Schedule

Monday

Paul Hacking - Homological mirror symmetry for projective K3 surfaces

Joint work with Ailsa Keating (Cambridge). We prove the homological mirror symmetry conjecture of Kontsevich for projective K3 surfaces. In particular we prove the Lekili--Ueda conjecture: the complement of a smooth hyperplane section on a K3 surface is mirror to a degenerate K3 surface of type III in Kulikov's classification. Our proof builds on prior work of Seidel, Sheridan, and Ganatra--Pardon--Shende. I will not assume familiarity with the Fukaya category. arxiv:2503.05680

Kristin DeVleming - The Noether-Lefschetz locus in families of threefolds

Abstract: For a fixed threefold X and very ample line bundle H, the Noether-Lefschetz locus parametrizes surfaces in X which are linearly equivalent to H and have Picard rank greater than that of X. We discuss the behavior of special components of the Noether-Lefschetz locus as we deform the pair (X, H), particularly when X is a singular Fano threefold. This is joint work with A. Grassi and J. Rana.

Tom Coates - TBA

Tuesday

Rahul Pandharipande - The quantum cohomology of the Hilbert scheme of points 

I will present results old and new about the quantum cohomology and the higher genus enumerative geometry of the Hilbert scheme of points in the plane. I will highlight the remarkable number of connections to different streams of mathematics. 

Cristina Manolache -  Higher genus reduced Gromov--Witten invariants

Gromov--Witten (GW) invariants ideally give counts of curves of genus g in a given variety. However, GW invariants with g greater than one, have a more subtle enumerative meaning: curves of lower genus also contribute to GW invariants. In genus one this problem was corrected by Vakil and Zinger, who defined more enumerative numbers called "reduced GW invariants". More recently Hu, Li and Niu gave a construction of reduced GW invariants in genus two. I will use desingularisations of sheaves to define reduced Gromov--Witten invariants in all genera. This is work with A. Cobos Rabano, E. Mann and R. Picciotto. 

Helge Ruddat - Smoothing singular toric Fanos

This talk provides an overview of Alessio Corti’s contributions, including our joint work, to the program of constructing smoothings for singular toric Fano varieties. A key achievement of this program is the unifying production of all 105 smooth Fano threefolds from singular toric varieties by a single method. The construction of (all?) Q-Gorenstein Fano 3-folds and smooth Fano 4-folds is within reach. I will highlight Alessio’s recent work about zero-mutable Laurent polynomials, explaining their role in constructing log structures and their application to studying the topology of complex singularities, such as the well-known but challenging Tom and Jerry singularities.

Renata Picciotto - TBA

Carolina Araujo - Automorphisms of quartic surfaces and Cremona transformations 

In this talk, I will address the following question, attributed to Gizatullin: "Which automorphisms of a smooth quartic surface in projective 3-space are restrictions of Cremona transformations of the ambient space?" Corti and Kaloghiros have introduced a general framework that is extremely useful for approaching this problem, namely, a special version of the Sarkisov program for Calabi-Yau pairs. I will report on recent progress on Gizatullin’s problem obtained using this theory, in collaborations with Alessio Corti and Alex Massarenti, and with Daniela Paiva and Sokratis Zikas. 

Fabrizio Catanese - Numerically and cohomologically trivial  Automorphisms of  compact Kaehler manifolds and  surfaces

I shall report on the results of several papers, in collaboration with Wenfei Liu, Matthias Schuett, Christian Gleissner, Davide Frapporti. An automorphism of a cKM X  is  said to be numerically trivial (in Aut_Q(X)) if it acts trivially on rational cohomology, and cohomologically trivial (in Aut_Z(X)) if it acts trivially on integral cohomology. The interest for these notions stems from the theory of period maps and from Teichmueller theory, and there has been, since the 70’s, much work of several  authors for the case of algebraic surfaces S, and for the group Aut_Q(S). For surfaces, I will illustrate how the answer depends on the Kodaira dimension, and on minimality-nonminimality assumptions. For properly elliptic surfaces we showed: for \chi >0 all 2-generated abelian groups appear as Aut_Q(S), but there are upper bounds for N : = |Aut_Q(S)|, depending  on the bigenus P_2(S) or on the irregularity q(S).  Our results are sharp in the isotrivial case. As Noether said, curves were created by God, and surfaces by the devil: several authors  claimed that in this situation there are no numerically trivial automorphisms if \chi , p_g >0... For general type, with Frapporti (work in progress) we got an example with  |Aut_Q| = 192, while for Aut_Z we can only got up to  now  |Aut_Z| = 2.

Wednesday

Ana-Maria Castravet - Gale duality, blow-ups and moduli spaces

I will discuss joint work with Carolina Araujo, Inder Kaur and Diletta Martinelli about the birational geometry of blow-ups of projective spaces at points in general position.  We will explore Gale duality, a correspondence between sets of n=r+s+2  points in projective spaces P^r and P^s. For small values of s, this duality has a remarkable geometric manifestation: the blow-up of P^r at n points can be realized as a moduli space of vector bundles on the blow-up of P^s at the Gale dual points.  

Elana Kalashnikov - Tableaux Littlewood—Richardson rules for 2-step flags

The Abelian/non-Abelian correspondence gives rise to a natural basis for the cohomology of flag varieties, which - except for Grassmannians - is distinct from the Schubert basis. I will describe this basis and its multiplication rules, and explain how to relate it to the Schubert basis for two-step flag varieties. I will then explain how this leads to new tableaux Littlewood--Richardson rules for many products of Schubert classes. This is joint work (separately) with Wei Gu and Linda Chen.

