Шрамов Константин Александрович

Three embeddings of the Klein simple group into the Cremona group of rank three

2012 · ARTICLE · en

We study the action of the Klein simple group PSL2(F7 ) consisting of 168 elements on two rational threefolds: the three-dimensional projective space and a smooth Fano threefold X of anticanonical degree 22 and index 1. We show that the Cremona group of rank three has at least three non-conjugate subgroups isomorphic to PSL2(F7 ). As a by-product, we prove that X admits a Kähler–Einstein metric, and we construct a smooth polarized K3 surface of degree 22 with an action of the group PSL2( F7 ). Unless explicitly stated otherwise, varieties are assumed to be projective, normal and complex.

Nine-dimensional exceptional quotient singularities exist

2012 · CHAPTER · en

We prove that nine-dimensional exceptional quotient singularities exist.

On Unitary Submodules in the Polynomial Representations of Rational Cherednik Algebras

2012 · ARTICLE · en

We consider representations of rational Cherednik algebras that are particular ideals in the ring of polynomials. We investigate convergence of the integrals that express the Gaussian inner product on these representations. We derive that the integrals converge for the minimal submodules in types B and D for the singular values suggested by Cherednik with at most one exception; hence the corresponding modules are unitary. The analogous result on unitarity of the minimal submodules in type A was obtained by Etingof and Stoica; we give a different proof of convergence of the Gaussian product in this case. We also obtain partial results on unitarity of the minimal submodule in the case of exceptional Coxeter groups and group B with unequal parameters.

Jordan property for Cremona groups

2012 · PREPRINT · en

Six-dimensional exceptional quotient singularities

2011 · ARTICLE · en

We classify six-dimensional exceptional quotient singularities and show that seven-dimensional exceptional quotient singularities do not exist. Inter alia we prove that the irreducible six-dimensional projective representation of the sporadic simple Hall–Janko group gives rise to an exceptional quotient singularity.

On exceptional quotient singularities

2011 · ARTICLE · en

Weakly-exceptional singularities in higher dimensions

2011 · PREPRINT · en

We show that infinitely many Gorenstein weakly-exceptional quotient singularities exist in all dimensions, we prove a weak-exceptionality criterion for five-dimensional quotient singularities, and we find a sufficient condition for being weakly-exceptional for six-dimensional quotient singularities. The proof is naturally linked to various classical geometrical constructions related to subvarieties of small degree in projective spaces, in particular Bordiga surfaces and Bordiga threefolds.

Exceptional del Pezzo hypersurfaces

2010 · ARTICLE · en

О бирациональной жесткости и Q-факториальности особого двойного накрытия квадрики с ветвлением в дивизоре степени 4

2008 · ARTICLE · ru

Лог-канонические пороги неособых трехмерных многообразий Фано

2008 · ARTICLE · ru