Organizer: James McKernan, University of California, San Diego
The MPS Conference on Higher Dimensional Geometry, February 2022 focused on recent progress in higher-dimensional geometry and its interaction with other fields.
Topics included K-stability and recent progress on the construction of the moduli space of K-stable Fano varieties, the minimal model program in mixed characteristic and the higher dimensional Cremona group.
Dori Bejleri, Harvard University
Jérémy Blanc, Universität Basel
Kristin DeVleming, University of Massachusetts
Lena Ji, University of Michigan
Ludmil Katzarkov, University of Miami
Joaquin Moraga, Princeton University
Mircea Mustaţă, University of Michigan
David Stapleton, University of Michigan
Jakob Witaszek, University of Michigan
Ziquan Zhuang, Massachusetts Institute of Technology
Susanna Zimmermann, University of Angers
Wednesday, February 23
9:30 AM Mircea Mustaţă - The Du Bois complex and the minimal exponent of hypersurface singularities 11:00 AM Ziquan Zhuang - Boundedness of singularities and minimal log discrepancies of Kollár components 1:00 PM Kristin DeVleming - K stability and birational geometry of moduli spaces of quartic K3 surfaces 2:30 PM Jakob Witaszek - Classification of algebraic varieties in positive and mixed characteristic 4:00 PM Problem Session (organizer)
Thursday, February 24
9:30 AM Susanna Zimmermann - Algebraic groups acting birationnally on the plane over a non-closed field 11:00 AM Ludmil Katzarkov - Zamolodchikov's c theorems and nonrationality 1:00 PM Dori Bejleri - Wall crossing for moduli of stable log varieties 2:30 PM David Stapleton - Mori's Conjecture, Plane Curves, and Markov Numbers 4:00 PM Poster Session
Friday, February 25
9:30 AM Joaquin Moraga - Fundamental group and reductive quotients of klt singularities 11:00 AM Lena Ji - Rationality of conic bundle threefolds over non-closed fields 1:00 PM Jeremy Blanc - Non-simplicity of Bir(X)
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.
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.
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.
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.
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.
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.
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.
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.