2015 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 | 因子収縮写像 / unprojectioon |
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| Outline of Annual Research Achievements |
Thomas Ducat, a JSPS postdoctoral fellow, has been studying terminal divisorial extractions that extract a divisor from a singular curve centred in a smooth threefold. The method works by writing down explicit equations for the curve, blowing up the curve and then using type I Gorenstein unprojection to contract any divisors that appear in the central fibre.
He is currently writing up a description of all the cases of his construction that use at most one unprojection after the initial blow up and therefore appear in low codimension (codimension 3). The problem divides into lots of repetitive cases and he has found the analysis of the singularities appearing on the source variety to be quite tricky. He thinks it will be hard to give a very clear necessary and sufficient condition on the curve for the divisorial extraction to have only terminal singularities, as there are lots of degenerate cases to consider.
He has also been considering the case of divisorial extractions for which the general elephant containing the curve has a Type A Du Val singularity. They come in large infinite families and are clearly related to (generalised) rank 2 cluster algebras. In a recent paper Hacking, Tevelev and Urzua studied some special types of type A flipping contractions and showed how to parameterise families of flips by constructing a universal family of flipping contractions over an affine toric surface (which is only of locally finite type). He has been trying to understand their construction and how he can generalise their approach to apply it to his problem.
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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 although some problems, such as analysing the singularities on the varieties he has constructed, have been harder than he anticipated.
Part of the reason for his progress have been the excellent facilities and research environment which he has benefited from at RIMS, his host institution. The secretaries at the International Office at RIMS have also been incredibly helpful in making arrangements for him and helping him in practical day-to-day issues, so that he is more free to concentrate on his research.
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| Strategy for Future Research Activity |
Ducat's current plan is to finish writing up his work on divisorial extractions in low codimension, aiming to have a finished preprint.
After that, his primary focus will be on generalising this work of Hacking, Tevelev and Urzua to his setting. It is clear that a completely explicit approach (constructing all the generators and relations of the rings in question) will be too difficult in general given the number of families of examples that exist in arbitrarily large codimension. However he hopes that this way of parameterising them will simplify the problem considerably, without the need for such explicit computations. Hopefully it will also generalise to the case of a singular curve contained in a singular 3-fold.
He would also like to think of any applications that his constructions (at least the ones in small codimension) may have. The initial calculation, by his PhD supervisor Miles Reid, that started this research project appeared in a construction by Prokhorov and Reid of a Q-Fano 3-fold of index 2.
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