Seminar in KIAS

August 2018, Seoul

Title: Semistability of del Pezzo surfaces, good models and reductions

Abstract:
Kollar has introduces a notion of stability of a hypersurface in projective space over a ring. This notion can be used to find good reductions of hypersurfaces to finite characteristic. It also has application to birational geometry. If a cubic fibration over an affine curve is a semistable cubic surface over the coordinate ring of the base, then the fibration is a Mori fiber space and the total space has only Gorenstein singularities. This is an improvment over an existing result of Corti, whose version gave models with also Gorenstein but worse singularities. On the other hand, Corti’s method allowed to prove similar result for del Pezzo fibrations of degree 2. I will talk about extending the results of Kollar to del Pezzo surfaces of degree 2 and 1. As a corollary we improve Corti’s result and prove an analogue in degree 1.

Seminar in KIAS

August 2018, Seoul

Title:Birational geometry of rationally connected varieties.

Abstract:The talk is about the problem of classification of algebraic varieties. Rationally connected varieties are the most understood of the classes of algebraic varieties. In dimension 2 the classification is known but already in dimension three there are many problems to overcome: more cases and singularities. I will remind of the basic notions of the minimal model program and of some results on classification in dimension 2 and 3. Then I will introduce the notion of birational rigidity and of the good model and discuss strategy of dealing with the difficulties of dimension 3.

Research seminar in Baureuth University

April 2018, Bayreuth

Title:Stability over rings and good models of del Pezzo fibrations.

Abstract:This talk is motivated by the following problem, given a three-dimensional Mori fiber space, can we find a birational to it model with nice singularities? Sarkisov proved that for a conic bundle there exists a smooth model. For del Pezzo fibrations smooth model may not exist in case of degree <4. Corti has shown that there are Gorenstein (resp. 2-Gorenstein) models for del Pezzo fibrations of degree 3 (resp. 2). He proved it by constructing explicit birational maps improving singularities. Kollar improved his result in degree 3 using geometric invariant theory. I discuss what are the issues in adapting Kollar's approach for degrees 1 and 2 and how to work around them.

Seminar in Higher School of Economics

January 2018, Moscow

Title:Существование хороших моделей расслоений на поверхности дель Пеццо.

Abstract: Программа минимальных моделей позволяет найти хорошего представителя в бирациональном классе алгебраического многообразия. В размерности 3 и выше этот представитель, как правило, не единственен. Хотелось бы понять, какой представитель лучший и как его найти. Известно, что у расслоений на коники над поверхностями есть стандартная модель, которая, в частности, гладкая. Совместно с Максимом Федорчуком и Хамидом Ахмадинежадом я занимаюсь этим вопросом для расслоений на поверхности над кривой.
Геометрическая теория инвариантов позволяет построить многообразия классифицирующие другие объекты. Первый шаг – определить, какие именно объекты являются “хорошими”. Я расскажу, как использовать геометрическую теорию инвариантов для определения “хороших” расслоений на поверхности дель Пеццо и как построить бирациональное отображение в эту хорошую модель.

The Shokurovs: Workshop for birationalists

December 2017, Pohang

Title:Stability over rings and good models of del Pezzo fibrations

Abstract:This talk is motivated by the following problem, given a three-dimensional Mori fiber space, can we find a birational to it model with nice singularities? Sarkisov proved that for a cubic bundles there exists a smooth model. For del Pezzo fibrations smooth model may not exist in case degree <4. Corti has shown that there are Gorenstein (resp. 2-Gorenstein) models for del Pezzo fibrations of degree 3 (resp. 2). He proved it by constructing explicit birational maps improving singularities. Kollar improved his result in degree 3 using GIT. I discuss what are the issues in adapting Kollar's approach for degrees 1 and 2 and how to work around them.

Iskovskih seminar in Steklov Institute

September 2017, Moscow, Russia

Title: Stability over rings and good models of del Pezzo fibrations.

Abstract: This talk is motivated by the following problem, given a three-dimnsional Mori fiber space, can we find a birational to it model with nice singularities? Sarkisov proved that for a cubic bundles there exists a smooth model. For del Pezzo fibrations smooth model may not exist in case degree <4. Corti has shown that there are Gorenstein (resp. 2-Gorenstein) models for del Pezzo fibrations of degree 3 (resp. 2). He proved it by constructing explicit birational maps improving singularities. Kollar improved his result in degree 3 using GIT. I discuss what are the issues in adapting Kollar's approach for degrees 1 and 2 and how to work around them.

EDGE days

June 2017, Edinburgh

Title: Birational rigidity of orbifold del Pezzo fibrations

Geometry seminar in Loughborough University

May 2017, UK

Title: Rationality of del Pezzo fibrations and the Cremona group

Abstract: I will talk about embeddings of PSL_2(7) into the Cremona group of rank 3. Study of embeddings of a finite group G into the Cremona group is equivalent to study of G-equivariant birational geometry of rational GQ-Mori fiber spaces. Thus to classify embeddings of PSL_2(7) one has to classify rational PSL_2(7)Q-Mori fiber spaces up to PSL_2(7)-equivariant birational equivalence. I will discuss classification of PSL_2(7)Q-del Pezzo fibrations and their birational rigidity, in particular, rationality.