On del Pezzo elliptic varieties of degree $\leq 4$

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 $Y$ be a del Pezzo variety of degree $d\leq 4$ and dimension $n\geq 3$, let $H$ be an ample class such that $-K_Y=(n-1)H$ and let $Z\subset Y$ be a $0$-dimensional subscheme of length $d$ such that the subsystem of elements of $|H|$ with base locus $Z$ gives a rational morphism $\pi_Z\colon Y\dashrightarrow{\mathbb P}^{n-1}$. Denote by $\pi\colon X\to {\mathbb P}^{n-1}$ the elliptic fibration obtained by resolving the indeterminacy locus of $\pi_Z$. Extending the results of [arXiv:1305.3340] we study the geometry of the variety $X$ and we prove that the Mordell-Weil group of $\pi$ is finite if and only if the Cox ring of $X$ is finitely generated.