Del Pezzo elliptic fibrations

Luca Ugaglia
Università degli studi di Palermo
Given an elliptic fibration
\[\pi:X\to W\]
, i.e. a morphism whose general fiber is a genus one curve, the set of rational sections turns out to be a group (once we fix the origin), called the Mordell Weil group of
First aim of this talk is to describe the Mordell Weil group of the so called del Pezzo elliptic fibrations, i.e. elliptic fibrations obtained via the following construction. We take
to be a complex del Pezzo variety of degree
\[d\leq 4\]
and dimension
\[n\geq 3\]
, and let
be an ample class such that
. A linear subsystem of
with zero dimensional base locus of length
defines a rational map onto
, whose general pseudofibers are genus one curves. Resolving the indeterminacy of this map we obtain an elliptic fibration
Relying on the above result we then prove that the Mordell Weil group of a del Pezzo elliptic fibration
is finite if and only if the Cox ring [1]
\[\mathcal{R}(X)=\bigoplus_{[D]\in{\rm Pic}(X)} H^0(X,\mathcal{O}_X(D))\]
 of the variety
is a finitely generated
-algebra [2, 3].
This is a joint work with J. Hausen, A. Laface and A. L. Tironi.
[1] I. Arzhantsev, U. Derenthal, J. Hausen, and A. Laface, Cox rings, Cambridge Studies in Advanced Mathematics, vol. 144, Cambridge University Press, Cambridge, 2015.
[2] J. Hausen, A. Laface, A.L. Tironi and L. Ugaglia: On cubic elliptic varieties, Transactions of the American Mathematical Society, 368 (2016), no. 1, 689–708.
[3] A. Laface, A. L. Tironi and L. Ugaglia: Del Pezzo elliptic varieties of degree
\[d\leq 4\]
, arXiv:1509.09220.