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Let E be an smooth elliptic curve in a projective plane.

Suppose that X is the projective cone over E in a projective space of dimension three.

What is the Picard group of X ?

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The restriction map $\operatorname{Pic}(\mathbb{P}^3) \rightarrow \operatorname{Pic}(X)$ is an isomorphism. This is actually true for any cone provided the depth at the vertex is $>1$: see Proposition 3.10 in Jaffe's Space curves which are the intersection of a cone with another surface, Duke Math. J. 57, no. 3, 859-876 (1988).

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