Del Pezzo orbifold surfaces and toric degenerations

Lead Research Organisation: University of Warwick
Department Name: Mathematics


Edwin Kutas has studied the possible toric degenerations of del Pezzo surface's with orbifold points. He has, with Daniel Cavey (Nottingham PhD student), produced an algorithm to categorise all toric degenerations of del Pezzo surfaces with given singularity content. In addition to this, calculations have been done to prove that various classes of singularity contents cannot occur as a toric degenerations. This problem fits in closely with the EPSRC program grant 3C in G, and leads to natural follow-up questions and further classifications relating to toric degenerations.


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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509796/1 01/10/2016 30/09/2021
1789814 Studentship EP/N509796/1 03/10/2016 30/09/2020 Edwin Kutas
Description A log del Pezzo is a surface with particular bounds on how much it curves. In the case where this surface is smooth, a classical cassification of these surfaces exist. In the case where the surface is not smooth, no known algorithm currently can classify them. It is natural within the field to try and classify these surfaces up to certain. We have constructed algorithms which

a) Classify the toric degenerations of log del Pezzos with given invariants.
b) Classify the complexity one log del Pezzos, these are non general log del Pezzos., however they are still sufficiently general to be of interest.
c) I have an algorithm which 'hunts' for a toric degeneration or a complexity one degeneration of a log del Pezzo defined by purely combinatorial methods.
Exploitation Route There is much interest in the Mirror symmetry of Fanos (these are like log del Pezzos, but in higher dimensions). There are many ways of finding Mirrors, probably the most commonly used one is finding a Toric degeneration. This fits very nicely part algorithm a. Proving that there is a toric degeneration in general is non trivial, and the only way that can currently do this is, to the best of my knowledge algorithm c. There are well known log del Pezzos which do not admit Toric degenerations. There is only currently only one family of these log del Pezzos for which a mirror has been constructed. This construction was, somewhat ad hoc, and the hope is that by finding complexity one degenerations a more general algorithm may be found.
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