# The Shokurovs: workshop for birationalists

December 18–22, 2017

IBS Center for Geometry and Physics, Pohang, South Korea

## Organizers

• Ivan Cheltsov (University of Edinburgh)
• Sung Rak Choi (Yonsei University)
• Jihun Park (IBS Center for Geometry and Physics, POSTECH)

## Venue

• POSTECH Mathematical Science Building #404

## Lectures Schedule

December 18 (Mon) December 19 (Tue) December 20 (Wed) December 21 (Thu) December 22 (Fri)
10:00 – 11:00 Birkar Alexeev Prokhorov Hwang Lee
11:00 – 11:30 Break
11:30 – 12:30 Ambro Choi Cheltsov Dubouloz Shokurov
12:30 – 14:30 Lunch
14:30 – 15:30 Chen Cutrone Garcia Won
15:30 – 16:00 Tea Time
16:00 – 17:00 Hu Katzarkov Karzhemanov Krylov

## Title & Abstract

Speaker
Valery Alexeev (University of Georgia)
Title
Abstract
In a joint work with Alan Thompson, we construct compact moduli spaces corresponding to the ADE root lattices.
Speaker
Florin Ambro (Institute of Mathematics of the Romanian Academy)
Title
Canonical rings of NC varieties
Abstract
I will discuss the finite generation, and the existence of minimal models, for projective varieties with at most normal crossings singularities. Based on joint work with J. Kollar.

Speaker
Caucher Birkar (University of Cambridge)
Title
Some problems in birational geometry
Abstract
Singularities play a fundamental role in modern birational geometry as they naturally appear in many contexts. In this talk I will discuss some problems in birational geometry focusing on singularities and their interaction with global geometry.
Speaker
Ivan Cheltsov (University of Edinburgh)
Title
Finite collineation groups and birationally rigidity
Abstract
I will report on my recent work with Shramov on equivariant birational geometry of the projective space.

Speaker
Yifei Chen (Chinese Academy of Sciences)
Title
Some topics on birational geometry in positive characteristic
Abstract
I will introduce my observations of birational geometry in positive characteristic, including subadditivity of Kodaira dimension, canonical bundle formula and semi-positivity based on my joint work with Caucher Birkar, Lei Zhang and Yi Gu.
Speaker
Sung Rak Choi (Yonsei University)
Title
Okounkov bodies of pseudoeffectieve divisors
Abstract
I will survey our recent results on Okounkov bodies obtained through the collaboration with Y.Hyun, J.Park, and J.Won. After introducing the constructions of Okounkov bodies for pseudoeffective divisors, I will explain how various positivity properties of divisors are encoded in the Okounkov bodies.

Speaker
Joseph Cutrone (Goucher College)
Title
A Weak Fano Threefold Arising as a Blowup of a Curve of Genus 5 and Degree 8 on $P^3$
Abstract
This article constructs the smooth weak Fano threefold of Picard number two with small anti- canonical map that arises as a blowup of a smooth curve of genus 5 and degree 8 independent of prior classification results.
Speaker
Title
Flexible open subsets of del Pezzo fibrations
Abstract
By the work of Pukhlikov, "most" three-dimensional fibrations in del Pezzo surfaces of degree lower than or equal to 3 have non rational total spaces. In this talk, I will discuss special del Pezzo fibrations which in contrast contain many affine cylinders over a rational base, some of these even arising as projective completions of the affine 3-space. (Joint work with T. Kishimoto, Saitama Univ.)

Speaker
Jesus Martinez Garcia (University of Bath)
Title
Compactification of the moduli space of certain K3 surfaces
Abstract
We study the moduli space of K3 surfaces obtained as minimal resolutions of double covers of the plane branched at the union of a quartic curve and two distinct lines. We construct and describe such a compactification using Geometric Invariant Theory. Then, we show that our compactification is isomorphic to the Baily-Borel compactification of the arithmetic quotient of the period domain of such K3 surfaces by certain arithmetic group.
Speaker
Zhengyu Hu (National Taiwan University)
Title
Log canonical pairs with boundaries containing ample divisors
Abstract
Let $(X,B)$ be a projective log canonical pair. Assume that $B \geq A$ for some ample divisor $A \geq 0$. We prove that $(X,B)$ has a good minimal model or a Mori fiber space. As an application we prove that a log Fano type variety $X$ with $\mathbb{Q}$-factorial log canonical singularities is a Mori dream space.

