Yu. G. Prokhorov

Yu. G. Prokhorov
Russian Academy of Sciences | RAS · Steklov Mathematical Institute

PhD

About

235
Publications
7,025
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2,510
Citations
Additional affiliations
November 2010 - present
National Research University Higher School of Economics
Position
  • Research Associate
November 1989 - present
Lomonosov Moscow State University
Position
  • Professor (Full)

Publications

Publications (235)
Article
We prove rationality criteria over nonclosed fields of characteristic $0$ for five out of six types of geometrically rational Fano threefolds of Picard number $1$ and geometric Picard number bigger than $1$ . For the last type of such threefolds, we provide a unirationality criterion and construct examples of unirational but not stably rational var...
Article
We classify nonrational Fano threefolds X with terminal Gorenstein singularities such that rkPic(X)=1, (-KX)3≥8, and rkCl(X)≤2.
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We study del Pezzo varieties, higher-dimensional analogues of del Pezzo surfaces. In particular, we introduce ADE classification of del Pezzo varieties, show that in type A the dimension of non-conical del Pezzo varieties is bounded by $12 - d - r$, where $d$ is the degree and $r$ is the rank of the class group, and classify maximal del Pezzo varie...
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The automorphism groups of the Fano–Mukai fourfold of genus 10 were studied in our previous paper (Prokhorov and Zaidenberg in Eur J Math 4(3):1197–1263, 2018). In particular, we found the neutral components of these groups. In the present paper we finish the description of the discrete parts. Up to isomorphism, there are three special Fano–Mukai f...
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The paper is devoted to biography and scientific achievements of professor Victor Nikolaevich Latyshev
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We classify nonrational Fano threefolds $X$ with terminal Gorenstein singularities such that $\mathrm{\rk}\, \mathrm{\Pic}(X)=1$, $(-K_X)^3\ge 8$, and $\mathrm{\rk}\, \mathrm{\Cl}(X)\le 2$.
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We study embeddings of symmetric groups to the space Cremona group.
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We show that the Cremona group of rank $2$ over a finite field is Jordan, and provide an upper bound for its Jordan constant which is sharp when the number of elements in the field is different from $2$, $4$, and $8$.
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We obtain a sufficient condition for a Fano threefold with terminal singularities to have a conic bundle structure.
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We prove the Jordan property for groups of bimeromorphic selfmaps of three-dimensional compact K\"ahler varieties of zero Kodaira dimension, and for three-dimensional compact K\"ahler varieties of positive Kodaira dimension and irregularity.
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We classify quasi-simple finite groups of essential dimension 3.
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We survey new results on finite groups of birational transformations of algebraic varieties.
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We give a brief review on recent developments in the three-dimensional minimal model program.
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The automorphism groups of the Fano-Mukai fourfold of genus 10 were studied in our previous paper [arXiv:1706.04926]. In particular, we found in [arXiv:1706.04926] the neutral components of these groups. In the present paper we finish the description of the discrete parts. Up to isomorphism, there are two special Fano--Mukai fourfold of genus 10 wi...
Preprint
We prove rationality criteria over algebraically non-closed fields of characteristic $0$ for five out of six types of geometrically rational Fano threefolds of Picard number $1$ and geometric Picard number bigger than $1$. For the last type of such threefolds we provide a unirationality criterion and prove stable non-rationality under additional as...
Article
Let be a germ of a threefold with terminal singularities along a connected reduced complete curve with a contraction such that and is -ample. Assume that each irreducible component of contains at most one point of index 2}$> . We prove that a general member is a normal surface with Du Val singularities. Bibliography: 16 titles.
Article
The purpose of the survey is to systematize a vast amount of information about the minimal model program for varieties with group actions. We discuss the basic methods of the theory and give sketches of the proofs of some principal results. Bibliography: 243 titles.
Article
Пусть $(X, C)$ - росток трехмерного многообразия $X$ с терминальными особенностями вдоль связной приведенной полной кривой $C$, допускающего стягивание $f\colon (X, C) \to (Z, o)$ такое, что $C = f^{-1} (o)_{\mathrm{red}}$ и $-K_X$ является $f$-обильным. Предположим, что каждая неприводимая компонента $C$ содержит не более одной точки индекса $>2$....
