MPS Conference on Higher Dimensional Geometry, February 2022

Dori Bejleri
Harvard University

Wall crossing for moduli of stable log varieties

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
Universität Basel

Non-simplicity of Bir(X)

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.
 

Lena Ji
University of Michigan

Rationality of conic bundle threefolds over non-closed fields

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
University of Miami

Zamolodchikov’s c theorems and nonrationality

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

Joaquín Moraga
Princeton University

Fundamental group and reductive quotients of klt singularities

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.
 

Jakub Witaszek
University of Michigan

Classification of algebraic varieties in positive and mixed characteristic

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
Massachusetts Institute of Technology

Boundedness of singularities and minimal log discrepancies of Kollár components

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 Kollár components.
 

Susanna Zimmermann
University of Angers

Algebraic groups acting birationally on the plane over a non-closed field

There are many algebraic groups acting birationally 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.