Some of the titles will link to the slides used during the conference.

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Adebisi Agboola (UC Santa Barbara)

Line bundles on abelian schemes and p-adic heights

Let F be a number field with ring of integers O_F, and suppose that E/O_F is an abelian scheme. If p is a prime of ordinary reduction of E, to what extent is an element of Pic^{0}(E) determined by its restriction to p-power torsion subgroup schemes of E?

I shall discuss an answer to this question that involves a new construction of the p-adic height pairing associated to E. This is joint work in progress with F. Castella and M. Ciperiani.

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Valery Alexeev (University of Georgia)

Compact moduli of K3 and Enriques surfaces

I will report on recent advances in understanding geometric, KSBA stable pair compactifications of K3 and Enriques surfaces.

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Florin Ambro (Simion Stoilow Institute)

On toric Fano fibrations

We discuss the classification of germs of toric Fano fibrations, extending work of A. Borisov in the case of Q-factorial toric singularities. As an application, we verify in the toric case a conjecture of V. Shokurov on the existence of complements with bounded index and prescribed singularities.

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Lucia Caporaso (Roma Tre University)

Hypertangency of plane curves and algebraic hyperbolicity

The talk will be about joint work in progress with Amos Turchet, investigating pairs of plane curves intersecting in at most two points.

The principal motivation comes from a series of conjectures of Vojta and others on the hyperbolicity of the complement (in the projective plane) of a curve of degree at least 4 with normal crossing singularities. We apply our results to some cases of these conjectures.

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Alessio Corti (Imperial College London and Washington University in St. Louis)

Smoothing admissible toric Fano 3-folds

I will describe a sufficient criterion for smoothing a toric Fano 3-fold. I will also discuss how the criterion fits in the Fano/LG correspondence. This is work with Paul Hacking and Andrea Petracci.

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Kristin DeVleming (UMass Amherst)

K-moduli of a family of conic bundle threefolds

Recently, there has been significant progress in understanding the K-moduli spaces of Fano varieties and log Fano pairs (X,cD). When D is a rational multiple of the anticanonical divisor of X, the K-moduli spaces of log Fano pairs (X,cD) admit a wall crossing framework as c varies and there is a finite collection of rational values of c where the K-moduli spaces change. With Lena Ji, Patrick Kennedy-Hunt, and Ming Hao Quek, we explore the K-moduli spaces in an example where D is not proportional to the anticanonical divisor. We study the K-moduli space of pairs (P1xP2, cD) where D is a (2,2) divisor and prove that there is exactly one irrational value of c where the moduli spaces change. We further relate these moduli spaces to several related spaces: the GIT of (2,2) divisors in P1xP2, K-moduli of the conic bundle threefold that is the double cover of P1xP2 branched over D, and various moduli spaces of quartic plane curves arising as the discriminant of these conic bundles.

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Christopher Hacon (University of Utah)

The work of James McKernan

I will survey some of James' most important contributions to algebraic geometry.

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Elham Izadi (UC San Diego)

Hyperkahler manifolds and Lagrangian fibrations

Hyperkahler manifolds are one of the main classes of manifolds appearing in Berger’s classification of holonomy groups of Riemannian manifolds. It is known that for any non-constant map f from a hyperkahler manifold of dimension 2n, the generic fibers of f are either finite or abelian varieties of dimension n. The latter are Lagrangian fibrations. I will discuss some open problems and some results concerning Lagrangian fibrations on hyperkahler manifolds.

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Anne-Sophie Kaloghiros (Brunel University)

On K-moduli spaces of Fano 3-folds

In this talk, I will discuss the components of K-moduli spaces associated to some of the 105 deformation families of smooth Fano 3-folds. I will discuss all 1-dimensional components of K-moduli spaces of Fano 3-folds and a 3-dimensional component parametrising K-polystable Fano 3-folds with a smoothing which is a (1,1,1,1) divisor in (P^1)^4.

 

This talk is based on joint projects with Abban, Cheltsov, Denisova, Etxabarri-Alberdi, Jiao, Martinez-Garcia and Papazachariou, and with Cheltsov, Fedorchuk and Fujita.

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Yujiro Kawamata (University of Tokyo)

On derived McKay correspondence between non-commutative deformations of commutative and non-commutative minimal resolutions

We consider an example of derived McKay correspondence between non-commutative (NC) deformations in the case of surface singularities of type An. We construct a versal NC deformation of the commutative crepant resolution and compare it with the versal NC deformation of the non-commutative crepant resolution. We show the derived McKay correspondence in the case n=1.

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Sándor Kovács (University of Washington)

On higher Du Bois singularities

Du Bois singularities were first defined and studied by Steenbrink, motivated by Hodge theoretic interests. Soon after it was recognized by Kollár that, due to their good deformation theoretic properties, Du Bois singularities play an important role in the moduli theory of higher dimensional varieties. Recently, returning to the initial Hodge theoretic motivation, Friedman and Laza and a bit later Mustață and Popa started the systematic study of higher degree analogues of Du Bois singularities.
In this talk, I will discuss a generalization of a result of Mustață and Popa as well as the deformation invariance of higher Du Bois singularities. 
 

