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The cone conjecture for some rational elliptic threefolds
A central problem of modern minimal model theory is to describe the various cones of
divisors associated to a projective variety. For Fano varieties the nef cone and movable cone
are rational polyhedral by the cone theorem [4, Theorem 3.7] and the theorem of Birkar–
Cascini–Hacon–McKernan [1]. For more general varieties the picture is much less clear: these
cones need not be rational polyhedral, and can even have uncountably many extremal rays.
The Morrison-Kawamata cone conjecture [8, 3, 13] describes the action of automorphisms
on the cone of nef divisors and the action of pseudo-automorphisms on the cone of movable
divisors, in the case of a Calabi-Yau variety, a Calabi-Yau fibre space, or a Calabi-Yau pair.
Although these cones need not be rational polyhedral, the conjecture predicts that they
should have a rational polyhedral fundamental domain for the action of the appropriate
group. It is not clear where these automorphisms or pseudo-automorphisms should come
from; nevertheless, the conjecture has been proved in various contexts by Sterk–Looijenga–
Namikawa [11, 9] Kawamata [3], and Totaro [14].
In this paper we give some new evidence for the conjecture, by verifying it for some
threefolds which are blowups of P3 in the base locus of a net (that is, a 2-dimensional linear
system) of quadrics.
History
School
- Science
Department
- Mathematical Sciences
Published in
Mathematische ZeitschriftVolume
272Issue
1-2Pages
589 - 605Citation
PRENDERGAST-SMITH, A., 2012. The cone conjecture for some rational elliptic threefolds. Mathematische Zeitschrift, 272 (1-2), pp. 589 - 605.Publisher
© Springer-VerlagVersion
- SMUR (Submitted Manuscript Under Review)
Publisher statement
This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/Publication date
2012Notes
This article was submitted for publication in the journal, Mathematische Zeitschrift [© Springer-Verlag]. The final publication is available at Springer via http://dx.doi.org/10.1007/s00209-011-0951-2ISSN
0025-5874eISSN
1432-1823Publisher version
Language
- en