Miles Reid - The Tate-Oort group scheme TO_n for n >= 1

Joint with LIAO Yiming. We construct the Tate-Oort group scheme TO_n. For the prime case TO_p see https://mreid.warwick.ac.uk/TOp. This is the cyclic group ZZ/n _as you have never seen it before_: it lives over an arithmetic base; over primes not dividing n, it is an etale form of ZZ/n. However, where p^r divides n, its degenerations include the nonreduced group schemes mu_{p^i} and alpha_{p^i} for i < r, alongside etale forms of ZZ/(n/p^r). Our construction is straightforward, based on simple modifications of the celebrated Cauchy-Liouville-Mirimanoff polynomials.

Thursday

Hiroshi Iritani - Mirror theorem and shift operators 

The genus-zero Gromov-Witten invariants of a smooth projective variety can be encoded in an infinite-dimensional Lagrangian submanifold, known as the Givental cone, within the loop space of the cohomology group. A Givental-style mirror theorem states that a certain explicit cohomology-valued hypergeometric series, called the I-function, lies in this Givental cone. In joint work with Coates, Corti and Tseng, we established a Givental-style mirror theorem for toric Deligne-Mumford stacks. In this talk, I will explore the connection between Givental-style mirror theorems and shift operators for equivariant parameters, and I will reframe our results using this perspective. This is partly based on joint work with Fumihiko Sanda. 

Mark Gross - Wall-crossing for moduli spaces of coherent sheaves on P^2

Pierrick Bousseau showed that wall-crossing for Bridgeland stability conditions on P^2 was governed by the same scattering diagram which governs mirror symmetry for P^2. The latter scattering diagram was studied in some detail by Thomas Prince. We make a deeper dive into the structure of this very complicated scattering diagram, and use it to understand when various moduli spaces, including the Hilbert scheme of points on P^2, undergo birational changes under variation of Bridgeland stability conditions. This is joint work with Fatemeh Rezaee. 

Burt Totaro - Terminal 3-folds that are not Cohen-Macaulay

An important local vanishing theorem for the minimal model program is the fact that klt singularities in characteristic zero are Cohen-Macaulay. In contrast, even in the narrow setting of terminal singularities of dimension 3, we show that Cohen-Macaulayness can fail in characteristic p or mixed characteristic (0,p) for p equal to 2, 3, or 5. This is optimal, by work of Arvidsson-Bernasconi-Lacini. (These examples help to explain why the MMP remains an open question for 3-folds in characteristic 2 or 3.) The examples are quotients of regular schemes by the cyclic group G of order p. In characteristic p or mixed characteristic (0,p), such quotients can exhibit a wide range of behavior.

Angelo Vistoli -  p-subgroups of relative Brauer groups for transcendental extensions

I am going to report on joint work with Giulio Bresciani and Zinovy Reichstein. If k is  a field and M is a locally factorial variety over k with field of rational functions k(M), I will discuss the kernel K(M) of the pullback map on Brauer groups Br(k) \to Br(k(M)). It is known that K(M) is a torsion group with bounded exponent; however, it may be infinite. We prove a general bound on finite p-subgroups of K(M), which implies, in particular, that if M contains a subscheme X \subseteq M, which is proper over k, and such that \chi(X, O_X) = \pm 1, then K(M) is finite. This has applications to p-subgroups of groups of birational transformations of rationally connected varieties, which I will mention if I have any time left (this is unlikely to happen).

Cinzia Casagrande - Classifying Fano 4-folds with large Picard number

We will present some classification results for (smooth, complex) Fano 4-folds X with Picard number rho(X)>6. First of all, if rho(X)>9, then X is a product of del Pezzo surfaces; this is sharp, since we know one family of Fano 4-folds with rho(X)=9 that is not a product of surfaces. In the range rho(X)=7,8,9, we will explain some partial classification results, based on a detailed and explicit study of the geometry of X using birational geometry in the framework of the MMP. In particular, if rho(X)>6 and X has no small contractions, then either X is a product of surfaces, or rho(X)=7,8,9 and X is a blow-up of a cubic 4-fold along rho(X)-1 planes that intersect pairwise at a point.

Michael McQuillan - TBA

Short talks

Zhaoyang Liu - Morphisms between Grassmannians 

Tianzhi Yang - Field of moduli and arithmetic of quotient singularities

Federico Tufo - Homogeneous varieties approach to quiver zero loci Fano fourfolds

Thamarai Valli Venkatachalam - Classification of Q-Fano 3-folds

Friday

Lucia Caporaso - TBA 

Dan Abramovich - Resolution of singularities on foliated orbifolds

We resolve singularities of a subvariety X of a smooth variety Y endowed with a foliation F, through weighted blowups aligned with F. Stacks and logarithmic structures will feature prominently, yet explicitly and concretely, with examples, as befits the event. This is work with A. Belotto da Silva, M. Temkin and J. Włodarczyk. 

Paola Frediani - Asymptotic directions in the moduli space of curves. 

I will report on a joint work with E. Colombo and G.P. Pirola, where we study asymptotic directions in the tangent bundle of the moduli space of curves of genus g, namely those tangent directions that are annihilated by the second fundamental form of the Torelli map. I will show that there exist examples of asymptotic directions for any g >3. I will present one main result saying that if the rank of a tangent direction at [C] (with respect to the infinitesimal deformation map) is less than the Clifford index of the curve C, then the tangent direction is not asymptotic. Finally I will determine all asymptotic directions of rank 1 and give an almost complete description of asymptotic directions of rank 2.