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.)
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1$\begingroup$ The class group of $X$ (which is equal to its Picard group if $X$ is locally factorial) is described precisely in Hartshorne, Exercise II.6.3 — it is far from trivial. $\endgroup$– abxCommented Feb 8, 2023 at 5:38
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$\begingroup$ But $Pic(X)$ still can be trivial even though $cl(X)$ isn't, right? $\endgroup$– John Z.Commented Feb 8, 2023 at 6:07
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$\begingroup$ @abx: I guess a cone is very rarely locally factorial, isn't it? $\endgroup$– SashaCommented Feb 8, 2023 at 8:21
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$\begingroup$ @Sasha: Well, I am not sure how rare it is (I am definitely not an expert). The cone over a smooth complete intersection of dimension $\geq 3$, or over a Grassmanian, is locally factorial (but the Picard group is indeed trivial in those cases). $\endgroup$– abxCommented Feb 8, 2023 at 9:02
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2$\begingroup$ @abx: Let $\tilde{X}$ be the blowup of the cone $X$ at the vertex. Then the exceptional divisor is isomorphic to the base of the cone, the pullback map $\mathrm{Pic}(X) \to \mathrm{Pic}(\tilde{X})$ is injective and its image is contained in the subgroup of line bundles trivial on the exceptional divisor. On the other hand, the projection from $\tilde{X}$ to the base of the cone is the total space of a line bundle, hence this subgroup is trivial. I think this proves that $\mathrm{Pic}(X)$ is trivial for any affine cone. $\endgroup$– SashaCommented Feb 8, 2023 at 9:24
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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 $\mathrm{Pic}(X) = 0$.
