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dc.contributor.advisorGasparim, Elizabeth
dc.contributor.advisorFigueroa-O'Farrill, Jose
dc.contributor.advisorWemyss, Michael
dc.contributor.advisorCheltsov, Ivan
dc.contributor.authorMartinez Garcia, Jesus
dc.contributor.authorGarcia, Jesus Martinez
dc.date.accessioned2013-11-08T15:41:09Z
dc.date.available2013-11-08T15:41:09Z
dc.date.issued2013-11-28
dc.identifier.urihttp://hdl.handle.net/1842/8090
dc.description.abstractThe global log canonical threshold, algebraic counterpart to Tian's alpha-invariant, plays an important role when studying the geometry of Fano varieties. In particular, Tian showed that Fano manifolds with big alpha-invariant can be equipped with a Kahler-Einstein metric. In recent years Donaldson drafted a programme to precisely determine when a smooth Fano variety X admits a Kahler-Einstein metric. It was conjectured that the existence of such a metric is equivalent to X being K-stable, an algebraic-geometric property. A crucial step in Donaldson's programme consists on finding a Kahler-Einstein metric with edge singularities of small angle along a smooth anticanonical boundary. Jeffres, Mazzeo and Rubinstein showed that a dynamic version of the alpha-invariant could be used to find such metrics. The global log canonical threshold measures how anticanonical pairs fail to be log canonical. In this thesis we compute the global log canonical threshold of del Pezzo surfaces in various settings. First we extend Cheltsov's computation of the global log canonical threshold of complex del Pezzo surfaces to non-singular del Pezzo surfaces over a ground field which is algebraically closed and has arbitrary characteristic. Then we study which anticanonical pairs fail to be log canonical. In particular, we give a very explicit classifiation of very singular anticanonical pairs for del Pezzo surfaces of degree smaller or equal than 3. We conjecture under which circumstances such a classifcation is plausible for an arbitrary Fano variety and derive several consequences. As an application, we compute the dynamic alpha-invariant on smooth del Pezzo surfaces of small degree, where the boundary is any smooth elliptic curve C. Our main result is a computation of the dynamic alpha-invariant on all smooth del Pezzo surfaces with boundary any smooth elliptic curve C. The values of the alpha-invariant depend on the choice of C. We apply our computation to find Kahler-Einstein metrics with edge singularities of angle β along C.en_US
dc.contributor.sponsorFundación Caja Madriden_US
dc.contributor.sponsorEngineering and Physical Sciences Research Council (EPSRC)en_US
dc.language.isoenen_US
dc.publisherThe University of Edinburghen_US
dc.relation.hasversionJ. Martinez-Garcia, Log canonical thresholds of Del Pezzo Surfaces in characteristic p, ArXiv e-prints (March 2012), 1203.0995.en_US
dc.subjectKahler-Einstein metricsen_US
dc.subjectdel Pezzo surfacesen_US
dc.subjectK-stablilityen_US
dc.titleDynamic alpha-invariants of del Pezzo surfaces with boundaryen_US
dc.typeThesis or Dissertationen_US
dc.type.qualificationlevelDoctoralen_US
dc.type.qualificationnamePhD Doctor of Philosophyen_US


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