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Del Pezzo fibrations and rank 3 Cox rings

Ahmadinezhad, Hamid (2011) Del Pezzo fibrations and rank 3 Cox rings. Doctor of Philosophy (PhD) thesis, University of Kent. (doi:10.22024/UniKent/01.02.94155) (KAR id:94155)

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One possible output of the minimal model program is a Mori fibre space. These varieties in 3 dimensions are Fano varieties, del Pezzo fibrations and conic bundles. Uniqueness of this output, the so-called rigidity of a Mori fibre space, is a question which arises naturally. In many cases, it has been proven for a general Fano 3-fold to be rigid. Del Pezzo fibrations over the rational curve have been studied in higher degrees and consequently it is known that if deg > 3 then the del Pezzo fibration is nonrigid.

The goal of this thesis is to study rigidity and nonrigidity of low degree del Pezzo fibrations. We give a construction of these objects and classify the nonrigid ones whose link to the other model is obtained by the ambient space. This, in particular, provides many examples of nonrigid degree 2 del Pezzo fibrations which are not necessarily smooth. It is known that the study of rigidity for degree 3 del Pezzo fibrations is subject to consideration of Corti-KoMr stability condition. A first attempt to generalise this stability notion for lower degree fibrations is given in this thesis. The relation between these stability conditions and Sarkisov program is also studied in an explicit way. This requires techniques of working with rank 3 Cox rings which we develop. In particular, the notion of well-formedness for Cox rings is introduced as a generalisation of well- formedness of weighted projective spaces. We also construct families of cubic surface fibration in dimension 4 and study their nonrigidity in a similar way.

Item Type: Thesis (Doctor of Philosophy (PhD))
Thesis advisor: Brown, Gavin D.
DOI/Identification number: 10.22024/UniKent/01.02.94155
Additional information: This thesis has been digitised by EThOS, the British Library digitisation service, for purposes of preservation and dissemination. It was uploaded to KAR on 25 April 2022 in order to hold its content and record within University of Kent systems. It is available Open Access using a Creative Commons Attribution, Non-commercial, No Derivatives ( licence so that the thesis and its author, can benefit from opportunities for increased readership and citation. This was done in line with University of Kent policies ( If you feel that your rights are compromised by open access to this thesis, or if you would like more information about its availability, please contact us at and we will seriously consider your claim under the terms of our Take-Down Policy (
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
SWORD Depositor: SWORD Copy
Depositing User: SWORD Copy
Date Deposited: 02 Sep 2022 14:56 UTC
Last Modified: 02 Sep 2022 14:56 UTC
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