May 2023

**Abstract**: We classify non-factorial nodal Fano threefolds with 1 node and class group of rank 2.

May 2023

**Abstract**: We classify non-factorial nodal Fano threefolds with 1 node and class group of rank 2.

April 2023, Gokova, Turkey

The session will bring together algebraic geometers from the UK, Europe and Asia to work together on higher dimensional Fano varieties. The focus will be made on their birational geometry, derived categories, K-stability and applications to Cremona groups. In particular, the following explicit problems will be considered during the research session: K-stability of smooth Fano threefolds of Picard rank 2 and degree 26, geometric version of Kollar’s conjecture about birational rigidity and field extension, classification of maximally non-factorial nodal Fano threefolds after Pavic and Shinder, boundedness of the image of the Lin-Shinder invariant.

August 2022, Pohang

**Title:**Families of simple subgroups in the Cremona group arising from del Pezzo fibrations

**Abstract:**

Cremona group of rank n is the group of birational self-maps of the projective space of dimension n. For any subgroup G of Cremona group there is a rational variety on which G acts regularly. This allows to translate the study of subgroups of Cremona group into the study of G-equivariant geometry of rational varieties. In this talk I will describe some continuous families of rational threefolds with an action of alternating group of rank 5. I will also explain why the corresponding subgroups of the Cremona group are not pair-wise conjugate.

May 2022

**Abstract**: The 4*n*^2-inequality for smooth points plays an important role in the proofs of birational (super)rigidity. The main aim of this paper is to generalize such an inequality to terminal singular points of type *cA*_1, and obtain a 2*n*^2-inequality for *cA_*1 points. As applications, we prove birational (super)rigidity of sextic double solids, many other prime Fano 3-fold weighted complete intersections, and del Pezzo fibrations of degree 1 over P1 satisfying the *K*^2-condition, all of which have at most terminal *cA*1 singularities and terminal quotient singularities. These give first examples of birationally (super)rigid Fano 3-folds and del Pezzo fibrations admitting a *cA*1 point which is not an ordinary double point.

March 2021, Saitama

**Title:**Families of simple subgroups in the Cremona group arising from del Pezzo fibrations

**Abstract:**

Cremona group of rank nis the group of birational self-maps of the projective space of dimension n. For any subgroup Gof Cremona group there is a rational variety on which Gacts regularly. This allows to translate the study of subgroups of Cremona group into study of G-equivariant geometry of rational varieties. In this talk I will describe some continuous families of rational threefolds with an action of alternating group of rank 5. I will also explain why the corresponding subgroups of the Cremona group are not pair-wise conjugate.

December 2019

Stability of fibrations over one-dimensional bases

with H. Ahmadinezhad and M. Fedorchuk

**Abstract:**

A Mori fiber space is called birationally rigid if, roughly speaking, it has only one Mori fiber space structure. I will talk about birational rigidity of singular del Pezzo fibrations. I will explain the essence of the method of proving birational rigidity: Noether-Fano method, also known as method of maximal singularities. I will discuss the differences in applying this to Fano varieties and Mori fiber spaces with a positive dimensional base.

November 2019

**Title:**On embeddings of Klein simple group into the Cremona group.

**Abstract:**

The Cremona group is the group of the birational transformations of the projective space. It is known that for any finite subgroup G in the Cremona group there is a rational variety X on each G acts biregularly. By running G-MMP on X we get a rational GQ-Mori fiber space. The study of embeddings of G into the Cremona group is equivalent to study of rational GQ-Mori fiber spaces. I will talk about PSL_2(7)Q-del Pezzo fibrations, their rationality, and the relation to quotients of certain quartic threefolds.

This a joint work with Takuzo Okada. This talk is related to his talk at this conference.

June 2019, Shanghai

**Title:** Semistable Models of del Pezzo Fibrations, Birational Rigidity, and Universal K-Condition

**Abstract:**

Del Pezzo fibrations are one of the types of the Mori Fiber Space output of the MMP. There may be many models for the del Pezzo fibration and we would like to work with the best one. For example it is known that for conic bundles there exists a model with a smooth total space. I will describe a construction of parameter space of del Pezzo surfaces of degree 1 and 2. Using this parameter space I define what are the best models of del Pezzo fibrations of degrees 1 and 2. Then I show the existence of a good birational model.

May 2019, Providence

**Title:** Parameter spaces of del Pezzo surfaces and birational geometry of del Pezzo fibrations

**Abstract:**

Del Pezzo fibrations are one of the types of the Mori Fiber Space output of the MMP. There may be many models for the del Pezzo fibration and we would like to work with the best one. For example it is known that for conic bundles there exists a model with a smooth total space. I will describe a construction of parameter space of del Pezzo surfaces of degree 1 and 2 together with a notion of stability. Then I define what are the best models of del Pezzo fibrations of degrees 1 and 2 and show the existence of a good birational model.

May 2019, Boston

**Title:** Birational rigidity of low degree del Pezzo fibrations

**Abstract:**

A Mori fiber space is called birationally rigid if, roughly speaking, it has only one Mori fiber space structure. I will talk about birational rigidity of singular del Pezzo fibrations. I will explain the essence of the method of proving birational rigidity: Noether-Fano method, also known as method of maximal singularities. I will discuss the differences in applying this to Fano varieties and Mori fiber spaces with a positive dimensional base.