On del Pezzo elliptic varieties of degree $\leq 4$
Antonio Laface, Andrea Luigi Tironi, Luca Ugaglia

TL;DR
This paper investigates the geometry of certain elliptic varieties derived from del Pezzo varieties of degree up to 4, establishing a connection between the finiteness of the Mordell-Weil group and the Cox ring's finite generation.
Contribution
It extends previous results to analyze the geometry of these elliptic varieties and proves the equivalence between Mordell-Weil group finiteness and Cox ring finite generation.
Findings
Mordell-Weil group is finite iff Cox ring is finitely generated.
Provides geometric descriptions of elliptic fibrations from del Pezzo varieties.
Extends prior work to a broader class of varieties.
Abstract
\special{html:<a href="hrefstring">} Let be a del Pezzo variety of degree and dimension , let be an ample class such that and let be a -dimensional subscheme of length such that the subsystem of elements of with base locus gives a rational morphism . Denote by the elliptic fibration obtained by resolving the indeterminacy locus of . Extending the results of [arXiv:1305.3340] we study the geometry of the variety and we prove that the Mordell-Weil group of is finite if and only if the Cox ring of is finitely generated.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Algebraic structures and combinatorial models
