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.

Algebraic geometry seminar in Higher School of Economics

Februray 2017, Moscow, Russia

Title: Stable rationality of del Pezzo fibrations.

Abstract: We say that f:X \to Z is a del Pezzo fibration if a generic fiber of f is a del Pezzo surface.

I will:

  • Discuss stable rationality of del Pezzo fibrations of low degree.
  • Show that a very general del Pezzo fibration of degrees 1,2, or 3, such that its anticanonical class is not ample, is not stably rational.
  • Make a small survey of known results on stable rationality and about how can one improve the result on del Pezzo fibrations in dimension three.
  • Discuss the mthod of reduction to finite characteristic and how to apply it for del Pezzo fibrations.

Shavarevich seminar in Steklov Institute

Februray 2017, Moscow, Russia

Title: Rationality of del Pezzo fibrations and the Cremona group

Abstract: I will talk about embeddings of PSL2(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 PSL2(7) one has to classify rational PSL2(7) Q-Mori fiber spaces up to PSL2(7)-equivariant birational equivalence.
I will discuss classification of PSL2(7) Q-del Pezzo fibrations and their birational rigidity, in particular, rationality.

Liverpool algebraic geometry seminar

Februray 2017, Moscow, Russia

Title: Birational geometry of del Pezzo fibrations

Abstract: Del Pezzo brations appear as minimal models of rationally connected varieties. The rationality of smooth del Pezzo brations is a well studied question but smooth brations are not dense in moduli. Little is known about the rationality of the singular models. We prove
birational rigidity, hence non-rationality, of del Pezzo brations with simple non-Gorenstein singularities satisfying the famous K2-condition. We then apply this result to study embeddings of PSL2(7) into the Cremona group.

Manchester geometry seminar

October 2016, Manchester, UK

Title: Rationally connected non Fano type varieties

Abstract: The class of varieties of Fano type is a generalization of Fano varieties which is very well behaved under the MMP. It is known that all varieties of Fano type are rationally connected. The converse is true in a sense in dimension 2. I will give counter examples to the converse statement in dimension 3 and higher and use the techniques of birational rigidity and MMP.

EDGE days

June, 2016, Edinburgh, UK

Title: Rationality and Stable rationality of Del Pezzo fibrations

Abstract: I consider stable rationality of double covers of Pm-bundles over projective spaces branched over divisors of high enough degree. In particular I study del Pezzo fibrations of degree 2 with a smooth total space. I prove that a very general del Pezzo fibration of degree 2 is not stably rational. Then I discuss the case of singular total space, which singularities does it make sense to consider, and discuss my non rationality result for this class of varieties.

Algebraic geometry seminar in John Hopkins University

March, 2016, Baltimore, USA

Title: Classification and birational rigidity of del Pezzo fibrations with an action of the Klein simple group

Abstract: Study of embeddings of a finite group G into the Cremona group is equivalent to study of G-birational geometry of rational G-Mori fiber spaces. A good place to start study finite subgroups is a study of simple subgroup. We prove that any del Pezzo fibration over projective line with an action of the Klein simple group is either a direct product or a certain singular del Pezzo fibration Xn of degree 2. It is known that del Pezzo fibrations of degree 2 satisfying the K2-condition are birationally superrigid. I extend this result to singular del Pezzo fibrations and prove that Xn are superrigid, in particular not rational, for n>2.

Algebraic geometry seminar in New-York University

March, 2016, New-York, USA

Title: Classification and birational rigidity of del Pezzo fibrations with an action of the Klein simple group

Abstract: Study of embeddings of a finite group G into the Cremona group is equivalent to study of G-birational geometry of rational G-Mori fiber spaces. A good place to start studying finite subgroups in Cremona group is a study of simple subgroups. I prove that any del Pezzo fibration over projective line with an action of the Klein simple group is either a direct product or a certain singular del Pezzo fibration Xn of degree 2. It is known that del Pezzo fibrations of degree 2 satisfying the K2-condition are birationally superrigid. I extend this result to singular del Pezzo fibrations and prove that Xn are superrigid, in particular not rational, for n>2.

Edinburgh Geometry seminar

March, 2016, Edinburgh, UK

Title: Classification and birational rigidity of del Pezzo fibrations with an action of the Klein simple group

Abstract: Cremona group of rank b is the group of birational transformations of the projective n-space. One was to study Cremona group is to study its finite subgroups. This problem can be translated to the geometric language: instead of subgroups of Cremona group isomorphic to a group G we can study rational G-Mori fiber spaces. This idea works particularly well for simple subgroups of Cremona group. I prove that any del Pezzo fibration over projective line with an action of the Klein simple group is either a direct product or a certain singular del Pezzo fibration Xn of degree 2. It is known that del Pezzo fibrations of degree 2 satisfying the K2-condition are birationally superrigid. I extend this result to singular del Pezzo fibrations and prove that Xn are superrigid, in particular not rational, for n>2.