Constructions and obstructions in birational geometry

5-9 November 2018

ICMS, The Bayes Centre, 47 Potterrow, Edinburgh EH8 9BT

Scientific Organisers

• Arend Bayer, University of Edinburgh
• Ivan Cheltsov, University of Edinburgh
• Liana Heuberger, University of Warwick
• Evgeny Shinder, University of Sheffield

This timely workshop focuses on the fast-developing field of Birational Geometry. Bringing together some of the most well-known algebraic geometers in recent times, the aim of the workshop includes reviewing the most important recent developments in birational geometry, relating to the explicit methods and toric varieties and degenerations on the Constructions side, and Chow groups, derived categories and the Grothendieck ring of varieties on the Obstructions side, and subsequently initiating the discussion between experts. The chosen themes cover a broad spectrum of research currently being carried out in birational geometry, both by UK-based mathematicians and by their international collaborators.  The workshop is dedicated William Edge, born on November 8th, 1904, who pioneered research in Algebraic Geometry in Britain and worked at the University of Edinburgh for most of his life.

This workshop is organised in partnership with The Clay Mathematical Institute and the European Research Council (ERC)

Photographs and Videos

Photographs from the workshop are available via the ICMS Flckr account, the link is here.

The video of the Clay Institute Lecture is available via the Clay Institute video library, the link is here.

Speakers

The organisers are delighted to announce that Claire Voisin, Collège de France, will participate in the conference and we are grateful to the Clay Mathematical Institute for their generous support which has made this possible. Claire Voisin is widely recognised as one of the most remarkable and stimulating mathematicians of our time.

Arrangements

It is anticipated that the workshop will commence with Registration early on the morning of Monday 5 November and finish around lunchtime on Friday 9 November 2018. Please take this into consideration when making your travel arrangements.

Invitations

Invited speakers were sent an email invitation from ICMS in August 2018.

Public Applications

The registration period for the public application has now closed and all applicants will be notified of decisions by Monday 15 October.

UK Visas

If you are travelling from outside of the UK you may require an entry visa. A European visa does not guarantee entry to the UK. Please use this link to the UK Visas site to find out if you need a visa and if so how to apply for one.

Travel

Please note that it is your responsibility to have adequate travel insurance to cover medical and other emergencies that may occur on your trip.

Please see the Visiting ICMS page for details of our location.

A taxi directly from the airport will cost approximately 20.00 to 25.00 GBP to the city centre for a one-way journey.

There is also a bus service direct from the airport to the city centre which will cost 4.50 GBP single or 7.50 GBP return - the Airlink 100. This is a frequent service (every 10 minutes during peak times) and will bring you close to Waverley Railway Station, only a short walk to the workshop venue.

Lothian buses charge £1.70 for a single, £4.00 for a day ticket. Please note that the exact fare is required and no change is given.

If travelling by train, please note that Edinburgh has several railway stations - Waverley Railway Station being the main station and closest to the workshop venue. If you alight at Edinburgh Waverley, the workshop venue is an easy 15 minute walk away

ICMS does NOT use third parties to arrange accommodation.  If you are approached by a third party (e.g. Business Travel Management) asking for booking details, please ignore.  If you have any concerns please contact ICMS.

Accommodation

If you are booking your own accommodation for this workshop, here are a few suggestions below which are close to ICMS and reasonably priced.

• Masson House Pollock Halls of Residence, The University of Edinburgh, 18 Holyrood Park Road, Edinburgh EH16 5AY 0800 028 7118 (UK only) - +44 (0)131 651 2189 - University of Edinburgh accommodation. Pollock Halls is about a 20-25 minute walk
• Ibis Hotel on South Bridge (not Hunter Square) A 2 minute walk from ICMS.
• Motel One Edinburgh (Please note that Motel One has two hotels in the city centre - Motel One Royal or Motel One Princes.  Both are within 5 to 10 minutes walk from ICMS)
• Jurys Inn Edinburgh, 43 Jeffrey Street, Edinburgh EH1 1DH +44 (0)131 200 3300 - A 5 minute walk from ICMS.
• Edinburgh City Hotel, 79 Lauriston Place, Edinburgh, EH3 9HZ  +44 (0)131 622 7979 - Also around a 5 minute walk from ICMS
• St Christophers Hostel, 9-13 Market Street, Edinburgh EH1 1DE  (approx 25.00 or less depending on offers available)+44 (0)20 7407 1856 - bookings@st-christophers.co.uk This is Youth Hostel accommodation.

There are of course many more hotels, guest houses, hostels and self-catering flats in Edinburgh. The Visit Scotland page is a good place to start looking for them.

Programme

This programme is provisional and may be subject to change.

