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 special varieties, motivated by his conjectures on the potential density of rational and integral points.

In the first lecture we discuss the class of varieties over a number field that have a potentially dense set of integral points. We will explain some fundamental properties of this class, for instance its stability under finite etale covers, and present many examples ranging from abelian varieties to symmetric products of surfaces. We will also describe our joint work with Bartsch and Levin; see https://arxiv.org/abs/2412.14931.

The second lecture focuses on the question of which varieties should admit a potentially dense set of integral points. The first proposed answer to this question, known as the Weakly Special Conjecture, turns out to contradict the abc conjecture. I will explain how this contradiction arises by studying a specific family of Enriques surfaces over the affine line. We will see how nowhere reduced fibres in fibrations impose tangency conditions on integral points, naturally leading to Campana’s theory of orbifold pairs. This lecture is based on joint work with Bartsch, Campana, and Wittenberg; see https://arxiv.org/abs/2410.06643.

In the third lecture we will disprove the Weakly Special Conjecture over function fields, both in characteristic zero and in positive characteristic. Our argument relies on Campana’s Orbifold Mordell conjecture for orbifold curves over function fields, which extends the classical theorem of Manin-Grauert and Samuel. This result is due to Campana in characteristic zero and to Kebekus-Pereira-Smeets in positive characteristic. This is joint work in progress with Bartsch and Wittenberg.

In the final lecture we will explore integral points on general type orbifold pairs. We will see that recent work of Xie and Yuan on the geometric Bombieri-Lang conjecture for ramified covers of abelian varieties can be adapted to prove the following extension. Let X be a projective variety over a function field K of characteristic zero, and assume X admits a surjective flat morphism to a geometrically traceless abelian variety A over K with at least one nowhere reduced fibre. Then X(L) is not dense for any finite field extension L over K. This statement follows from the geometric Bombieri-Lang conjecture for orbifold divisors on abelian varieties and is established in joint work in progress with G. Gao and Rousseau.

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 (Title and abstract TBA)

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.

Marta Pieropan (Title and abstract TBA)

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: Symmetric products and puncturing Campana-special varieties.

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.