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London-Tokyo Workshop In Birational Geometry


Location: Imperial College - Huxley Building - Room 140

Date: Tuesday, 29 May 2018 - Friday, 1 June 2018

Organisers: Paolo Cascini, Yoshinori Gongyo and Shunsuke Takagi


  • Fabio Bernasconi (London)
  • Sho Ejiri (Osaka)
  • Katsuhisa Furukawa (Tokyo)
  • Takeru Fukuoka (Tokyo)
  • Kenta Hashizume (Kyoto)
  • Zhengyu Hu (Taipei)
  • Akihiro Kanemitsu (Kyoto)
  • Yohsuke Matsuzawa (Tokyo)
  • Mirko Mauri (London)
  • Enrica Mazzon (London)
  • Andrei Negut (Boston)
  • Kenta Sato (Tokyo)
  • Nick Shepherd-Barron (London)
  • Calum Spicer (London)
  • Hiromu Tanaka (Tokyo)
  • Jakub Witaszek (London)


Tuesday, 29 May

10:00 - 10:50: Akihiro Kanemitsu

11:00 - 11:50: Takeru Fukuoka

1:30 - 2:20: Calum Spicer

3:00 - 3:50: Nick Shepherd-Barron

Wednesday, 30 May

1:30 - 2:20: Katsuhisa Furukawa

3:00 - 3:50: Jakub Witaszek

4:00 - 4:50: Kenta Sato

Thursday, 31 May

10:00 - 10:50: Mirko Mauri

11:00 - 11:50: Enrica Mazzon

1:00 - 1:50: Andrei Negut

2:00 - 2:50: Zhengyu Hu

3:30 - 4:20: Kenta Hashizume

4:30 - 5:20: Yohsuke Matsuzawa

Friday, 1 June

10:00 - 10:50: Sho Ejiri

11:00 - 11:50: Fabio Bernasconi

1:30 - 2:20: Hiromu Tanaka

Titles and Abstracts:

Fabio Bernasconi (London)

Title: Pathologies for Fano varieties and singularities in low characteristic.

Abstract: In this talk I will review the cone construction for Fano-type varieties and I will explain how this allows to construct pathological examples of singularities in positive characteristic. As an application, I will construct log del Pezzo surfaces in characteristic two and three violating Kodaira vanishing, thus deducing the existence of klt not Cohen-Macaulay threefold singularities and I will use some recent examples of Totaro to construct non-normal plt centres.

Sho Ejiri (Osaka)

Title: Nef anti-canonical divisors and rationally connected fibrations.

Abstract:  In this talk, we will first consider an algebraic fiber space whose relative anti-canonical divisor is nef. We will then prove that the Iitaka-Kodaira dimension of the relative anti-canonical divisor is at most the dimension of a general fiber. Next, we will use this result to study the maximal rationally connected fibration of a projective manifold X whose anti-canonical divisor is nef. This will tell us that X has a dense open subset covered by rationally connected varieties of dimension at least k, where k is the Iitaka-Kodaira dimension of the anti-canonical divisor on X. This is joint work with Yoshinori Gongyo. 

Takeru Fukuoka (Tokyo)

Title: Relative linear extensions of sextic del Pezzo fibrations and its applications.

Abstract:  It is well-known that every del Pezzo surface can be embedded into a certain Fano variety as a (weighted) complete intersection. When we consider del Pezzo fibrations, which are Mori fiber spaces with smooth 3-dimensional total spaces over curves by the definition, it is natural to ask how to relativize such embeddings for del Pezzo fibrations. A main result of this talk will shows that there is a relativization of such embeddings for sextic del Pezzo fibrations. We will also discuss about some applications of this result. 

Katsuhisa Furukawa (Tokyo)

Title: Cubic hypersurfaces with positive dual defects.

Abstract:  The dual defect of a projective variety $X$ in $P^N$ is the difference between $N-1$ and the dimension of the dual variety $X^*$ of $X$. Zak classified cubic hypersurfaces $X$ with positive dual defects in the case when $X^*$ is smooth; such $X$ is given as the secant of a Severi variety or an intersection of it and some hyperplanes. Recently, Hwang gave a characterization of secants of Severi varieties in terms of cubic hypersurfaces with nonzero Hessians and nonzero prolongations, without the assumption that $X^*$ is smooth. I will talk about my progressing work for this topic. 

Kenta Hashizume (Kyoto)

Title: A class of singularity of arbitrary pairs and log canonicalizations.

Abstract:  In the birational geometry, the class of lc pairs is a very important framework to prove a lot of theorems. In general, when we deal with pairs, we assume that the log canonical divisor is $\mathbb{R}$-Cartier. Using this property, we define discrepancy and lc singularity. In this talk, I extend the notion of discrepancy to arbitrary pairs of a normal variety and an effective $\mathbb{R}$-divisor on it, and define pseudo-lc singularity by using it. By giving examples of pseudo-lc pairs which are not lc, I show that pseudo-lc singularity is a strictly extended notion of lc singularity. Also, I explain that any pseudo-lc pair admits a small log canonicalization. 

Zhengyu Hu (Taipei)

Title: Complements on log Fano varieties.

Abstract: The theory of complements was introduced by Shokurov and developed by Prokhorov, Shokurov and more recently by Birkar with an application in the proof of BAB conjecture. In this talk I will discuss the theory of complements on log canonical Fano varieties. Let (X, B) be a set of log canonical Fano pairs of dimension ≤ 3 with the coefficients of the boundary divisors B belonging to a hyperstandard set. Then we prove the boundedness of their Q-complements. This is a joint work with Caucher Birkar and Yanning Xu.