PDF of abstracts

Monday 5 November 2018

 09:00-09:50 Registration & Tea/Coffee 09:50-10:00 Welcome Talk 10:00-11:00 Susanna Zimmermann (Université d'Angers) Quotients of Cremona groups 11:00-11:30 Tea/Coffee break 11:30-12:30 Enrica Floris (Université de Poitiers) On the b-Semiampleness conjecture 12:30-14:00 Lunch 14:00-15:00 Hamid Ahmadinezhad (Loughborough University) Models of 3-fold Mori fibre spaces 15:15-16:15 Cinzia Casagrande (Università di Torino) Fano 4-folds with rational fibrations 16:15-16:45 Tea/Coffee break 16:45-17:45 Piotr Pokora (Polish Academy of Sciences,Cracow) On the bounded negativity conjecture 17:45-18:45 Informal wine reception

Tuesday 6 November 2018

 09:30-10:30 Nicolas Addington (University of Oregon) Exoflops 10:30-11:00 Tea/Coffee break 11:00-12:00 Claire Voisin (Collège de France) - Clay Institute Lecture Decomposition of the  diagonal and nilpotence 12:00-14:00 Lunch 14:00-15:00 Alessio Corti (Imperial College London) Volume-preserving birational maps of PP3 15:15-16:15 John Ottem (University of Oslo) The integral Hodge conjecture for curve classes 16:15-16:45 Tea/Coffee break 16:45-17:45 Marta Pieropan (École Polytechnique Fédérale de Lausanne) On rationally connected varieties over C_1 fields of characteristic 0

Wednesday 7 November 2018

 09:15-10:15 Rita Pardini (Universita' di Pisa) Higher dimensional Clifford-Severi equalities 10:15-10:45 Tea/Coffee break 10:45-11:45 Gavin Brown (University of Warwick) Flops and contraction algebras 12:00-13:00 Anne-Sophie Kaloghiros (Brunel University London) Maximal intersection Calabi-Yau pairs 13:00 Free afternoon

Thursday 8 November
# THERE WILL BE A FIRE ALARM TEST AT 11:30 TODAY. PLEASE REMAIN SEATED. THERE IS NO NEED TO EVACUATE.

 09:30-10:30 Jean-Louis Colliot-Thélène (CNRS and Université Paris-Sud Paris-Saclay) A few results on cubic  surfaces and hypersurfaces 10:30-11:00 Tea/Coffee break 11:00-12:00 Ekaterina Amerik (Université de Paris-Sud,Orsay) Contraction loci on hyper-Kaehler manifolds in families 12:00-13:45 Lunch 13:45-14:45 tba 15:00-16:00 Elisa Postinghel (Loughborough University) Mutations of Newton-Okounkov polytopes and classification of Fano manifolds 16:00-16:30 Tea/Coffee break 16:30-17:30 Francesco Zucconi (Universitá degli Studi di Udine) Rationality of special sub-loci in the moduli space of spin curves 19:00 Workshop dinner at Blonde Restaurant, 75 St. Leonard’s Street, Edinburgh

Friday 9 November 2018

 09:30-10:30 Sergey Gorchinskiy (Steklov Mathematical Institute of Russian Academy of Sciences and NRU HSE) Categorical measures for varieties with finite group actions 10:30-11:00 Tea/Coffee break 11:00-12:00 Diane Maclagan (University of Warwick) Tree compactifications of the moduli space of genus zero curves 12:00-13:00 Lev Borisov (Rutgers University) Explicit equations of ball quotients 13:00 Close of workshop

PDF of abstracts

Talks and abstracts

Exoflops
The derived category of a complete intersection is equivalent to the category of matrix factorizations of a certain function on the total space of a vector bundle over the ambient space.  The complete intersection is smooth if and only if the critical locus of the function is compact.  I will present a construction through which a resolution of singularities of the complete intersection corresponds to a compactification of the critical locus of the function, which can be very interesting in examples.  Kuznetsov and Lunts's categorical crepant resolution will make an appearance.  (Some of you heard me talk about this phenomenon in 2015, but now I understand it much better.  This is joint work with Paul Aspinwall and Ed Segal.)

Models of 3-fold Mori fibre spaces
I will talk about a new approach to explicit classification of Mori fibre spaces in dimension three. Special attention is given to improving models of low degree del Pezzo surfaces over curves; a joint work with Maksym Fedorchuk and Igor Krylov.

Ekaterina Amerik
Contraction loci on hyperkaehler manifolds in families
This is a joint work with Misha Verbitsky. Let X be an irreducible holomorphic symplecticmanifold and z a Beauville-Bogomolov negative (1,1)-class. Using the ergodicity of the monodromy action, we prove some deformation-invariance statements for the locus covered by rational curves of class z in X.