Speaker
Jun-Muk Hwang (Korea Institute for Advanced Study)
Title
Recognizing $G/P$ by its varieties of minimal rational tangents
Abstract
Let $G/P$ be a rational homogeneous space of Picard number 1 and let $X$ be a Fano manifold of Picard number 1. The question we are interested in is: if the variety of minimal rational tangents at a general point of $X$ is isomorphic to that of $G/P$, is $X$ biholomorphic to $G/P$? An affirmative answer is given for $G/P$ associated with a long root in the works of Mok and Hong-Hwang in 2008. In these cases, Tanaka theory for parabolic geometry was used to handle the underlying differential geometric structures. The question has been open for the other $G/P$, i.e., symplectic Grassmannians or two cases of $F_4$-homogeneous spaces. The main difficulty in the remaining cases is that the underlying geometric structure is no longer a parabolic geometry and certain degenerate structures may occur. In a collaboration with Qifeng Li, we overcome this difficulty by constructing a Cartan connection associated to the geometric structure under the assumption that certain vector bundles arising from Spencer complexes do not have nonzero sections. Using this construction, we settle the case of symplectic Grassmannians.
Speaker
lya Karzhemanov (Moscow Institute of Physics and Technology)
Title
Some properties of (almost) invertible rational maps
Abstract
I will present, old and new, families of rational self-maps of Pn, including surjective ones and those arising from homaloidal polynomials. I will discuss some, old and new, properties of these maps.

Speaker
Ludmil Katzarkov (University of Miami, University of Vienna, Higher School of Economics)
Title
Geometric Rieman Roch for categories
Abstract
We will introduce a noncommutative version of Gromov Whitten invariants. Applications will be considered.
Speaker
Igor Krylov (University of Bayreuth)
Title
Stability over rings and good models of del Pezzo fibrations
Abstract
This talk is motivated by the following problem, given a three-dimensional Mori fiber space, can we find a birational to it model with nice singularities? Sarkisov proved that for a cubic bundles there exists a smooth model. For del Pezzo fibrations smooth model may not exist in case degree <4. Corti has shown that there are Gorenstein (resp. 2-Gorenstein) models for del Pezzo fibrations of degree 3 (resp. 2). He proved it by constructing explicit birational maps improving singularities. Kollar improved his result in degree 3 using GIT. I discuss what are the issues in adapting Kollar's approach for degrees 1 and 2 and how to work around them.

Speaker
Yongnam Lee (Korea Advanced Institute of Science and Technology)
Title
On function fields of general hypersurface sections of Fano 3-folds
Abstract
I will present on my recent work with Gian Pietro Pirola on dominant rational maps from general hypersurface sections of Fano 3-folds.
Speaker
Yuri Prokhorov (Steklov Mathematical Institute, Moscow State Lomonosov University, Higher School of Economics)
Title
Degenerations of del Pezzo surfaces in Q-Gorenstein families
Abstract
I am going to discuss log canonical surface singularities admitting smoothings in Q-Gorenstein families. As an application I present the classification of log canonical surfaces of Picard rank 1 which are Q-Gorenstein degenerations of smooth del Pezzo.

Speaker
Vyacheslav Shokurov (Johns Hopkins University)
Title
Abstract
I will explain that the complement conjecture follows from some other standart conjectures and already nonconjectures. In particular, the complemt conjecture holds in dimension 2 and in dimension 3 it follows from the acc of mld's near 1.
Speaker
Joonyeong Won (IBS Center for Geometry and Physics)
Title
K-stability for smooth del Pezzo surfaces
Abstract
We introduce some invariants for K-stability of Fano varieties then show how we can estimate the invariants to determine K-stability of smooth del Pezzo surfaces.

## Registration

Online registration is available here until December 1, 2017.

## How to get to POSTECH

Visit here for information about how to get to Pohang.

For the location of the conference site, click here for POSTECH Campus Map.

## Accommodation

We regret to say that we cannot support your travel and local expenses unless the conference promised to pay.

To visit the website of the hotel, please click the name of the hotel.

POSCO Int'l Center Hotel Yeongildae Apple Tree Hotel Philos Hotel Benikea Hotel Pohang Eco Hotel 88,000 KRW 110,000 KRW 55,000 KRW 80,000 KRW 70,000 KRW 90,000 KRW (1 Person)100,000 KRW (2 Persons) 88,000 KRW 132,000 KRW 60,000 KRW 80,000 KRW 80,000 KRW 100,000 KRW120,000 KRW (Deluxe) 13,200 KRW 12,000 KRW free 11,000 KRW free free 10 min. by walk 15 min. by car 15 min. by car 15 min. by car 20 min. by car 20 min. by car +82-54-279-8500 +82-54-280-8900 +82-54-241-1234 +82-54-250-2000 +82-54-282-2700 +82-54-282-8787

* The above rate is as of April, 2017 (VAT included)

** The rate may vary

## Visa

Please visit here and if you need to apply for the visa to attend the conference, please contact geometry@ibs.re.kr. We would request a copy of your passport and up-to-date CV for review.

## Contact

workshop@ibs.re.kr