2016 Fiscal Year Annual Research Report
| Project/Area Number |
15F15771
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| Research Institution | Kyoto University |
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Principal Investigator |
川北 真之 京都大学, 数理解析研究所, 准教授 (10378961)
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| Co-Investigator(Kenkyū-buntansha) |
DUCAT THOMAS 京都大学, 数理解析研究所, 外国人特別研究員
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| Project Period (FY) |
2015-11-09 – 2018-03-31
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| Keywords | Mori theory / divisorial contractions / unprojection / cluster algebras |
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| Outline of Annual Research Achievements |
Thomas Ducat has been continuing to work on terminal 3-fold divisorial extractions which extract a divisor from a singular curve. He has now managed to produce a large number of examples (several hundred) in the case that the general elephant has a type A Du Val singularity, which is the main case of the problem. He has also been able to extend his unprojection method to cover the more general case of divisorial extractions from curves contained in singular threefolds.
He has recently uploaded two preprints on the arXiv, both of which have been submitted to journals for refereeing. The first preprint is about the classification of divisorial contractions to singular curves. It covers the case when the divisorial extraction exists in low codimension, i.e. when the relative graded ring construction only requires a small number of generators. Even in this case the analysis is very complicated and it is now clear that a thorough description in this level of generality will probably not be possible in the general case. The second preprint applies these results to give an explicit construction of some Sarkisov links which start with a divisorial extraction from a singular curve. This provides some new constructions for some Q-Fano 3-folds appearing in the Graded Ring database and the first examples of Sarkisov links that include divisorial contractions to non-local complete intersection curves.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
Ducat has been making progress on his research problem. In individual cases his method of constructing divisorial extraction by unprojection has been very successful and practical, although he has found it difficult to make it an effective form of proof in the general case. However the number of examples that he has been able to produce has given him very good insight into the structure of these divisorial extractions so it should be possible now to formulate some reasonable conjectures and find an alternative method of proof.
Also one new avenue of research has arisen from this project. The graded rings appearing in these type A divisorial extractions (and in type A 3-fold flipping contractions) have a cluster algebra structure, and he has been investigating cluster algebras in general. These give rise to a class of Gorenstein rings with a classification according to the Cartan classification of root systems. They have a large amount of symmetry and will have many useful applications as key varieties in constructing algebraic varieties and flipping/divisorial neighbourhoods.
The working environment at RIMS has been excellent and he has been fortunate enough to benefit from a very generous JSPS research-in-aid grant. This has covered travel expenses for him to visit international conferences and a research visit from his colleague Dr. Seung-Jo Jung. He has also been able to use the grant to purchase textbooks which have helped him in his research.
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| Strategy for Future Research Activity |
Ducat currently has an unpublished table containing all of his examples of type A divisorial extractions from singular curves. He intends to finish writing up a proof of the results in this table in a form which can be published.
He intends to carry on working on the applications of such results in the Sarkisov program and classification of Q-Fano 3-folds (and Mori fibre spaces more generally). The results of the second preprint mentioned in section 1 should be easy to generalise and he has been talking to Hamid Ahmadinezhad about this and other applications in studying birational rigidity of 3-fold Mori fibre spaces.
The connection to cluster algebras should also be very fruitful. He has a half-written preprint explaining the rank 2 and 3 cases which he intends to finish writing up and publish. He has been working on applications of these cluster algebra to explicit birational geometry with Stephen Coughlan. For instance they can use them to construct many new examples of Q-Fano 3-folds and hopefully some surface of general type too. (They already appear in a construction by Reid of a Godeaux surface with fundamental group Z/3Z.)
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Research Products
(7 results)
