Yuri Prokhorov - Academia.edu (original) (raw)

Papers by Yuri Prokhorov

arXiv: Algebraic Geometry, May 2, 2019

We study the complexity of birational self-maps of a projective threefold X by looking at the bir... more 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 a fibration into cubic surfaces. Similarly, we prove that the gonality of the curves is unbounded if and only if X is birational to a conic bundle. Contents 1. Introduction 1 2. The case of conic bundles 4 3. Reminders on the Sarkisov program 5 4. Bounding the gonality and genus of curves 9 5. Del Pezzo fibrations of degree 3 14 References 16

A conic bundle is a proper flat morphism π : X → Z of non-singular varieties such that it is of r... more A conic bundle is a proper flat morphism π : X → Z of non-singular varieties such that it is of relative dimension 1 and the anticanonical divisor −KX is relatively ample. We say that a variety X has a conic bundle structure if there exists a conic bundle π : X ′ → Z and a birational map X 99K X . Varieties with conic bundle structure play a very important role in the birational classification of algebraic varieties of negative Kodaira dimension. For example, any variety with rational curve fibration has a conic bundle structure [Sar82]. For these varieties there is well-developed techniques to solve rationality problems [Sar82], [Sho84], [Isk87], [Pro18]. Another important class of varieties of negative Kodaira dimension is the class of Q-Fano varieties. Recall that a projective variety X is called Q-Fano if it has only terminal Qfactorial singularities, the Picard number ρ(X) equals 1, and the anticanonical class −KX is ample. In fact, these two classes overlap. Moreover, Q-Fano v...

Algebraic Geometry, 2021

We classify del Pezzo surfaces with Du Val singularities that have infinite automorphism groups, ... more We classify del Pezzo surfaces with Du Val singularities that have infinite automorphism groups, and describe the connected components of their automorphisms groups. Throughout this paper, we always assume that all varieties are projective and defined over an algebraically closed field k of characteristic 0. 2. Del Pezzo surfaces with Du Val singularities Let X be a Du Val del Pezzo surface with d := K 2 X. Then d is known as the degree of the surface X. Let µ : X → X be the minimal resolution of singularities. Then K X ∼ µ * K X , so that X is a weak del Pezzo surface, that is, the anticanonical divisor −K X is nef and big. By the Noether formula d = 10 − ρ(X) 9 and by the genus formula every irreducible curve on X with negative self-intersection number is either (−1

We give a brief review on recent developments in the three-dimensional minimal model program. In ... more We give a brief review on recent developments in the three-dimensional minimal model program. In this note we give a brief review on recent developments in the three-dimensional minimal model program (MMP for short). Certainly, this is not a complete survey of all advances in this area. For example, we do not discuss the minimal models of varieties of non-negative Kodaira dimension, as well as, applications to birational geometry and moduli spaces. The aim of the MMP is to find a good representative in a fixed birational equivalence class. Starting with an arbitrary smooth projective variety one can perform a finite number of certain elementary transformations, called divisorial contractions and flips, and at the end obtain a variety which is simpler in some sense. Most parts of the MMP are completed in arbitrary dimension. One of the basic remaining problems is the following: Describe all the intermediate steps and the outcome of the MMP. The MMP makes sense only in dimensions ≥ 2 ...

Izvestiya: Mathematics, 2020

We classify uniruled compact Kähler threefolds whose groups of bimeromorphic selfmaps do not have... more We classify uniruled compact Kähler threefolds whose groups of bimeromorphic selfmaps do not have the Jordan property.

Proceedings of the Steklov Institute of Mathematics, 2019

We classify some special classes of non-rational Fano threefolds with terminal singularities. In ... more We classify some special classes of non-rational Fano threefolds with terminal singularities. In particular, all such hyperelliptic and trigonal varieties are found.

We show that the Cremona group of rank 2 over a finite field is Jordan, and provide an upper boun... more 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.

Mathematical Notes, 2019

We study automorphism groups of Moishezon threefolds and show that such groups are always Jordan.