Lev Borisov
Explicit equations of ball quotients.
I will talk about recent work on finding explicit equations of quotients of the two-dimensional complex ball, specifically a fake projective plane (joint with J. Keum) and Cartwright-Steger surface (joint with S. Yeung).

Gavin Brown
Flops and contraction algebras
I will describe an on-going project with Michael Wemyss (Glasgow) in which we construct smooth 3-fold flops from the ingredients of Donovan-Wemyss contraction algebras, which are certain finite-dimensional non-commutative algebras related to the flopping geometry, and their Calabi-Yau potentials. Our methods are enough to construct and distinguish certain flops that conventional invariants cannot.

Cinzia Casagrande
Fano 4-folds with rational fibrations
Smooth complex Fano 4-folds are not classified, and we still lack a good understanding of their general properties. In the talk we will focus on Fano 4-folds with large second Betti number b_2, studied via birational geometry and the detailed study of their contractions and rational contractions. We recall that a contraction is a morphism with connected fibers onto a normal projective variety, and a rational contraction is given by a sequence of flips followed by a (regular) contraction. The main result that we present is the following: let X be a Fano 4-fold having a rational contraction X --> Y of fiber type (with dim Y > 0). Then either X is a product of surfaces, or b_2(X) is at most 17, or Y is P^1 or P^2.

Jean-Louis Colliot-Thélène
A few results on cubic  surfaces and hypersurfaces
Two results on the generation of Chow groups of 1-cycles by lines; Chow-zero triviality of partially diagonal cubic hypersurfaces; stable rationality versus rationality for surfaces over a quasi-finite field.

Alessio Corti
Volume-preserving birational maps of PP3
After a short summary of the Sarkisov program for Mori fibred Calabi–Yau pairs, I report on joint work with Araujo and Massarenti on volume preserving birational maps of PP3.

Enrica Floris
On the b-semi-ampleness conjecture
An lc-trivial fibration f:(X,B)- >Y is a fibration such that the log-canonical divisor of the pair (X,B) is trivial along the fibres off. As in the case of the canonical bundle formula for elliptic fibrations, the log canonical divisor can be written as the sum of the pullback of three divisors: the canonical divisor of Y; a divisor, called discriminant, which contains informations on the singular fibres; a divisor, called moduli part that contains informations on the variation in moduli of the fibres. The moduli part is conjectured to be semiample. Ambro proved the conjecture when the base Y is a curve. In this talk we will explain how to prove that the restriction of the moduli part to a hypersurface is semiample assuming the conjecture in lower dimension. This is a joint work with Vladimir Lazić.

Sergey  Gorchinskiy
Categorical measures for varieties with finite group actions
The talk is based on a common work with D. Bergh, M. Larsen, and V. Lunts. Given a variety with a finite group action, we compare categorical measures of the corresponding quotient stack and the extended quotient. Using weak factorization for orbifolds, we show that for a wide range of cases, these two measures coincide. This implies, in particular, a conjecture of Galkin and Shinder on categorical and motivic zeta-functions of varieties. We provide examples showing that, in general, these two measures are not equal.

Anne-Sophie Kaloghiros
Maximal intersection Calabi-Yau pairs
A Calabi-Yau (CY) pair (X, D) consists of a normal projective variety and a reduced anti-canonical integral divisor D on it. Such pairs come up in a variety of contexts; for example, cluster varieties are obtained by glueing CY pairs by crepant birational maps. Recent developments in mirror symmetry suggest that cluster varieties are “natural" mirror partners of Fano varieties with a toric degeneration, and that understanding the birational geometry of CY pairs is an important step in the study of mirror symmetry and moduli of Fano varieties. In this talk, I will discuss the birational geometry of CY pairs and some 3-fold examples.

Alexander Kuznetsov
Semiorthogonal decompositions of singular surfaces
It is well known that any smooth projective toric surface has a full exceptional collection. I will talk about a generalization of this fact for singular surfaces. First, if the class group of Weil divisors of the surface is torsion free (for instance, this holds for all weighted projective planes), I will construct a semiorthogonal decomposition of the derived category with components equivalent to derived categories of modules over certain local finite dimensional algebras. When the class group has torsion, a similar semiorthogonal decomposition will be constructed for an appropriately twisted derived category. Many of these results extend to non-necessarily toric rational surfaces. This is a joint work with Joseph Karmazyn and Evgeny Shinder.

Diane Maclagan
Tree compactifications of the moduli space of genus zero curves
The moduli space \overline{M}_{0,n} of stable genus zero curves with n marked points can be constructed as the closure of M_{0,n} inside a toric variety.  The fan of the toric variety is moduli space of phylogenetic trees.  I will discuss joint work with Dustin Cartwright to construct other compactifications of M_{0,n} by varying the toric variety using variants of phylogenetic trees.  These compactifications include many of the standard alternative compactifications of M_{0,n}.

