New Preprint

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.

Birational Geometry, Kahler-Einstein Metrics, and Degenerations

November 2019, Pohang

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.

Birational Geometry, Kahler-Einstein Metrics, and Degenerations

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.

Algebraic Geometry seminar in Brown University

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.

Algebraic Geometry seminar in Harvard

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.

Algebraic Geometry seminar in NYU

May 2019, New York

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.

Algebraic Geometry seminar in Johns Hopkins University

April 2019, Baltimore

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.

Algebraic Geometry near-Boston Conference

April 2019, Boston

Title: Parameter spaces of del Pezzo fibrations and birational geometry

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.