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

Abstract

\special{html: } 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 $π_Z\colon Y\dashrightarrow{\mathbb P}^{n-1}$. Denote by $π\colon X\to {\mathbb P}^{n-1}$ the elliptic fibration obtained by resolving the indeterminacy locus of $π_Z$. Extending the results of [arXiv:1305.3340] we study the geometry of the variety $X$ and we prove that the Mordell-Weil group of $π$ is finite if and only if the Cox ring of $X$ is finitely generated.