DSpace Collection:
https://dspace.lboro.ac.uk/2134/90
2015-07-31T05:17:16Z
2015-07-31T05:17:16Z
The cone conjecture for some rational elliptic threefolds
Prendergast-Smith, Artie
https://dspace.lboro.ac.uk/2134/18240
2015-07-13T13:02:09Z
2012-01-01T00:00:00Z
Title: The cone conjecture for some rational elliptic threefolds
Authors: Prendergast-Smith, Artie
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 [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.
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
2012-01-01T00:00:00Z
Extremal rational elliptic threefolds
Prendergast-Smith, Artie
https://dspace.lboro.ac.uk/2134/18238
2015-07-14T15:33:20Z
2010-01-01T00:00:00Z
Title: Extremal rational elliptic threefolds
Authors: Prendergast-Smith, Artie
Abstract: An elliptic fibration is a proper morphism f : X → Y of normal projective varieties whose
generic fibre E is a regular curve of genus 1. The Mordell–Weil rank of such a fibration is
defined to be the rank of the finitely generated abelian group Pic0 E of degree-0 line bundles
on E. In particular, f is called extremal if its Mordell–Weil rank is 0.
The simplest nontrivial elliptic fibration is a rational elliptic surface f : X → P1. There
is a complete classification of extremal rational elliptic surfaces, due to Miranda–Persson
in characteristic 0 [14] and W. Lang in positive characteristic [12, 13]. (See also Cossec–
Dolgachev [4, Section 5.6].) The purpose of the present paper is to produce a corresponding
classification of a certain class of extremal rational elliptic threefolds.
Description: This article was published in Michigan Mathematical Journal and is available here with the kind permission of the publisher..
2010-01-01T00:00:00Z
The cone conjecture for abelian varieties
Prendergast-Smith, Artie
https://dspace.lboro.ac.uk/2134/18236
2015-07-14T08:44:44Z
2012-01-01T00:00:00Z
Title: The cone conjecture for abelian varieties
Authors: Prendergast-Smith, Artie
Abstract: The purpose of this paper is to write down a complete proof of the Morrison-Kawamata cone conjecture for abelian varieties. The conjecture predicts, roughly speaking, that for a large class of varieties (including all smooth varieties with numerically trivial canonical bundle) the automorphism group acts on the nef cone with rational polyhedral fundamental domain. (See Section 1 for a precise statement.) The conjecture has been proved in dimension 2 by Sterk-Looijenga, Namikawa, Kawamata, and Totaro [Ste85, Nam85, Kaw97, Tot 10], but in higher dimensions little is known in general. Abelian varieties provide one setting in which the conjecture is tractable, because in this case the nef cone and the automorphism group can both be viewed as living inside a larger object, namely the real endomorphism algebra. In this paper we combine this fact with known results for arithmetic group actions on convex cones to produce a proof of the conjecture for abelian varieties.
Description: This article was published in the Journal of Mathematical Sciences [University of Tokyo] and is available here with the kind permission of the publisher.
2012-01-01T00:00:00Z
Conceptualising a university mathematics teaching practice in an activity theory perspective
Treffert-Thomas, Stephanie
https://dspace.lboro.ac.uk/2134/18115
2015-07-02T08:51:00Z
2015-01-01T00:00:00Z
Title: Conceptualising a university mathematics teaching practice in an activity theory perspective
Authors: Treffert-Thomas, Stephanie
Abstract: In this article I present a theorisation of a university mathematics teaching practice,
based on a research study into the teaching of linear algebra in a first year mathematics
undergraduate course. The research was largely qualitative and consisted of
data collected in interviews with the lecturer and in observations of his lectures.
Using Leontiev’s (1981) activity theory framework I categorised the teaching of linear
algebra on three levels: activity-motive, actions-goals and operations-conditions.
Each level of analysis provided insights into the lecturer’s teaching approach, his
motivation, his intentions and his strategies in relation to his teaching. I developed
a model of the teaching process that relates goals as expressed by the lecturer in
interviews to
the strategies that he designed for his teaching.
Description: This article was published in the journal, Nordic Studies in Mathematics Education (NOMAD) and the definitive version is available at: http://ncm.gu.se/node/7726
2015-01-01T00:00:00Z