Yen-An Chen

Title: Boundedness of toric Fano foliations

Abstract: In recent years, there are significant developments of the minimal model program for foliated varieties. It is intriguing to ask if Fano foliations form a bounded family. It is anticipated that Borisov-Alexeev-Borisov conjecture also holds in the context of foliations. In this talk, I will discuss the boundedness of the toric Fano adjoint foliated structure with mild singularities. This is a joint work with Chih-Wei Chang. 

Masatomo Sawahara

Title: Rationality and cylindricity of k-forms of singular del Pezzo surfaces

Abstract: Cylinders in normal projective varieties have attracted considerable attention due to their close relationship with unipotent group actions on affine algebraic varieties. Let k be a field of characteristic zero. It is known that the cylindricity of lower-dimensional varieties over k corresponds to the existence of vertical cylinders in higher-dimensional varieties admitting a fibration structure, where the former appears as the generic fiber. In previous work, Dubouloz–Kishimoto and the speaker determined conditions for the existence of cylinders in Picard rank one k-forms of canonical del Pezzo surfaces. Such surfaces arise as generic fibers of Mori fiber spaces of relative dimension two. In this talk, we give a criterion for the rationality and cylindricity of k-forms of a singular del Pezzo surface obtained as a blow-up of a weighted projective plane. This is joint work with In-Kyun Kim and Dae-Won Lee.

Nasu Hirokazu

Title: Obstructions to deforming space curves lying on a complete intersection of quadrics.

Abstract: The Hilbert schemes of rational curves on Fano varieties have been widely studied, because rational curves on Fano varieties tend to deform well and play an important role in the classification of Fano varieties. In contrast, much less is known about the Hilbert schemes of curves of higher genus. This is mainly because deformation obstructions are harder to control, and the geometry of the Hilbert scheme becomes more complicated. In particular, when obstructions do not vanish, it is often difficult to determine even the local dimension of the Hilbert scheme. Even for the three-dimensional projective space P^3, the situation can be subtle: Mumford (1962) showed that the Hilbert scheme of smooth connected curves of degree 14 and genus 24 has a non-reduced irreducible component. In this talk, starting from Mumford's example, we explain how to construct similar non-reduced irreducible components of the Hilbert schemes of curves in P^4 and P}^5, using a quartic del Pezzo surface S_{2,2} in P^4 and a degree-8 (genus-5) K3 surface S_{2,2,2} in P^5.

Tiago Duarte Guerreiro

Title: Mori dreamness of blowups of P^3 along a curve

Abstract: Mori dream spaces are a special kind of varieties introduced by Hu and Keel in 2000 that enjoy very good properties with respect to the minimal model program. In this talk we explore when blowups of P^3 along smooth curves are Mori dream spaces, generalizing an early example of A. Küronya.  This is joint work with Sokratis Zikas.

Dasol Jeong

Title: Conical Kähler-Einstein metric on K-unstable del Pezzo surface

Abstract: For a Fano manifold X, the greatest Ricci lower bound R(X), arising from the continuity method, plays a key role for the study of Kähler-Einstein metrics. In particular, the existence of Kähler-Einstein metric is equivalent to R(X)=1. On the other hand, Yau-Tian-Donaldson conjecture was solved using Kähler-Einstein metric with singularities along (pluri)anticanonical divisor D. Motivated by the formal similarity between the equations arising in the continuity method and those defining conical Kähler–Einstein metrics, Donaldson conjectured that R(X) coincides with the supremum R(X,D) of cone angles along anticanonical divisors D on X. However, Székelyhidi provided counterexamples in the surface cases. Note that there are only two K-unstable smooth del Pezzo surfaces S_1 and S_2, that are blowups of P^2 at one point and two points, respectively. In this talk, I will briefly review the history and introduce several tools such as K-stability and delta-invariants. Then, I will explain how to find R(S_i,C_i), where C_i are smooth anticanonical curves on S_i, for i=1,2.

Elena Denisova

Title: K-moduli of Fano threefolds of Picard rank 3 and degree 20.

Abstract: (Joint work with T. Papazachariou)
K-moduli spaces provide a canonical parametrisation of K-polystable Fano varieties, but they are rarely accessible in concrete terms. In this talk I will describe an explicit example in dimension three. I consider the Fano threefolds in Mori-Mukai family №3.5, which can be realised as blow-ups of P1xP2 along curves of bidegree (5,2). I will explain how the K-stability of these threefolds is determined by the classical GIT stability of the corresponding curves. This leads to an explicit description of the K-moduli space as a GIT quotient and yields a full classification of all K-stable, K-polystable and K-semistable members of the family.

