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|Title: ||The cone conjecture for some rational elliptic threefolds|
|Authors: ||Prendergast-Smith, Artie|
|Issue Date: ||2012|
|Publisher: ||© Springer-Verlag|
|Citation: ||PRENDERGAST-SMITH, A., 2012. The cone conjecture for some rational elliptic threefolds. Mathematische Zeitschrift, 272 (1-2), pp. 589 - 605.|
|Abstract: ||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 . 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 , and Totaro .
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.|
|Description: ||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/s00209-011-0951-2|
|Version: ||Submitted for publication|
|Publisher Link: ||http://dx.doi.org/10.1007/s00209-011-0951-2|
|Appears in Collections:||Published Articles (Maths)|
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