Iskovskih seminar in Steklov Institute

November 2019, Moscow

Title: The variety of semistable del Pezzo surfaces

Abstract:
Kollar introduced a generalization of GIT stability for hypersurfaces: stability over rings. This notion is useful for finding good reductions to finite characterstic and for finding good (semistable) birational models of fibrations. I will talk about extending this notion to del Pezzo surfaces of degree 1 and 2. These surfaces are hypersurfaces in weighted projective spaces, where GIT techniques do not work. On the other hand embeddings into the projective spaces are not complete intersections and are difficult to work with. I will talk about generalizations of stability in both of these settings. In particular, I will talk about parameter spaces of del Pezzo surfaces of degrees 1 and 2

Workshop on Algebraic Geometry in Gunsan

October 2018, Gunsan

Title: Stability of del Pezzo surfaces over rings

Abstract:
Kollar introduced a generalization of GIT stability for hypersurfaces: stability over rings. This notion is useful for finding good reductions to finite characterstic and for finding good (semistable) birational models of fibrations. I will talk about extending this notion to del Pezzo surfaces of degree 1 and 2. These surfaces are hypersurfaces in weighted projective spaces, where GIT techniques do not work. On the other hand embeddings into the projective spaces are not complete intersections and are difficult to work with. I will talk about generalizations of stability in both of these settings. In particular, I will talk about parameter spaces of del Pezzo surfaces of degrees 1 and 2.

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.