Program

Schedule

Titles and abstracts

Minicourses

Brendan Hassett: Rationality, stable rationality, and the geometry of threefolds.

This mini-course will discuss constructions of and obstructions to rationality and stable rationality for threefolds. Techniques will include the intermediate Jacobian, Brauer groups, and the decomposition of the diagonal. We will focus first on complex varieties but will mention generalizations to non-closed fields. If time allows, we will touch on new results on cubic threefolds due to Engel, de Gaay Fortman, and Schreieder and others.

Ariyan Javanpeykar: Campana’s special varieties and his theory of orbifold pairs.

In this course we will introduce Campana’s notion of special varieties, motivated by his conjectures on potential density of rational and integral points.  

The first lecture will focus on the Orbifold Mordell Conjecture, a vast generalization of Faltings’s theorem for curves of genus at least two to Campana’s orbifold curves, which remains open. This conjecture naturally leads to the notions of weakly special varieties and the Weakly Special Conjecture. I will present new examples of weakly special but non-special varieties, based on joint work with Finn Bartsch, Frédéric Campana, and Olivier Wittenberg. If time permits, I will also discuss new results on higher-dimensional generalizations of Orbifold Mordell over function fields, from joint work with Guoquan Gao and Erwan Rousseau.

A problem sheet.

Junliang Shen: The D-equivalence conjecture and hyperholomorphic bundles I, II, III.

The D-equivalence conjecture of Bondal-Orlov predicts that birational smooth projective Calabi-Yau varieties are derived equivalent. Although this conjecture is wide open in general, in recent years much progress of this conjecture has been made for hyper-Kahler/holomorphic symplectic varieties (Namikawa, Kawamata, Kaledin, Halpern-Leistner, etc.) through various techniques. The purpose of this lecture series is to discuss the recent proof of the D-equivalence conjecture for hyper-Kahler varieties of K3[n]-type, based on my joint work with Davesh Maulik, Qizheng Yin, and Ruxuan Zhang. A key ingredient in the proof is the theory of hyperholomorphic bundles (by Verbitsky, Buskin, and Markman), where the transcendental aspect of the hyper-Kahler structure plays a crucial role. If time permits, further applications will be discussed.

Sho Tanimoto: Homological sieve and Manin’s conjecture.

In these lectures, we introduce a geometric and topological approach to counting problems over global function fields, and this method is called homological sieve. In particular, we explain its application to Manin’s conjecture for split quartic del Pezzo surfaces defined over Fq with q being large. This is a nice combination of algebraic geometry (birational geometry of moduli spaces of rational curves), arithmetic geometry (simplicial schemes, their homotopy theory, and Grothendieck-Lefschetz trace formula), algebraic topology (the inclusion-exclusion principle and the Vassiliev type method of the bar complex), and some elementary analytic number theory. If time permits, we also explain an application of homological sieve to Cohen-Jones-Segal conjecture. A part of lectures is joint work with R. Das, B. Lehmann, and P. Tosteson with a help by W. Sawin and M. Shursterman.

Research talks

Lucia Caporaso: Hypertangency of curves on surfaces and log-hyperbolicity

Well known conjectures (of Green-Griffiths, Kobayashi, Lang-Vojta, ... ) predict that a surface of general type contains finitely many rational curves. Such conjectures extend to a  log-surface (S,B), and are open in general. We shall discuss the case where B is  a reducible curve reporting on recent progress, including some joint work with Amos Turchet.

Annalisa Grossi: In search of maximal branes on Hyper-Kähler manifolds.

Abstract: Given an involution on a complex variety, the Smith-Thom inequality says that the total \mathbb{F}_2-Betti number of the fixed locus is no greater than the total \mathbb{F}_2-Betti number of the ambient variety. The involution is called maximal when the equality is achieved. On a Hyper-Kähler manifold X a holomorphic or a anti-holomorphic involution is referred to as a brane involution. While examples of non-compact hyper-Kähler manifolds admitting maximal branes are known, the compact case is more intriguing. In particular, although there exist some K3 surfaces admitting maximal brane involutions, the main result that I will show you is the non-existence of maximal branes on Hyper-Kähler manifolds deformation equivalent to the Hilbert scheme of points on a K3 surface. This talk is based on a joint work with S. Billi, L. Fu and V. Kharlamov.

Mirko Mauri: Baily–Borel Compactifications of Period Images and the b-Semiampleness Conjecture

Semiampleness criteria are subtle foundational statements in algebraic geometry. In this talk, I present a new semiampleness result for the Hodge bundle of certain Calabi–Yau variations of Hodge structure. This result is a key ingredient in the proof of two long-standing conjectures: Griffiths’ conjecture on functorial compactifications of images of period maps, and the b-semiampleness conjecture in birational geometry. The proof crucially relies on o-minimal GAGA, marking the first use of o-minimality techniques in birational geometry. This is a joint work with Benjamin Bakker, Stefano Filipazzi, and Jacob Tsimerman.

Claudio Onorati: The SYZ conjecture for singular moduli spaces of sheaves on K3 surfaces.

I will report about my recent joint work with Angel Rios Ortiz on the SYZ conjecture for a special class of singular symplectic varieties. The SYZ conjecture predicts that nef and isotropic line bundles are associated to lagrangian fibrations. After having recalled some generalities about symplectic varieties and the SYZ conjecture, I will state the main result and explain the main ideas behind its proof.

Sara Mehidi (Utrecht University) and Marta Pieropan (Utrecht University): Lifting rational points along log smooth morphisms

Determining the image of the set of rational points under a morphism of varieties is a very natural and difficult question. The case where the morphism is geometrically surjective has been studied extensively. From Campana's theory of orbifolds, it follows that over a number field, the image of the set of rational points is contained in the set of Campana points for the orbifold base of the morphism. This first approximation of the image of the set of rational points is refined by Abramovich's theory of firmaments. This talk presents our joint work with Herr and Poiret that addresses the same questions in the setting of log schemes and provides a proof of a claim by Abramovich about lifting firm points under toroidal morphisms.

Alessio Sammartano: Components and singularities of Hilbert schemes of points

The Hilbert scheme of points in affine n-dimensional space (or in a smooth n-dimensional variety) parametrizes finite subschemes of a fixed length. Several basic questions about the Hilbert scheme of points remain open, especially about its irreducible components, their singularities and birational geometry. I will give an overview of the geometry of this parameter space, detailing how its behavior changes as we vary the ambient dimension, and focusing on the recent progress in this area.

Lightning Talks

Finn Bartsch: The Arithmetic Puncturing Problem.

Enhao Feng: Geometric Manin’s Conjecture for genus one curve on smooth cubic threefolds.

Boaz Moerman: M-points of bounded height.

Simone Pesatori: Singular rational curves on Enriques surfaces.

Francesca Rizzo: Linear spaces in Gushel–Mukai varieties and EPW cubes.

Elena Sammarco: New nonspecial divisors in the moduli space of cubic fourfolds.

Soumya Sankar: Rationality of conic bundle threefolds over non-algebraically closed fields.

Tianzhi Yang: Twisted representations and arithmetic of quotient singularities.