Abstracts Week 3
Dori Bejleri
Title: The cotangent complex for pairs
Abstract: I will describe a new approach to the deformation theory of pairs (X, Z)
using derived algebraic geometry with applications toward moduli spaces of higher dimensional log pairs. This is joint work in progress with G. Inchiostro and E. Elmanto.

Harold Blum
Title: Moduli of Fano varieties with complement
Abstract: While the theories of KSBA stability and K-stability have been successful in constructing compact moduli spaces of canonically polarized varieties and Fanovarieties, respectively, the case of K-trivial varieties remains less well understood. I will
discuss a new approach to this problem in the case of pairs (X, D), where X is a Fanovariety and D is an anticanonical Q-divisor, in which we consider all semi-log-canonicaldegenerations. One challenge of this approach is that the moduli stack can be unbounded.
Nevertheless, if we consider pairs (X, D), where X is degeneration of P2 , we show that
there exists a projective moduli space. This is joint work with K. Ascher, D. Bejleri, K.
DeVleming, G. Inchiostro, Y. Liu, and X. Wang.

Rankeya Datta
Title: Mittag-Leffler modules and Frobenius
Abstract: A central conjecture in prime characteristic commutative algebra is thattest elements exist for the class of excellent domains. The conjecture holds for F -finite
noetherian domains, domains that are essentially of finite type over excellent local ringsand excellent domains with universally injective Frobenius. However, the problem remains elusive in general, even in interesting geometric settings. For example, due torecent constructions of Murayama and myself, we do not know if the rigid analytic analogof the coordinate ring of an affine variety admits test elements over an arbitrary nonArchimedean field of prime characteristic. Sharp proposed an approach to tackle theconjecture for rings of this latter type via a technical notion called Frobenius intersection
flatness. This notion was first introduced by Hochster and Huneke in their efforts to show
the existence of test elements. In unrelated seminal work on the descent of projectivity,
Raynaud and Gruson introduced the notion of a Mittag-Leffler module. In this talk wewill connect the intersection flatness condition to the Mittag-Leffler condition. Consequently, we will show how the existence of test elements in the rigid analytic and related
settings reduces to a simple question about the openness of universally injective loci for1

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certain maps of modules. This talk is based on joint work with Neil Epstein and Kevin
Tucker.

Tommaso de Fernex
Title: Geometry of arc spaces
Abstract: The arc space of a variety is an infinite dimensional scheme whose geometric structure captures, in a way that is yet to be fully understood, certain featuresof the singularities of the variety. Focusing on its local rings and invariants of theserings such as embedding dimension and codimension, we explore the local structure of arcspaces. Arc spaces are defined as the inverse limit of jet schemes and are often studied via
their truncation maps; here we will present an alternative approach using projections to
infinite dimensional affine spaces and finiteness properties of arc fibers. The talk is basedon joint work with Christopher Chiu and Roi Docampo.

Mattias Jonsson
Title: Divisorial stability
Abstract: Let (X, B) be a complex projective klt pair. We define a notion ofdivisorial stability for an ample numerical class on X, defined in terms of an invariant
on convex combinations of divisiorial valuations. For the class of an ample line bundle,divisorial stability coincides with K-stability for filtrations, as studied by C. Li, and is thus
a stronger notion than (uniform) K-stability, and a sufficient condition for the existenceof an cscK metric when X is smooth. For a Fano variety, divisorial stability is the sameas uniform K-stability. We prove that divisorial stability is an open condition on thenumerical class. This is joint work with S. Boucksom.

Joaquin Moraga
Title: Fundamental groups of algebraic singularities
Abstract: In this talk, we will review the history of fundamental groups of algebraic
singularities and explain some new developments. We start by reviewing classic results
regarding normal, quotient, rational, and Cohen-Macaulay singularities. Then, we will
discuss more modern classes of singularities: log terminal and log canonical. We will
explain some new results regarding fundamental groups of lc singularities. For instance,in dimension 3, no free group of rank at least 2 is the fundamental group of an lc singularity.
On the other hand, in dimension 4, every free group is the fundamental group of an lcsingularity.

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Yusuke Nakamura
Title: Minimal log discrepancies of quotient singularities
Abstract: The minimal log discrepancy (MLD) is an invariant of singularity definedin the context of the minimal model program. In this talk, we will discuss the minimallog discrepancies of quotient singularities. I will explain that the PIA (precise inversionof adjunction) conjecture and Shokurov’s index conjecture hold for quotient singularities.
For the PIA conjecture, the theory of the arc space of a quotient singularity establishedby Denef and Loeser is an essential tool. For the index conjecture, Jordan’s theorem on
finite linear groups is essentially used. This is joint work with Kohsuke Shibata.

Zsolt Patakfalvi
Title: Varieties with nef anti-canonical have surjective Albanese
Abstract: I will present a joint work with Sho Ejiri showing that smooth projective
varieties with nef anti-canonical divisor have surjective Albanese morphism. The statement was conjectured in the Kähler setting by Demailly-Peternell-Schneider in 1993, andit was shown in characteristic zero by Zhang in 1996. Our contribution is that it also
holds in positive characteristic, and hence over any field. This is the first arbitrary dimensional positive characteristic result on varieties with nef anti-canonical divisor that is
not sensitive to wild behavior: wild action of Frobenius on cohomology, wild singularitiesof the general fibers over the Albanese image, etc. I will also mention a few corollaries
and generalizations.

Kenta SatoTitle: General hyperplane section of log canonical threefolds in positive characteristic
Abstract: In this talk, we prove that if a 3-dimensional quasi-projective variety X
over an algebraically closed field of characteristic p > 3 has only log canonical singularities,
then so does a general hyperplane section H of X. In the course of the proof, we give asufficient condition for log canonical surface singularities over a field to be geometricallylog canonical.

Teppei Takamatsu
Title: Fedder-type criteria for quasi-F-splitting
Abstract: In algebraic geometry in positive characteristic, singularities definedby the Frobenius map, including the notion of F -splitting, have played a crucial role.
Moreover, there are powerful criteria, known as Fedder’s criteria, for confirming such4properties. On the other hand, Yobuko introduced the notions of quasi-Frobenius-splitting
and F -split heights, which generalize and quantify the notion of F -splitting. In this talk,I will present a generalization of Fedder’s criteria to quasi-F -splittings and explain some
applications of such criteria. This talk is based on a joint paper with Tatsuro Kawakamiand Shou Yoshikawa.

Joe Waldron
Title: The log minimal model program for excellent threefolds
Abstract: The log minimal model program has recently been completed for klt
threefolds over regular excellent base schemes of residue characteristic p > 5. In thistalk I will survey the known results, together with some motivations and applications forworking in this more general setup.

Jakub Witaszek
Title: Singularities in mixed characteristic via the Riemann-Hilbert correspondence
Abstract: In my talk, I will start by reviewing how various properties of characteristic zero singularities can be understood topologically by ways of the Riemann-Hilbertcorrespondence. After that, I will explain how similar ideas can be applied in the studyof mixed characteristic singularities. This is based on a joint work (in progress) with
Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker, and Joe
Waldron.

Ziquan Zhuang
Title: Boundedness of singularities and minimal log discrepancies of Kollár components
Abstract: The local volume of a klt singularity is an interesting invariant thatcomes up in the local study of K-stability. It is conjectured that K-semistable Fano cone
singularities form a bounded family if their local volumes are bounded away from zero.
I’ll discuss some recent work on this conjecture through the minimal log discrepancies ofKollár components.