Rationality
Monday July 27th-Friday July 31st, 2020
Update: The videos of the conference can be found here.
Organisers:
Paolo Cascini, Ivan Cheltsov, James McKernan and Chenyang Xu
Speakers:
Dan Abramovich
Caucher Birkar
Jérémy Blanc
János Kollár
Robert Lazarsfeld
Emanuele Macrì
Alexander Perry
Giulia Saccà
Evgeny Shinder
Vyacheslav Shokurov
Burt Totaro
Yuri Tschinkel
Claire Voisin
Please contact one of the organisers if you need the link and password to connect with Zoom.
Schedule
In California time (add 3 hours for the East Coast, 8 hours for UK and 9 hours for Continental Europe)
27 July 2020
9:00-9:50 Yuri Tschinkel (Slides)
10:00-10:15 Coffee Break
10:15-11:05 Dan Abramovich (Slides)
11:15-12:05 János Kollár (Slides)
28 July 2020
9:00-9:50 Giulia Saccà
10:00-10:15 Coffee Break
10:15-10:30 Charlotte Ure (Slides)
10:30-10:45 Jihao Liu (Slides)
10:45-11:00 Andrea Petracci (Slides)
11:15-12:05 Burt Totaro
29 July 2020
9:00-9:50 Caucher Birkar
10:00-10:15 Coffee Break
10:15-11:05, Robert Lazarsfeld (Slides)
11:15-12:05 Vyacheslav Shokurov
30 July 2020
9:00-9:50 Jérémy Blanc (Slides)
10:00-10:15 Elden Elmanto
10:15-10:30 Cesar Hilario
10:30-10:45 Kai Huang
10:45-11:00 Yujie Luo (Slides)
11:15-12:05 Evgeny Shinder (Slides)
31 July 2020
9:00-9:50 Alexander Perry
10:00-10:15 Coffee Break
10:15-11:05 Emanuele Macrì
11:15-12:05 Claire Voisin
Titles and Abstracts
Dan Abramovich - Resolution and logarithmic resolution using weighted blowings up
Abstract: I report on work with Temkin and Wlodarczyk and work of Quek. Resolving singularities in families requires logarithmic geometry. Trying to do this canonically forces us to use stack-theoretic modifications. Surprisingly, stack-theoretic modifications provide an efficient iterative resolution method in which the worst singularities are blown up without regard to the history. To make exceptional divisors cooperate we need logarithmic geometry again.
Caucher Birkar - On non-rationality of divisors on 3-folds Fano fibrations
Abstract: It is well-known that a del Pezzo surface which is rational can degenerate into a non-rational surface. The natural question is how far the degeneration can be from being rational. We investigate this question in a much more general setting. Given a 3-fold Fano fibration, that is, a 3-fold X which is relatively Fano over a base Z, and given a prime divisor D on X contracted to a point on Z, we obtain bounds on the non-rationality of D under mild conditions. This is joint work with Konstantin Loginov.
Jérémy Blanc - Quotients of groups of birational transformations
Abstract: The non-simplicity of the Cremona group of dimension 2 was left open for a long time and proven in 2013 by Cantat and Lamy. In dimension higher, one can not only prove that the group is not simple but also that it is not perfect and find quotient isomorphic to Z/2Z or even to free products of Z/2Z. I will explain how one can prove this using the Sarkisov program and relations among Sarkisov links, and in particular for conic bundles (joint work with Lamy and Zimmermann). I will also explain some further applications to del Pezzo fibrations (joint work with Yasinsky).
Elden Elmanto - An application of rationality to (higher) algebraic K-theory.
Abstract: One of the first major achievement of Morel-Voevodsky's stable motivic homotopy theory was a clean construction of the motivic Atiyah-Hirzebruch spectral sequence converging from motivic cohomology to algebraic K-theory. This construction was broken into a package of conjectures by Voevodsky which were all later settled by Levine. In this talk, we give a short, alternative approach to Voevodsky's conjectures using the birational geometry of Hilbert schemes and the theory of framed motives. This is joint work with Tom Bachmann based on 1912.01595.
Cesar Hilario - Counterexamples to Bertini's theorem in characteristic p>0
Abstract: Bertini's theorem on moving singularities is a classical result in algebraic geometry. It states that in characteristic zero almost all the fibers of a dominant morphism between two smooth algebraic varieties are smooth; in other words, there do not exist fibrations by singular varieties with smooth total space. Unfortunately, Bertini's theorem fails in positive characteristic, as was first observed by Zariski in the 1940s. Investigating this failure naturally leads to the classification of its exceptions. By a theorem of Tate, a fibration by singular curves of arithmetic genus g in characteristic p > 0 may exist only if p <= 2g + 1. When g = 1 and g = 2, these fibrations have been studied by Queen, Borges Neto, Stohr and Simarra Canate. A birational classification of the case g = 3 was started by Stohr (p = 7; 5), and then continued by Salomao (p = 3). In this talk I intend to report on some progress in the case g = 3, p = 2. In fact, several examples show already that, under these hypotheses, very interesting geometric phenomena arise.
