Wednesday February 23 to Friday February 25 2022

Organisers:

Paolo Cascini, Ivan Cheltsov, James McKernan and Chenyang Xu

Speakers:

1. Dori Bejleri

2. Jérémy Blanc

3. Kristin DeVleming

4. Lena Ji

5. Ludmil Katzarkov

6. Joaquín Moraga

7. Mircea Mustaţă

8. David Stapleton

9. Jakub Witaszek

10. Ziquan Zhuang

11. Susanna Zimmermann

Schedule

Wednesday, February 23

9:30 - 10:20 Mircea Mustaţă

11:00 - 11:50 Ziquan Zhuang

1:00 - 1:50 Kristin DeVleming

2:30 - 3:20 Jérémy Blanc

4:00 - 5:00 Poster Session

Thursday, February 24

9:30 - 10:20 Susanna Zimmermann

11:00 - 11:50 Ludmil Katzarkov

1:00 - 1:50 Dori Bejleri

2:30 - 3:20 David Stapleton

4:00 - 5:00 Problem session

Friday, February 25

9:30 - 10:20 Joaquin Moraga

11:00 - 11:50 Lena Ji

1:00 - 1:50 Jakub Witaszek

Titles and Abstracts

Dori Bejleri - Wall crossing for moduli of stable log varieties.

Abstract: Stable log varieties or stable pairs (X,D) are the higher dimensional generalization of pointed stable curves. They form proper moduli spaces which compactify the moduli space of normal crossings, or more generally klt, pairs. These stable pair compactifications depend on a choice of parameters, namely the coefficients of the boundary divisor D. In this talk, after introducing the theory of stable log varieties, I will explain the wall-crossing behavior that governs how these compactifications change as one varies the coefficients. I will also discuss some examples and applications. This is joint work with Kenny Ascher, Giovanni Inchiostro, and Zsolt Patakfalvi.

Jérémy Blanc - Non-simplicity of Bir(X).

Abstract: The group of birational transformations of the projective space has been recently proven to be not simple in any dimension at least 2. Actually, it seems that Bir(X) is most of time either finite or not simple, if X is an algebraic variety. I will describe the cases of conic bundles, del Pezzo fibrations and surfaces.

Kristin DeVleming - K stability and birational geometry of moduli spaces of quartic K3 surfaces.

Abstract: Recently it has been shown that K-stability provides well-behaved moduli spaces of Fano varieties and log Fano pairs, and allows one to naturally interpolate between other geometric compactifications. I will discuss the picture for quartic K3 surfaces, relating compactifications coming from geometric invariant theory (GIT), Hodge theory, and K-stability via wall crossings in K-moduli. This is joint work with Kenneth Ascher and Yuchen Liu.

Lena Ji - Rationality of conic bundle threefolds over non-closed fields.

Abstract: The intermediate Jacobian is an obstruction to rationality in dimension 3, first introduced over the complex numbers by Clemens–Griffiths in their proof of the irrationality of the cubic threefold. The definition has since been extended to other fields by work of Murre and Benoist–Wittenberg. Over non-closed fields, Benoist–Wittenberg, formalizing earlier observations of Hassett–Tschinkel, defined certain torsors over the intermediate Jacobian and showed that they carry further obstructions to rationality. We show that this intermediate Jacobian torsor obstruction does not characterize rationality in the case of conic bundle threefolds with degree 4 discriminant locus. This work is joint with Sarah Frei, Soumya Sankar, Bianca Viray, and Isabel Vogt.

Ludmil Katzarkov - Zamolodchikov's c theorems and nonrationality.

Abstract: In this talk we propose a connection between Zamolodshikov's c type theorems and upersemicontinuity of spectra. Applications to nonrationality are demonstrated.

Joaquín Moraga - Fundamental group and reductive quotients of klt singularities.

Abstract: Kawamata log terminal singularities is a class of singularities that naturally arise in the minimal model program.These singularities play a fundamental role in Birational Geometry, Moduli Theory, Fano varieties, and algebraic K-stability. In this talk, we will review some recent developments regarding the fundamental group of klt singularities, and the quotient of klt singularities by the action of reductive groups.

Mircea Mustaţă - The Du Bois complex and the minimal exponent of hypersurface singularities.

Abstract: It is a well-known result that a hypersurface H in a smooth complex variety X has Du Bois singularities if and only if the pair (X,H) is log canonical (a condition which can be reformulated as saying that the minimal exponent of H is at least 1). After reviewing some basic facts about the Du Bois complex and the minimal exponent, I will describe some vanishing and non-vanishing results for the cohomology of higher graded pieces in the Du Bois complex when the minimal exponent of H is larger than 1. This is joint work with S. Olano, M. Popa, and J. Witaszek.

David Stapleton - Mori's Conjecture, Plane Curves, and Markov Numbers.

Abstract: Fix a degree d and a dimension n. When is every smooth projective limit of degree d and dimension n hypersurfaces a hypersurface? Mori conjectured that this is true when d is prime and n>2 (it is not difficult to produce limits that are not hypersurfaces when d is composite). Interestingly there are counterexamples when n=1 or 2. For example, Griffin showed there is a family of smooth quintic curves with a smooth limit that is not a plane curve. In this talk we propose a modification to Mori's conjecture in dimension one: if d is a prime number that does not appear in a Markov triple then any smooth limit of degree d plane curves is a degree d plane curve. We also propose an approach to studying this problem using Hacking's work on the KSBA compactification of the moduli space of plane curve pairs. This is joint work in preparation with Kristin DeVleming.

Jakub Witaszek - Classification of algebraic varieties in positive and mixed characteristic.

Abstract: In my talk I will describe recent developments in classifying algebraic varieties in arithmetic settings. These are partially based on recent breakthroughs in arithmetic geometry and commutative algebra.

Ziquan Zhuang - Boundedness of singularities and minimal log discrepancies of Kollár components.

Abstract: A few years ago, Chi Li introduced the local volume of a klt singularity in his work on K-stability. The local-global analogy between klt singularities and Fano varieties, together with recent study in K-stability lead to the conjecture that klt singularities whose local volumes are bounded away from zero are bounded up to special degeneration. In this talk, I will discuss some recent work on this conjecture through the minimal log discrepancies of Koll\'ar components.

Susanna Zimmermann - Algebraic groups acting birationnally on the plane over a non-closed field.

Abstract: There are many algebraic groups acting birationnally on a projective space, and for the complex plane have been mostly classified. In higher dimension, there are partial classifications in dimension 3. In this talk I will motivate the classification for infinite algebraic groups acting on the plane over a perfect field.