Ilya Karzhemanov

• PhD
• Professor (Associate) at Moscow Institute of Physics and Technology

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 aut...

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 $G$.

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 prove that the moduli space of smooth primitively polarized $\mathrm{K3}$ surfaces of genus 33 is unirational.

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 $\mathrm{R} \hookrightarrow H^{2}(S, \mathbb{Z})$ of some special lattice $\mathrm{R}$ of rank 2, are parameterized up to isomorphism by some 18-dimensional unirational algebraic variety. It is al...

Given a rational dominant map \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi : Y \dashrightarrow X$$\end{document} between two generic hypersurfaces \documentclas...

For a very general complex projective K3 surface S and a smooth projective surface A with trivial canonical class, we prove that there is no dominant rational map A → S, which is not an isomorphism.

We study the unirationality property of an algebraic variety X (over C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {C}$$\end{document}) versus the so-called...

We study surjective (not necessarily regular) rational endomorphisms $f$ of smooth del Pezzo surfaces $X$. We prove that under certain natural non\,-\,degeneracy condition $f$ can have degree bigger than $1$ only when $(-K_X^2) > 5$. Some structural properties of $f$ in the case $X = \p^2$ are also established.

We provide an explicit description of exceptional collection of maximal length in the derived category D b (Y) for a particular class of elliptic surfaces Y. The existence of non-trivial semiorthogonal complement (a phantom) of this collection is also established.

With any \emph{surjective rational map} $f: \p^n \dashrightarrow \p^n$ of the projective space we associate a numerical invariant (\emph{ML degree}) and compute it in terms of a naturally defined vector bundle $E_f \map \p^n$.

We prove rationality of the quotient \(\mathbb {C}^n / H_n\) for the finite Heisenberg group \(H_n\), any \(n \ge 1\), acting on \(\mathbb {C}^n\) via its irreducible representation.KeywordsHeisenberg groupQuotientLog pairToric varietyMS 2020 classification14E0814M2514E3014J81

In this note, we propose a projective-geometric version of quantization, based on the surjective rational self-maps of Pn. Relations with the classical subjects via several examples and key properties of these maps are provided.

Let $f: \mathbb{P}^3 \longrightarrow \mathbb{P}^3$ be a morphism given by the linear system $\mathcal{L}$ of quadrics. Using geometry of the Jacobian surface $\widetilde{S}$ associated with $\mathcal{L}$, we show that if $\widetilde{S}$ is smooth, then $f$ is not gradient (that is $f \ne \mathrm{grad}\,F$ for any cubic polynomial $F$).

Given a rational dominant map $\phi: Y \dashrightarrow X$ between two generic hypersurfaces $Y,X \subset \mathbb{P}^n$ of dimension $\ge 3$, we prove (under an addition assumption on $\phi$) a ``Noether--Fano type'' inequality $m_Y \ge m_X$ for certain (effectively computed) numerical invariants of $Y$ and $X$.

We prove rationality of the quotient C n /Hn for the finite Heisen-berg group H n , any n ≥ 1, acting on C n via its irreducible representation.

On the projective plane there is a unique cubic root of the canonical bundle and this root is acyclic. On fake projective planes such root exists and is unique if there are no 3-torsion divisors (and usually exists, but not unique, otherwise). Earlier we conjectured that any such cubic root must be acyclic. In the present note we give two short pro...

We prove that generic complex projective $\mathrm{K3}$ surface $S$ does not admit a dominant rational map $A \dasharrow S$, which is not an isomorphism, from a surface $A$ with trivial canonical class.

We compute irregularity of some algebraic surfaces which compactify a ball quotient.

We prove that there exist rational but not uniformly rational smooth algebraic varieties. The proof is based on computing certain numerical obstruction developed in the case of compactifications of affine spaces. We show that for some particular compactifications this obstruction behaves differently compared to the uniformly rational situation.

We construct normal rationally connected varieties (of arbitrarily large dimension) not containing any smooth rational curves.

We consider the so-called surjective rational maps. We study how the surjectivity property behaves in families of rational maps. Some (counter) examples are provided and a general result is proved.

We construct examples of Fano manifolds which are defined over a field of positive characteristic but not over C.

For any prime $p\ge 5$, we show that generic hypersurface $X_p\subset\p^p$ admits a non-trivial (rational) dominant self-map, of degree $>1$. A simple arithmetic application of this fact is also given.

We provide a complete classification of Fano threefolds $X$ with canonical Gorenstein singularities such that $(-K_{X})^{3} \geqslant 64$.

We show that the above-named property (after M. Larsen and V. Lunts) does not hold in general.

We prove that there exist rational but not uniformly rational smooth algebraic varieties. The proof is based on computing certain numerical obstruction, which we apply to compactifications of affine spaces. We show that in one particular case this obstruction behaves differently, compared to the uniformly rational situation.

We complete the study of rationality problem for hypersurfaces $X_t\subset\p^4$ of degree $4$ invariant under the action of the symmetric group $S_6$.

This is an expository article in which 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 the conjecture due to V.\,V. Shokurov on characterization of toric varieties. We also consider one generalization of this conjecture. It is shown that none of the characterizations holds in dimensions $\ge 3$ (in addition, we comment on the recent paper \cite{brown}, claiming a ``proof" of the conjecture, in the Appendix at the end). Some...

The main object of study of the present paper is the group $\au_n$ of \emph{unimodular automorphisms} of $\com^n$. Taking $\au_n$ as a working example, our intention was to develop an approach (or rather an edifice) which allows one to prove, for instance, the non-simplicity of $\au_n$ for all $n \geq 3$. More systematic and, perhaps, general expos...

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...

For any smooth quartic threefold in P4 we classify pencils on it whose general element is an irreducible surface birational to a surface of Kodaira dimension zero.

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 $n \gg 0$. Comment: 8 pages; the main result was changed and the proof was corrected

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...

We classify three-dimensional Fano varieties with canonical Gorenstein singularities of degree bigger than 64.