Alexander Perepechko

Title: Automorphisms of affine varieties: flexibility and unipotent group actions 

Abstract: A Ga-action on an affine variety X is an algebraic action of the additive group Ga(K) of the base field K. Such actions are the fundamental building blocks of the automorphism group Aut(X): they are responsible for its infinite-dimensional structure. In this talk, we explore two aspects: their geometric role in flexibility and higher transitivity, and the algebraic structure of the unipotent subgroups they generate. A variety is called flexible, if for each smooth point, its tangent space is spanned by tangent vectors to orbits of Ga-actions. A famous result states that the automorphism group of a flexible variety acts transitively on tuples of N smooth points for any finite N. We will also survey families of varieties known to be flexible and outline the connection of flexibility with the unirationality problem. A subgroup U of Aut(X) is called unipotent if each element of U is contained in a Ga-subgroup. In the case of an affine space A^n, the unipotent subgroup of triangular automorphisms serves as an infinite-dimensional analogue of upper-triangular matrices U(n) in the matrix group GL(n). It is well known that any maximal unipotent subgroup of GL(n) is conjugate to U(n). However, this statement fails for triangular automorphisms in Aut(A^n). Our key result is a structural description of maximal unipotent subgroups of Aut(X) that extends the notion of triangular automorphisms. It allows us to give an affirmative answer to the question by  H.Kraft and M.Zaidenberg (arXiv:2203.11356): we show that connected nested subgroups of Aut(X) are closed with respect to the ind-topology. Here a group is called nested if it is a limit of an ascending sequence of algebraic subgroups. The talk is prepared within the framework of the project “International academic cooperation” HSE University.

Ching-Jui Lai

Title: On slope unstable Fano varieties

Abstract: For Fano varieties, significant progress has been made recently in the study of K-stability, while the understanding of the weaker but more algebraic concept of (−K)-slope stability remains intricate. For instance, a conjecture attributed to Iskovskikh states that the tangent bundle of a Picard rank one Fano manifold is slope stable. Peternell-Wi´sniewski and Hwang proved this conjecture up to dimension five in 1998, but Kanemitsu later disproved it in 2021. To address this gap in understanding, we present a method that aims to characterize the geometry associated with the maximal destabilizing sheaf of the tangent sheaf of a Fano variety. This approach utilizes modern advancements in the foliated minimal model program. In dimension two, our approach leads to a complete classification of (−K)-slope unstable weak del Pezzo surfaces with canonical singularities. As by-products, we provide the first conceptual proof that P1 × P1 and F1 are the only (−K)-slope unstable nonsingular del Pezzo surfaces, recovering a classical result of Fahlaoui in 1989. We also uncover a phenomenon that does not occur for Fano manifolds: there exists a del Pezzo surface with type A singularities admitting a weak K\”ahler-Einstein metric, yet whose tangent sheaf is slope unstable. 

Erik Paemurru

Title: Local inequalities for cA_k singularities

Abstract: We generalise an intersection-theoretic local inequality of Fulton-Lazarsfeld to weighted blowups. Using this together with the classification of 3-dimensional divisorial contractions, we prove birational rigidity and therefore nonrationality of many families of terminal Fano 3-folds. This is joint work with Igor Krylov and Takuzo Okada.

Seung-Joo Lee

Title: Addressing explicit bounds - dialogues between physics and geometry

Abstract: String theory leads to a huge number of supersymmetric effective theories, at least as many as the number of topologically distinct Calabi-Yau varieties, forming a vast "landscape" of string theory. Despite the vastness of the latter, interestingly and perhaps surprisingly, one can address universal constraints applying to the string landscape in its entirety. Such constraints sometimes lead to explicit, uniform bounds on concrete physical quantities of effective theories, which may be interpreted as those on the corresponding concrete topological quantities of Calabi-Yau varieties. In this talk, focusing on (real) 6-dimensional F-theory models, or equivalently, on genus-one fibered Calabi-Yau varieties of dimension 3, we will exemplify such dialogues between physics and geometry. For physicists, what we aim to constrain are particle spectra and discrete symmetries of supergravity theories, and for geometers, the base Picard number and (multi-)sections of the genus-one fibrations. 

Kiwamu Watanabe

Title: Characterization of products of projective spaces via the nef complexity

Abstract: The nef complexity, introduced by Yoshinori Gongyo, is a numerical invariant that measures how far a Fano variety is from being a product of projective spaces. In this talk, I will explain a divisorial characterization stating that the nef complexity is always nonnegative, and it vanishes precisely for products of projective spaces. I will also discuss explicit computations, several applications, and further related developments. This is joint work with Joshua Enwright, Stefano Filipazzi, Yoshinori Gongyo, Joaquin Moraga, Roberto Svaldi, and Chengxi Wang.

Livia Campo

Title: K-stability of weighted hypersurfaces

Abstract: In this talk we discuss K-stability of weighted Fano hypersurfaces of dimension n>=3. The 3-fold case hasbeen established by [Kim-Okada-Won], [Sano-Tasin], [Campo-Okada]. In a joint work with Kento Fujita, Taro Sano, and Luca Tasin we studied the n-fold case with n>3 producing bounds for delta invariants of weighted Fano n-fold hypersurfaces embedded in certain weighted projective spaces. 

Igor Krylov

Title : Equivariant birational rigidity of quadric fibrations

Abstract : I will talk about equivariant birational rigidity of quadric bundles and the related results about the Cremona group: models for certain actions of groups acting on P^3 and the embeddings of Alternating group of rank 5 into the Cremona group.