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John Lesieutre (Penn State University)

More pathologies of the volume function

I will report on joint work with Valentino Tosatti and Simion Filip in which we show by example that for a pseudoeffective divisor D and ample A, the volume function vol(D+tA) for small values of t can exhibit various pathological behaviors.

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Michael McQuillan (University of Rome Tor Vergata)

Thurston Vanishing

Thurston's method of proof in establishing that rational maps $f$ of $\mathbb{P}^1_{\mathbb{C}}$ with pre-periodic critical points which aren't multiplication on an elliptic curve $\pm 1$ are rigid was generalised by Adam Epstein to an almost exhaustive local to global theory of deformations of dynamical systems. Indeed arguably its only fault was not to have identified a suitable topus of $f$-equivariant sheaves. Once one remedies this lacuna, natural extensions of Epstein's methods and new results in dynamical systems immediately suggest themselves. En passant we calculate the dualising complex of a real blow up, which is not only a sheaf but a remarkably algebraic one which presents notable opportunities for trivialising the analysis of Stokes' phenomenon. A pre-print is available at, https://arxiv.org/abs/2311.02518.

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Mircea Mustață (University of Michigan)

A refined log canonical threshold for local complete intersections

The minimal exponent of a hypersurface is a refinement of the log canonical threshold that gives a numerical measure of rational singularities. I this talk I will discuss an extension of the minimal exponent to local complete intersections and explain how it can be used to characterize the higher order version of Du Bois and rational singularities in this setting. This is based on joint work with Qianyu Chen, Brad Dirks, and Sebastian Olano.

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Rahul Pandharipande (ETH Zürich)

Cycles on the moduli space of abelian varieties

I will present results and conjectures related to algebraic cycles on the moduli space of principally polarized abelian varieties (with connections to parallel questions on the moduli spaces of curves and K3 surfaces).  Joint work with S. Canning, 

S. Molcho, and D. Oprea.

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Nicholas Shepherd-Barron (King's College London)

The cohomology of elliptic surfaces

A variational approach shows that the (1,1) part of the primitive cohomology of an elliptic surface has a natural orthonormal basis. This leads to a new kind of period matrix for these surfaces.

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Vyacheslav Shokurov (Johns Hopkins University)

Simple Fano varieties

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Calum Spicer (King's College London)

Flips and Flops and Foliations

Due to recent progress in the birational geometry of foliations, it is known that minimal models exist for foliations in a wide range of cases. The key technical ingredient in showing the existence of minimal models in these cases is showing the existence of flips.  In this talk we will consider a related surgery operation, the flop.

In contrast to the case of varieties, flops do not exist in general, however, we are able to demonstrate the existence of flops for co-rank one foliations.  We will explain a proof of the existence of flops in this case, with a focus on the interplay between the study of singularities from the perspective of the MMP and the classical study of foliation singularities going back to Poincare and Dulac.  Features joint work in progress with P. Cascini.

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Roberto Svaldi (University of Milan)

The birational structure of log Calabi-Yau pairs

In the first part of the talk I will discuss recent progress concerning our understanding of the birational structure of log Calabi-Yau pairs that was influenced and inspired by results/conjectures/idea of James McKernan and his collaborators.
In the second part, I will explain the solution to a conjecture of Shokurov characterizing toric varieties and toric singularities (which will still have to do with log Calabi-Yau pairs). This part will feature joint work with Brown, McKernan, Zong, and with Moraga.

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Burt Totaro (UC Los Angeles)

Some failures of vanishing theorems

The minimal model program for 3-folds has been developed only in characteristics p at least 5. A key difficulty at small primes is that the singularities occurring in the minimal model program need not be Cohen-Macaulay, as they are in characteristic zero. This is related to the fact that Kodaira vanishing fails even on Fano varieties.  We present the latest work in this direction.

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Jarosław Włodarczyk (Purdue University)

Resolution of Singularities of Varieties with Foliations by Torus Actions (jointly with D.Abramovich, A.Belotto and M.Temkin)

We discuss the resolution method by the operation of cobordant blow-ups as introduced in my paper “Resolution by torus actions” and its generalization in “Cox rings of morphisms and Resolution of singularities”. A similar notion was considered independently by Rydh in the stack-theoretic context.

 

The papers and the methods are based upon the ideas of the joint work with Abramovich and Temkin and a similar result by McQuillan on resolution in characteristic zero via stack-theoretic weighted blow-ups, and the paper of Abramovich-Quek on multiple weighted blow-ups.

 

The method gives the straightforward and efficient embedded resolution of varieties and the principalization of ideals, expressed in the language of torus actions and Rees algebras on smooth ambient varieties with simple normal crossings (SNC) divisors.

 

In the most recent paper written jointly with Abramovich, Belotto, and Temkin the method is applied in the context of foliated varieties leading, in particular, to the desingularization of the cross sections.

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