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Цель обзора - систематизировать обширную информацию о программе минимальных моделей для многообразий с действиями группы. Обсуждаются основные методы теории. Даны наброски доказательств некоторых принципиальных результатов. Библиография: 243 названия.
Article
We classify compact complex surfaces whose groups of bimeromorphic selfmaps have bounded finite subgroups. We also prove that the stabilizer of a point in the automorphism group of a compact complex surface of zero Kodaira dimension, as well as the stabilizer of a point in the automorphism group of an arbitrary compact Kähler manifold of nonnegativ...
Article
Классифицируются компактные комплексные поверхности, группы бимероморфных автоморфизмов которых имеют ограниченные конечные подгруппы. Кроме того доказано, что стабилизатор точки в группе автоморфизмов компактной комплексной поверхности нулевой кодаировой размерности, а также стабилизатор точки в группе автоморфизмов любого компактного кэлерова мно...
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This paper is a survey about cylinders in Fano varieties and related problems.
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We classify del Pezzo surfaces with Du Val singularities that have infinite automorphism groups, and describe the connected components of their automorphisms groups.
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We show that the affine cones over any Fano--Mukai fourfold of genus 10 are flexible; in particular, the automorphism group of such a cone acts highly transitively outside the vertex. Furthermore, any Fano--Mukai fourfold of genus 10, with one exception, admits a covering by open charts isomorphic to the affine four-space.
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We discuss birational properties of Mukai varieties, i.e., of higher-dimensional analogues of prime Fano threefolds of genus $g \in \{7,8,9,10\}$ over an arbitrary field $\mathsf{k}$ of zero characteristic. In the case of dimension $n \ge 4$ we prove that these varieties are $\mathsf{k}$-rational if and only if they have a $\mathsf{k}$-point except...
Preprint
Let $(X, C)$ be a germ of a threefold $X$ with terminal singularities along a connected reduced complete curve $C$ with a contraction $f : (X, C) \to (Z, o)$ such that $C = f^{-1} (o)_{\mathrm{red}}$ and $-K_X$ is $f$-ample. Assume that each irreducible component of $C$ contains at most one point of index $>2$. We prove that a general member $D\in...
Article
Мы классифицируем трехмерные унилинейчатые компактные кэлеровы многообразия, группа бимероморфных автоморфизмов которых не обладает свойством Жордана. Библиография: 53 наименования.
Article
Мы классифицируем некоторые специальные классы трехмерных нерациональных многообразий Фано с терминальными особенностями. В частности, найдены все такие гиперэллиптические и тригональные многообразия.
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We give necessary and sufficient conditions for unirationality and rationality of Fano threefolds of geometric Picard rank-1 over an arbitrary field of zero characteristic.
Article
We classify some special classes of nonrational Fano threefolds with terminal singularities. In particular, all such hyperelliptic and trigonal varieties are found.
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We classify uniruled compact K\"ahler threefolds whose groups of bimeromorphic selfmaps do not have Jordan property.
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We classify compact complex surfaces whose groups of bimeromorphic selfmaps have bounded finite subgroups. We also prove that the stabilizer of a point in the automorphism group of a compact complex surface of zero Kodaira dimension, as well as the stabilizer of a point in the automorphism group of an arbitrary compact Kaehler manifold of non-negat...
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We classify some special classes of non-rational Fano threefolds with terminal singularities. In particular, all such hyperelliptic and trigonal varieties are found.
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We show that automorphism groups of Moishezon threefolds are always Jordan.
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We study the complexity of birational self-maps of a projective threefold $X$ by looking at the birational type of surfaces contracted. These surfaces are birational to the product of the projective line with a smooth projective curve. We prove that the genus of the curves occuring is unbounded if and only if $X$ is birational to a conic bundle or...
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We prove that $\mathbb{Q}$-Fano threefolds of Fano index $\ge 8$ are rational.
Article
Росток экстремальной окрестности - это аналитический росток трехмерного многообразия с терминальными особенностями вдоль приведенной полной кривой, допускающий стягивание, слои которого не более чем одномерны. Цель настоящей статьи - дать обзор результатов, касающихся стягиваний с неприводимым центральным слоем, содержащих только одну негоренштейно...
