Let π: X → ℙn−1 be an elliptic fibration obtained by resolving the indeterminacy of the projection of a cubic hypersurface Y of ℙn+1 from a line L not contained in Y . We prove that the Mordell-Weil group of π 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.
Hausen, J., Laface, A., Tironi, A., Ugaglia, L. (2016). On cubic elliptic varieties. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 368(1), 689-708 [10.1090/tran/6353].
On cubic elliptic varieties
UGAGLIA, Luca
2016-01-01
Abstract
Let π: X → ℙn−1 be an elliptic fibration obtained by resolving the indeterminacy of the projection of a cubic hypersurface Y of ℙn+1 from a line L not contained in Y . We prove that the Mordell-Weil group of π 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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