John Ottem
The integral Hodge conjecture for curve classes
The Hodge conjecture predicts which rational cohomology classes on a smooth complex projective variety can be represented by linear combinations of complex subvarieties. The integral Hodge conjecture, the analogous conjecture for integral homology classes, is known to be false in general (the first counterexamples were given in dimension 7 by Atiyah and Hirzebruch). I'll give a short survey talk on this conjecture, and present some new counterexamples in dimension three. This is joint work with Olivier Benoist.

Rita Pardini
Higher dimensional Clifford-Severi equalities
Let X be a projective variety, let a:X-->A be a morphism to an abelian variety with dim a(X)=dim X. Given a line bundle L on X, we define its continuous rank h^0_a(L) as the generic value of h^0(L+P), where P is the pullback of an element of Pic^0(A). The slope \lambda(L) is the ratio vol(L)/h^0_a(L); the Clifford-Severi inequalities are lower bounds for the slope that generalize at the same time the classical Severi inequality for surfaces of general type and Clifford's theorem for line bundles on a curve.

I will describe a characterization of:

1. the triples (X,a,L) attaining the lower bound \lambda(L)=n! (1st Clifford-Severi line)
2. the triples (X,a,L) with K_X-L pseff attaining the lower bound \lambda(L)=2n! (2nd Clifford-Severi line).

This is joint work with M.A. Barja and L. Stoppino.

Marta Pieropan
On rationally connected varieties over C_1 fields of characteristic 0
In the 1950s Lang studied the properties of C_1 fields, that is, fields over which every hypersurface of degree at most n in a projective space of dimension n has a rational point. Later he conjectured that every smooth proper rationally connected variety over a C_1 field has a rational point. The conjecture is proven for finite fields (Esnault) and function fields of curves over algebraically closed fields (Graber-Harris-de Jong-Starr), but it is still open for the maximal unramified extensions of p-adic fields. I use birational geometry in characteristic 0 to reduce the conjecture to the problem of finding rational points on Fano varieties with terminal singularities.

Piotr Pokora
On the bounded negativity conjecture
The main aim of my talk is to present some recent developments on the bounded negativity conjecture. The conjecture states that for every smooth complex projective surface $X$ there exists an integer $b(X)$ such that for all irreducible and reduced curves $C \subset X$ one has $C^{2} \geq -b(X)$. There are some surfaces for which the BNC holds, for instance surfaces with the canonical divisor $-K_{X}$ being $\mathbb{Q}$-effective, but in general the question is open. It is notoriously difficult to predict where a certain blowing up of a given surface has bounded negativity, for instance it is not known whether the blowing up of the complex projective plane along $10$ very general points has bounded negativity. In my talk I am going to discuss the current state of the art in that field especially focusing on the effective bounds on the self-intersection numbers of certain classes of curve on blow ups of surfaces with Kodaira dimension $\leq 0$.

Elisa Postinghel
Mutations of Newton-Okounkov polytopes and classification of Fano manifolds
Mutations are certain operations of Laurent polynomials, or of their Newton polytopes, that induce deformations between the corresponding toric varieties, as showed by Ilten. Mutations of Fano polytopes are studied by Corti et al. in order to classify Fano manifolds up to deformation using the mirror symmetry approach. On the other hand, to a pair given by a projective algebraic variety and a line bundle, we can associate the so called Okounkov bodies, that are a generalisation of Newton polytopes. In this talk, I will show how to obtain mutations of Okounkov polytopes, and hence toric deformations, of certain Fanos. This is join work (in progress) with A. Laface and S. Urbinati.

Claire Voisin
Decomposition of the  diagonal and nilpotence
The notion  of   decomposition of the diagonal  of an algebraic variety  appears in the work of Bloch and Srinivas, at least with rational coefficients. With integral coefficients, it has recently played an important role in the study of rationality questions. In this talk I will present this notion and discuss  the relationships between the cohomological and Chow decompositions of the diagonal, the former being much better understood.

Susanna Zimmermann
Quotients of Cremona groups
I will explain how the Sarkisov program and relations between Sarkisov links allow us to construct a surjective homomorphism from the Cremona groups in higher dimension to a (non-trivial) abelian group, whose kernel contains an infinite number of normal subgroups of index 2.

Francesco Zucconi
Rationality of special sub-loci in the moduli space of spin curves
We will show how to use the birational geometry of some well-known Fano 3-folds to prove the rationality of the moduli space of spin hyperelliptic curves and of genus 4 spin curves with a unique trigonal series.

PDF of abstracts