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.