arXiv subjects
Publications and source records attributed to Ilya Karzhemanov.
We prove that there exists a number field $\fie$ and a smooth projective $\mathrm{K3}$ surface $S_{22}$ (of genus $12$) over $\fie$ such that the geometric Picard number of $S_{22}$ is equal to $1$ and the $\fie$-rational points of $S_{22}$ are Zariski dense.
We provide a complete classification of Fano threefolds X having canonical Gorenstein singularities and the anticanonical degree (-KX)^3 equal 64.
This is an expository article, which contributes to the Proceedings of the conference "Groups of Automorphisms in Birational and Affine Geometry", held in Trento in 2012. We propose that (rational) fibrations on the projective space $\p^n$ by (birationally) Abelian hypersurfaces, for an arbitrary $n\ge 2$, provide an obstruction to stable rationality of algebraic varieties. We discuss the evidence for this proposition and derive some (almost straightforward) corollaries from it.
We study unirational algebraic varieties and the fields of rational functions on them. We show that after adding a finite number of variables some of these fields admit an infinitely transitive model. The latter is an algebraic variety with the given field of rational functions and an infinitely transitive regular action of a group of algebraic automorphisms generated by unipotent algebraic subgroups. We expect that this property holds for all unirational varieties and in fact is a peculiar one for this class of algebraic varieties among those varieties which are rationally connected.
We study rationality problem for the quotient of $\mathbb{C}^4$ by a finite primitive group $G$ of Type (I). We prove that this quotient is a rational variety for any such $G$.
Smooth primitively polarized $\mathrm{K3}$ surfaces of genus 36 are studied. It is proved that all such surfaces $S$, for which there exists an embedding $\mathrm{R} \hookrightarrow \mathrm{Pic}(S)$ of some special lattice $\mathrm{R}$ of rank 2, are parameterized up to an isomorphism by some 18-dimensional unirational algebraic variety. More precisely, it is shown that a general $S$ is an anticanonical section of a (unique) Fano 3-fold with canonical Gorenstein singularities.
Let $(X,D)$ be log canonical pair such $\dim X = 3$ and the divisor $-(K_X + D)$ is nef and big. For a special class of such $(X,D)$'s we prove that the linear system $|-n(K_{X}+D)|$ is free for $n \gg 0$.
For every smooth quartic threefold, we classify all pencils on it whose general element is an irreducible surface birational to a smooth surface of Kodaira dimension zero.
We give a classification of Fano threefolds $X$ with canonical Gorenstein singularities such that $X$ possess a regular involution, which acts freely on some smooth surface in $|-K_X|$, and the linear system $|-K_X|$ gives a morphism which is not an embedding. From this classification one gets, in particular, a description of some natural class of Fano--Enriques threefolds.
We classify three-dimensional Fano varieties with canonical Gorenstein singularities of degree bigger than 64.