Del Pezzo elliptic fibrations

Luca Ugaglia
Università degli studi di Palermo
Abstract:
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
\[\pi\]
.
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
\[Y\]
to be a complex del Pezzo variety of degree
\[d\leq 4\]
and dimension
\[n\geq 3\]
, and let
\[H\]
be an ample class such that
\[-K_Y=(n-1)H\]
. A linear subsystem of
\[|H|\]
with zero dimensional base locus of length
\[d\]
defines a rational map onto
\[\mathbb{P}^{n-1}\]
, whose general pseudofibers are genus one curves. Resolving the indeterminacy of this map we obtain an elliptic fibration
\[\pi:X\to\mathbb{P}^{n-1}\]
.
Relying on the above result we then prove that the Mordell Weil group of a del Pezzo elliptic fibration
\[\pi:X\to\mathbb{P}^{n-1}\]
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
\[X\]
is a finitely generated
\[\mathbb{C}\]
-algebra [2, 3].
This is a joint work with J. Hausen, A. Laface and A. L. Tironi.
References
[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.