Let $k$ be an algebraically closed field. Let $X\subset \mathbb{A}^n\_{k}$ be a conical closed subvariety. In other words,
$\mathcal{O}(X)=k[x\_1,\cdots, x\_n]/I$, where $I$ is generated by homogeneous polynomials. Assume also that $X$ is a normal variety. In this general setting, is there anything known about the Picard group of $X$? For example, is it always true that
the Picard group of $X$ is trivial? (This is my naive hope.)



Picard group of a normal conical affine variety
Let $X$ be the affine cone over a normal projective variety $Y$.
Let
$$
\pi \colon \tilde{X} \to X
$$
be the blowup of $X$ at the vertex. Then $\tilde{X}$ comes with a projection
$$
p \colon \tilde{X} \to Y
$$
that identifies $\tilde{X}$ with the total space of a line bundle on $Y$ (the restriction to $Y$ of the tautological bundle from the ambient projective space). Moreover, the zero section of this line bundle coincides with the exceptional divisor $E \subset \tilde{X}$ of the blowup; in particular, $p \colon E \to Y$ is an isomorphism.


Now, the pullback map
$$
\pi^\* \colon \mathrm{Pic}(X) \to \mathrm{Pic}(\tilde{X})
$$
is injective and its image is contained in the subgroup
$$
\{ L \in \mathrm{Pic}(\tilde{X}) \mid L\vert\_E \cong \mathcal{O}\_E \}.
\tag{\*}
$$
On the other hand, the map
$$
p^\* \colon \mathrm{Pic}(Y) \to \mathrm{Pic}(\tilde{X})
$$
is an isomorphism, and since the composition of $p^\*$ and the restriction to $E$ is an isomorphism, it follows that the subgroup $(\*)$ is zero,
hence $\mathrm{Pic}(X) = 0$.