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We prove that automorphism groups of Inoue and primary Kodaira surfaces are Jordan.
Chapter
Let ( X , C ) be a germ of a threefold X with terminal singularities along an irreducible reduced complete curve C with a contraction f:(X,C)→(Z,o) such that C=f−1(o)red and − KX is f -ample. Assume that ( X , C ) contains a point of type (IIA). This chapter continues the study of such germs containing a point of type (IIA), started in our previous...
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We show that the only finite quasi-simple non-abelian groups that can faithfully act on rationally connected threefolds are the following groups: $\mathfrak{A}_5$, $\operatorname{PSL}_2(\mathbf{F}_7)$, $\mathfrak{A}_6$, $\operatorname{SL}_2(\mathbf{F}_8)$, $\mathfrak{A}_7$, $\operatorname{PSp}_4(\mathbf{F}_3)$, $\operatorname{SL}_2(\mathbf{F}_{7})$...
Preprint
An extremal curve germ is the analytic germ of a threefold with terminal singularities along a reduced complete curve admitting a contraction whose fibers have dimension at most one. The aim of the present paper is to review the results concerning those contractions whose central fiber is irreducible and contains only one non-Gorenstein point.
Article
This expository paper is concerned with the rationality problems for three-dimensional algebraic varieties with a conic bundle structure. We discuss the main methods of this theory. We sketch the proofs of certain principal results, and present some recent achievements. Many open problems are also stated.
Article
We give an explicit construction of prime Fano threefolds of genus 12 with a $G_m$-action, describe their isomorphism classes and automorphism groups. Comment: 14 pages, LaTeX, updated version, to appear in \'Epijournal de G\'eom\'etrie Alg\'ebrique, Vol. 2 (2018), Article Nr. 3
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We study automorphism groups and birational automorphism groups of compact complex surfaces. We show that the automorphism group of such surface $X$ is always Jordan, and the birational automorphism group is Jordan unless $X$ is birational to a product of an elliptic and a rational curve.
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Let $(X, C)$ be a germ of a threefold $X$ with terminal singularities along an irreducible reduced complete curve $C$ with a contraction $f: (X, C)\to (Z, o)$ such that $C=f^{-1}(o)_{red}$ and $-K_X$ is ample. Assume that $(X, C)$ contains a point of type (IIA). This paper continues our study of such germs containing a point of type (IIA) started i...
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It is known that the moduli space of smooth Fano-Mukai fourfolds $V_{18}$ of genus $10$ has dimension one. We show that any such fourfold is a completion of $\mathbb{C}^4$ in two different ways. Up to isomorphism, there is a unique fourfold $V_{18}^{\mathrm s}$ acted upon by $\operatorname{SL}_2(\mathbb{C})$. The group $\operatorname{Aut}(V_{18}^{\...
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We obtain upper bounds on the number of singular points of factorial terminal Fano threefolds.
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We classify del Pezzo surfaces of Picard number one with log canonical singularities admitting Q-Gorenstein smoothings.
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We determine three-dimensional algebraic varieties whose groups of birational selfmaps do not satisfy the Jordan property.
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We prove that if $X$ is a rationally connected threefold and $G$ is a $p$-subgroup in the group of birational selfmaps of $X$, then $G$ is an abelian group generated by at most $3$ elements provided that $p\ge 17$. We also prove a similar result for $p\ge 11$ under an assumption that $G$ acts on a (Gorenstein) $G$-Fano threefold, and show that the...
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We give explicit bounds for Jordan constants of groups of birational automorphisms of rationally connected threefolds over fields of zero characteristic, in particular, for Cremona groups of ranks 2 and 3.
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We discuss various results on Hilbert schemes of lines and conics and automorphism groups of smooth Fano threefolds with Picard rank 1. Besides a general review of facts well known to experts, the paper contains some new results, for instance, we give a description of the Hilbert scheme of conics on any smooth Fano threefold of index 1 and genus 10...
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The aim of this short note is to give a simple proof of the non-rationality of the double cover of the three-dimensional projective space branched over a sufficiently general quartic.
