the generic point $v$ of $V$ must be not parafactorial. I leave the answer since I think the error is instructive.