Akihiro Kanemitsu (Kyoto)

Title: Classification of Mukai pairs with dimension 4 and rank 2.

Abstract:  A Mukai pair (X,E) of dimension n and rank r consists of a Fano n-fold X and an ample r-bundle E such that c_1(X) = c_1(E). In this talk, we give the complete classification of Mukai pairs of dimension 4 and rank 2 whose base manifold X has Picard number one. As a consequence, the classification of ruled Fano 5-folds with index two are given, which was partially done by C. Novelli and G. Occhetta in 2007.

Yohsuke Matsuzawa (Tokyo)

Title: The function field analogue of arithmetic degrees.

Abstract: The notion of arithmetic degree which is defined by using Weil height function and mainly studied over number fields is also defined over function fields. Height of a point is a certain intersection number on a model, so we can study the arithmetic degrees in purely geometric way. I will discuss several results on arithmetic degrees of self-rational maps over 1-dimensional function fields. This is joint work with Kaoru Sano and Takahiro Shibata.

Mirko Mauri (London)

Title: Dual complexes of log Calabi-Yau pairs and Mori fibre spaces.

Abstract: Dual complexes are CW-complexes, encoding the combinatorial data of how the irreducible components of a simple normal crossing pair intersect. They have been finding useful applications for instance in the study of degenerations of projective varieties, mirror symmetry and nonabelian Hodge theory. In particular, Kollár and Xu have asked  whether the dual complex of a log Calabi-Yau pair is always a sphere or a finite quotient of a sphere. It is natural to check first if this holds on the end products of minimal model programs. In this talk, we will give a positive answer for Mori fibre spaces of Picard rank two.

Enrica Mazzon (London)

Title: Berkovich approach to degenerations of hyper-Kähler varieties

Abstract: To a degeneration of varieties, we can associate the dual intersection complex, a topological space that encodes the combinatoric of the central fiber and reflects the geometry of the generic fiber. In this talk I will show how the techniques of Berkovich geometry give an insight in the study of the dual complexes. In this way, we are able to determine the homeomorphism type of the dual complex of some degenerations of hyper-Kähler varieties. The results are in accordance with the predictions of mirror symmetry, and the recent work about the rational homology of dual complexes of degenerations of hyper-Kähler varieties, due to Kollár, Laza, Saccà and Voisin. This is joint work with Morgan Brown.

Andrei Negut (Boston)

Title: Certain geometric Hall algebras and their categorification

Abstract:  A classical result of Nakajima and Grojnowski says that the infinite-dimensional Heisenberg algebra acts on the cohomology groups of Hilbert schemes on smooth projective surfaces. In this talk I'll generalize this result: from the Heisenberg algebra to the elliptic Hall algebra, from cohomology to derived categories and from Hilbert schemes to smooth moduli spaces of stable sheaves. Doing these generalizations requires completely new technology, and I'll point out some of the interesting geometric objects that arise.

Kenta Sato (Tokyo) 

Title: On the Ascending chain condition for F-pure thresholds.

Abstract:  For a germ of a variety in positive characteristic and a non-zero ideal sheaf on the variety, we can define the F-pure threshold of the ideal by using Frobenius morphisms, which measures the singularities of the pair. In this talk, I will show that the set of all F-pure thresholds with  fixed embedding dimension satisfies the ascending chain condition. This is a positive characteristic analogue of the "ascending chain  condition for log canonical thresholds" in characteristic 0, which was recently proved by Hacon, McKernan, and Xu.

Nick Shepherd-Barron (London)

Title: del Pezzo surfaces and effective Torelli in genus three.

Abstract: The tropes and singularities of a Kummer surface determine a curve of genus two in terms of elementary projective geometry. In this talk we describe a similar picture in genus three. This is joint work with M. Fryers.

Calum Spicer (London)

Title: Threefold foliations.

Abstract: In recent years there has been a growing body of work on the birational geometry of foliations, and in the construction of minimal models of foliations in particular. Nevertheless, the resulting theory deviates from the classical picture in several important and interesting respects, notably in the failure of abundance. We will discuss some recent developments on these questions in the threefold case. Joint with Paolo Cascini.

Hiromu Tanaka (Tokyo)

Title: On log Fano varieties in positive characteristic.

Abstract:  The notion of log Fano varieties is a generalisation of smooth Fano varieties, which is known as one of important classes in minimal model program. However, log Fano varieties behave pathologically in characteristic p. In this talk, we first summarise fundamental properties of log Fano varieties in characteristic zero. Then we will consider which properties should still hold true in characteristic p. 

Jakub Witaszek (London)

Title: Log non-vanishing conjecture for threefolds in positive characteristic

Abstract: One of very important tools used in the characteristic zero birational geometry is the canonical bundle formula describing the behaviour of canonical divisors under log Calabi-Yau fibrations. Unfortunately, in general the formula is false in positive characteristic, even in a case of a smooth fibration from a surface to a curve. The goal of this talk is to embody an idea that the canonical bundle formula can be partially recovered, if we replace the base of a log Calabi-Yau fibration by a purely inseparable cover. As a corollary, we will show that the log non-vanishing conjecture holds for threefolds in characteristic p>5.



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