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We show that, for a $\mathbb Q$-Fano threefold $X$ of Fano index 7, the inequality $\dim |-K_X| \ge 15$ implies that $X$ is isomorphic to one of the following varieties $\mathbb P (1^2,2,3)$, $X_6 \subset \mathbb P (1,2^2,3,5)$ or $X_6 \subset \mathbb P (1,2,3^2,4)$.
Article
Let $(X, C)$ be a germ of a threefold $X$ with terminal singularities along an irreducible reduced complete curve $C$ with a contraction $f: (X, C)\to (Z, o)$ such that $C=f^{-1}(o)_{red}$ and $-K_X$ is ample. Assume that $(X, C)$ contains a point of type (IIA) and that a general member $H\in |O_X|$ containing $C$ is normal. We classify such germs...
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We study singular Fano threefolds of type $V_{22}$.
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We study Fano threefolds with~terminal singularities admitting a "minimal" action of a finite group. We prove that under certain additional assumptions such a variety does not contain planes. We also obtain an upper bounds of the number of singular points of certain Fano threefolds with~terminal factorial singularities.
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We prove that, except for a few cases, stable linearizability of finite subgroups of the plane Cremona group implies linearizability.
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In a previous paper we established that for any del Pezzo surface Y of degree at least 4, the affine cone X over Y embedded via a pluri-anticanonical linear system admits an effective Ga-action. In particular, the group Aut(X) is infinite dimensional. In contrast, we show in this note that for a del Pezzo surface Y of degree at most 2 the generaliz...
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We construct four different families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of the form (Formula presented.) , where Z is a quasiprojective variety. The affine cones over such a fourfold admit effective (Formula presented.) -actions. Similar constructions of cylindrical Fano threefolds were...
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We give a sharp bound for orders of elementary abelian 2-groups of birational automorphisms of rationally connected threefolds.
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Assuming Borisov--Alexeev--Borisov conjecture and Minimal Model Program, we prove that finite subgroups of the automorphism group of a finitely generated field over Q have bounded orders. Further, we investigate which algebraic varieties have groups of birational selfmaps satisfying the Jordan property.
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We classify Fano threefolds with only Gorenstein terminal singularities and Picard number greater than 1, satisfying the additional assumption that the G-invariant part of the Weil divisor class group is of rank 1 with respect to an action of some group G.
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We discuss the problem of stable conjugacy of finite subgroups of Cremona groups. We show that the group $H^1(G,Pic(X))$ is a stable birational invariant and compute this group in some cases.
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We classify -Fano threefolds of Fano index and sufficiently big degree. Bibliography: 20 titles.
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Let X be a rationally connected three-dimensional algebraic variety and let τ be an element of order two in the group of its birational selfmaps. Suppose that there exists a non-uniruled divisorial component of the τ-fixed point locus. Using the equivariant minimal model program we give a rough classification of such elements.
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An affine algebraic variety X is called cylindrical if it contains a principal Zariski dense open cylinder U ≃ Z × A1. A polarized projective variety (Y, H) is called cylindrical if it contains a cylinder U = Y \ supp D, where D is an effective ℚ-divisor on Y such that [D] ∈ ℚ+[H] in Picℚ(Y). We show that cylindricity of a polarized projective vari...
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Assuming Borisov--Alexeev--Borisov conjecture, we prove that there is a constant $J=J(n)$ such that for any rationally connected variety $X$ of dimension $n$ and any finite subgroup $G\subset Bir(X)$ there exists a normal abelian subgroup $A\subset G$ of index at most $J$. In particular, we obtain that the Cremona group $Cr_3=Bir(P^3)$ enjoys the J...
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Let $X$ be a rationally connected three-dimensional algebraic variety and let $\tau$ be an element of order two in the group of its birational selfmaps. Suppose that there exists a non-uniruled divisorial component of the $\tau$-fixed point locus. Using the equivariant minimal model program we give a rough classification of such elements.
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We show that, for a Q-Fano threefold X of Fano index 2, the inequality dim |-1/2K_X| <= 4 holds with a single well understood family of varieties having dim |-1/2K_X| = 4.

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