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A special Cayley octad

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journal contribution
posted on 09.01.2017, 14:11 by Artie PrendergastArtie Prendergast
A Cayley octad is a set of 8 points in P3 which are the base locus of a net of quadrics. Blowing up the points of the octad gives a morphism to P2 defined by the net; the fibres of this morphism are intersections of two quadrics in the net, hence curves of genus 1. The generic fibre therefore has a group structure, and the action of this group on itself extends to a birational action on the whole variety. In particular, if the generic fibre has a large group of rational points, the birational automorphism group, and hence the birational geometry, of the variety must be complicated. It is natural to ask whether the converse is true: if the generic fibre has only a small group of rational points, is the birational geometry of the variety correspondingly simple? In this paper we study a special Cayley octad with the property that the generic fibre has finitely many rational points. In Section 1 we find that such an octad only exists in characteristic 2, and is unique up to projective transformations. Our main results then show that the simplicity of the generic fibre is indeed reected in the simplicity of the birational geometry of our blowup. In Section 2 we show that the cones of nef and movable divisors are rational polyhedral, as predicted by the Morrison{Kawamata conjecture. Finally, in Section 3 we prove that our blowup has the \best possible" birational geometric properties: it is a Mori dream space.

Funding

Partially supported by EPSRC First Grant EP/L026570/1.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Transactions of the American Mathematical Society

Volume

370

Issue

8

Pages

5359 - 5379

Citation

PRENDERGAST-SMITH, A., 2018. A special Cayley octad. Transactions of the American Mathematical Society, 370, pp.5359-5379.

Publisher

American Mathematical Society (© The Author)

Version

AM (Accepted Manuscript)

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/

Acceptance date

13/12/2016

Publication date

2017-12-20

Notes

First published in Transactions of the American Mathematical Society, 370, pp.5359-5379, published by the American Mathematical Society. © 2017 by A. Prendergast-Smith

ISSN

0002-9947

eISSN

1088-6850

Language

en

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