Kai Huang - A valuative criterion for K-semistability of log Fano cone singularities
Abstract: We define the delta invariant for log Fano cone singularities and show the corresponding valuative criterion for K-semistability. This is the generalization of the delta invariant and its valuative criterion for the log Fano case. We also show that there exists a quasi-monomial valuation computing the delta invariant when delta is not bigger than 1. This is also a generalization of the result in the log Fano case.
János Kollár - Relative MMP without Q-factoriality
Robert Lazarsfeld - Measures of irrationality for algebraic varieties
Abstract: I will survey a circle of ideas around the question of measuring and controlling “how irrational” are various sorts of varieties whose non-rationality is known.
Jihao Liu - Complements and ACC for minimal log discrepancies
Abstract: The theory of complements was introduced by Shokurov when he investigated log flips of threefolds, and plays an important role in many areas in birational geometry, e.g. boundedness of Fano varieties, log Calabi-Yau fibrations, K-stability theory, etc. In a recent work, we prove a complements conjecture of Shokurov, that is, the existence of n-complements for pairs with any DCC coefficients. We apply this result to the study of local singularities, and in particular, we show the ACC for minimal log discrepancies for exceptional singularities, which is the first result on the ACC conjecture for minimal log discrepancies for non-toric varieties for any DCC set and in arbitrary dimension. This is a joint work with J. Han and V.V. Shokurov.
Yujie Luo - On boundedness of divisors computing minimal log discrepancies for surfaces
Abstract: In this talk, we will briefly introduce our recent result: for any lc surface singularity $(X\ni x, B)$ whose boundary coefficients belong to a fixed DCC set, we can find an exceptional divisor E that computes its minimal log discrepancy, such that E is uniformly bounded, in the sense that a(E,X,0) <= N for some positive integer N depending only on the DCC set. This is joint work with Jingjun Han.
Emanuele Macrì - Special surfaces on special cubic fourfolds
Abstract: I will report on joint work in progress with Arend Bayer, Aaron Bertram, and Alex Perry on a possible characterization of Hassett divisors on the moduli space of cubic fourfolds by the property of containing special surfaces. We will discuss a construction of such special surfaces for infinitely many divisors and the relation with the work of Russo and Staglianò on rationality of such cubics in low discriminant.
Alexander Perry - The integral Hodge conjecture for two-dimensional Calabi-Yau categories
Abstract: I will formulate a version of the integral Hodge conjecture for categories, discuss its proof for categories which are suitably deformation equivalent to the derived category of a K3 or abelian surface, and explain how this implies cases of the usual integral Hodge conjecture for varieties of interest from the point of view of rationality.
Andrea Petracci - An obstructed K-polystable Fano 3-fold
Abstract: I will present an example of an obstructed K-polystable Gorenstein singular toric Fano 3-fold, which deforms to 3 different smooth Fano 3-folds. This example gives a non-smooth point both in the K-moduli stack and in the K-moduli space of Fano 3-folds of degree 12. This is joint work with Anne-Sophie Kaloghiros.
Giulia Saccà - Birational geometry of the intermediate Jacobian fibration
Abstract: A few years ago with Laza and Voisin we constructed a hyperkähler compactification of the intermediate Jacobian fibration associated to a general cubic fourfold. In this talk I will first show how such a compactification J(X) exists for any smooth cubic fourfold X and then discuss some aspects of the birational geometry of the fibration J(X) which are governed by any extra algebraic cohomology classes on J(X).
Evgeny Shinder - Birationality centers and Cremona groups
Abstract: I will introduce a framework to account for the ambiguity of stable birational types of a sequence of centers for birational transformations. I will explain in which settings the introduced invariants are nonvanishing, and give applications to the structure of Cremona groups over various fields. This is joint work in progress with Hsueh-Yung Lin.
Vyacheslav Shokurov - (Upper) moduli part of adjunction
Abstract: Generalities on muduli part of adjunction. Examples. Semiampleness conjecture and coarse moduli.
Burt Totaro - K3 surfaces and high-dimensional Fano varieties
Abstract: A calculation on a K3 surface determines whether the surface is contained in a Fano 3-fold, or in a higher-dimensional Fano variety. So one can experiment by computer to prove the existence of a high-dimensional Fano variety, and then try to find the variety by hand.
Yuri Tschinkel - Equivariant birational types
Abstract: I will discuss new invariants in equivariant birational geometry and some of their applications (joint work with A. Kresch).
Charlotte Ure - Brauer group of Elliptic Curves
Abstract: The Brauer group of a variety can detect both algebraic and arithmetic properties of the underlying object. In particular, the Brauer-Manin obstruction that lies in the Brauer group can obstruct the existence of rational points. In this talk, I will discuss an algorithm to compute the prime torsion of the Brauer group of an elliptic curve explicitly over various ground fields. This algorithm gives generators and relations of the torsion subgroup as tensor products of symbol algebras over the function field of the elliptic curve.
Claire Voisin - Decomposition of the diagonal
Abstract: I will discuss the notion of decomposition of the diagonal and some consequences. Many questions remain open on which obstructions to the existence of such a decomposition (hence to stable rationality) can appear on rationally connected varieties.