Weakly-exceptional singularities in higher dimensions
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Abstract.
We show that infinitely many Gorenstein weakly-exceptional quotient singularities exist in all dimensions, we prove a weak-exceptionality criterion for five-dimensional quotient singularities, and we find a sufficient condition for being weakly-exceptional for six-dimensional quotient singularities. The proof is naturally linked to various classical geometrical constructions related to subvarieties of small degree in projective spaces, in particular Bordiga surfaces and Bordiga threefolds.
Funding source: NSF
Award Identifier / Grant number: DMS-1001427, N.Sh.-4713.2010.1, RFFI 11-01-00336-a, RFFI 11-01-92613-KO-a, RFFI 08-01-00395-a, RFFI 11-01-00185-a
Funding source: AG Laboratory GU-HSE, RF
Award Identifier / Grant number: 11 11.G34.31.0023
The problem of finding a nice geometric criterion for a five- dimensional quotient singularity to be weakly-exceptional originated during the first author participation in the 18th Gökova Conference in Turkey. The first author would like to thank Selman Akbulut for inviting him to this beautiful place. The authors would like to thank Marco Andreatta, Eduardo Ballico, Pietro De Poi, Igor Dolgachev, Stephane Lamy, Jihun Park, Emilia Mezzetti, Yuri Prokhorov and Franchesco Russo for many fruitful discussions. We proved both Theorems 1.16 and 1.17 while participating in the Research in Groups program in the Center of International Research in Mathematics (Trento, Italy). We finished this paper at the Institute for the Physics and Mathematics of the Universe (Tokyo, Japan). We are really grateful to CIRM and IPMU for the beautiful working conditions. Special thanks goes to Sergey Galkin for his warm and encouraging support during our stay at IPMU. The work was also supported by the grants NSF DMS-1001427, N.Sh.-4713.2010.1, RFFI 11-01-00336-a, RFFI 11-01-92613-KO-a, RFFI 08-01-00395-a, RFFI 11-01-00185-a, and by AG Laboratory GU-HSE, RF government grant 11 11.G34.31.0023.
© 2014 by Walter de Gruyter Berlin/Boston
Articles in the same Issue
- Frontmatter
- Nonlinear PDE aspects of the tt* equations of Cecotti and Vafa
- On the number of integers in a generalized multiplication table
- Peripheral structures of relatively hyperbolic groups
- What is the total Betti number of a random real hypersurface?
- Parametrization of ideal classes in rings associated to binary forms
- Weakly-exceptional singularities in higher dimensions
- Erratum to A derived approach to geometric McKay correspondence in dimension three (J. reine angew. Math. 636 (2009), 193–236)
Articles in the same Issue
- Frontmatter
- Nonlinear PDE aspects of the tt* equations of Cecotti and Vafa
- On the number of integers in a generalized multiplication table
- Peripheral structures of relatively hyperbolic groups
- What is the total Betti number of a random real hypersurface?
- Parametrization of ideal classes in rings associated to binary forms
- Weakly-exceptional singularities in higher dimensions
- Erratum to A derived approach to geometric McKay correspondence in dimension three (J. reine angew. Math. 636 (2009), 193–236)
