On cubic elliptic varieties

Hausen, Jürgen; Laface, Antonio; Tironi, Andrea; Ugaglia, Luca

doi:10.1090/tran/6353

On cubic elliptic varieties
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by Jürgen Hausen, Antonio Laface, Andrea Luigi Tironi and Luca Ugaglia PDF

Trans. Amer. Math. Soc. 368 (2016), 689-708 Request permission

Abstract:

Let $\pi \colon X\to \mathbb {P}^{n-1}$ be an elliptic fibration obtained by resolving the indeterminacy of the projection of a cubic hypersurface $Y$ of $\mathbb {P}^{n+1}$ from a line $L$ not contained in $Y$. We prove that the Mordell-Weil group of $\pi$ is finite if and only if the Cox ring of $X$ is finitely generated. We also provide a presentation of the Cox ring of $X$ when it is finitely generated.

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