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. |