Mathematical Society of Japan Memoirs, 2001

Chapter 0. Introduction Chapter 1. Preliminary results 1.1. Singularities of pairs 1.2. Finite mo... more Chapter 0. Introduction Chapter 1. Preliminary results 1.1. Singularities of pairs 1.2. Finite morphisms and singularities of pairs 1.3. Log canonical covers Chapter 2. Inversion of adjunction 2.1. Two-dimensional toric singularities and log canonical singularities with a reduced boundary 2.2. Adjunction 2.3. Connectedness Lemma Chapter 3. Log terminal modifications 3.1. Log terminal modifications 3.2. Weighted blowups 3.3. Generalizations of Connectedness Lemma Chapter 4. Definition of complements and elementary properties 4.1. Introduction 4.2. Monotonicity 4.3. Birational properties of complements 4.4. Inductive properties of complements 4.5. Exceptionality Chapter 5. Log del Pezzo surfaces 5.1. Definitions and examples 5.2. Boundedness of log del Pezzos 5.3. On the existence of regular complements 5.4. Nonrational log del Pezzo surfaces Chapter 6. Birational contractions and two-dimensional log canonical singularities 6.1. Classification of two-dimensional log canonical singularities 6.2. Two-dimensional log terminal singularities as quotients ii CONTENTS iii Chapter 7. Contractions onto curves 7.1. Log conic bundles 7.2. Elliptic fibrations Chapter 8. Inductive complements 8.1. Examples 8.2. Nonrational case 8.3. The Main Inductive Theorem 8.4. Corollaries 8.5. Characterization of toric surfaces Chapter 9. Boundedness of exceptional complements 9.1. The main construction 9.2. Corollaries: Case of log Enriques surfaces 9.3. On the explicit bound of exceptional complements Chapter 10. On classification of exceptional complements: case δ ≥ 1 10.1. The inequlity δ ≤ 2 10.2. Case δ = 2 10.3. Examples Chapter 11. Appendix 11.1. Existence of complements 11.2. Minimal Model Program in dimension two Bibliography Index iv CONTENTS CHAPTER 0 * The proper transform is sometimes also called the birational or strict transform. † Note that our definition of ε-lt pairs is weaker than that given by Alexeev [A]: we do not claim that −d i > −1 + ε. * Shokurov pointed out that the stronger version of inequality (2.2) should be Weil(X)/Weil 0 ≥ d i − dim X, where Weil 0 ⊂ Weil(X) is the subgroup of all numerically trivial over Z (and Q-Cartier) divisors.

Sbornik: Mathematics, 2020

We classify compact complex surfaces whose groups of bimeromorphic selfmaps have bounded finite s... more 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 nonnegative Kodaira dimension, always has bounded finite subgroups. Bibliography: 23 titles.

Izvestiya: Mathematics, 2020

We classify uniruled compact Kähler threefolds whose groups of bimeromorphic selfmaps do not have... more We classify uniruled compact Kähler threefolds whose groups of bimeromorphic selfmaps do not have the Jordan property.

Compositio Mathematica, 2014

Assuming a particular case of the Borisov–Alexeev–Borisov conjecture, we prove that finite subgro... more Assuming a particular case of the Borisov–Alexeev–Borisov conjecture, we prove that finite subgroups of the automorphism group of a finitely generated field over mathbbQ\mathbb{Q}mathbbQ have bounded orders. Further, we investigate which algebraic varieties have groups of birational selfmaps satisfying the Jordan property. Unless explicitly stated otherwise, all varieties are assumed to be algebraic, geometrically irreducible and defined over an arbitrary field Bbbk\BbbkBbbk of characteristic zero.

Mathematical Notes, 2019

We study automorphism groups of Moishezon threefolds and show that such groups are always Jordan.

Sbornik Mathematics, 1995

String theories with branes can often be generalized by adding braneantibrane pairs. We explore t... more String theories with branes can often be generalized by adding braneantibrane pairs. We explore the cancellation of anomalies in this more general context, extending the familiar anomaly-cancelling mechanisms, both for ten-dimensional string theories with D-branes and for certain supersymmetric compactifications.

Tohoku Mathematical Journal, 2001

We verify a special case of V. V. Shokurov's conjecture about characterization of toric varieties... more We verify a special case of V. V. Shokurov's conjecture about characterization of toric varieties. More precisely, let (X, D = d i D i) be a threedimensional log variety such that K X + D is numerically trivial and (X, D) has only purely log terminal singularities. In this situation we prove the inequality d i ≤ rk Weil(X)/(algebraic equivalence) + dim(X). We describe such pairs for which the equality holds and show that all of them are toric.

Sbornik: Mathematics, 2007

Let U ⊂ P N be a projective variety which is not a cone and whose hyperplane sections are smooth ... more Let U ⊂ P N be a projective variety which is not a cone and whose hyperplane sections are smooth Enriques surfaces. We prove that the degree of such U is at most 32 and the bound is sharp.

arXiv: Algebraic Geometry, May 2, 2019

We study the complexity of birational self-maps of a projective threefold X by looking at the bir... more 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 a fibration into cubic surfaces. Similarly, we prove that the gonality of the curves is unbounded if and only if X is birational to a conic bundle. Contents 1. Introduction 1 2. The case of conic bundles 4 3. Reminders on the Sarkisov program 5 4. Bounding the gonality and genus of curves 9 5. Del Pezzo fibrations of degree 3 14 References 16

A conic bundle is a proper flat morphism π : X → Z of non-singular varieties such that it is of r... more A conic bundle is a proper flat morphism π : X → Z of non-singular varieties such that it is of relative dimension 1 and the anticanonical divisor −KX is relatively ample. We say that a variety X has a conic bundle structure if there exists a conic bundle π : X ′ → Z and a birational map X 99K X . Varieties with conic bundle structure play a very important role in the birational classification of algebraic varieties of negative Kodaira dimension. For example, any variety with rational curve fibration has a conic bundle structure [Sar82]. For these varieties there is well-developed techniques to solve rationality problems [Sar82], [Sho84], [Isk87], [Pro18]. Another important class of varieties of negative Kodaira dimension is the class of Q-Fano varieties. Recall that a projective variety X is called Q-Fano if it has only terminal Qfactorial singularities, the Picard number ρ(X) equals 1, and the anticanonical class −KX is ample. In fact, these two classes overlap. Moreover, Q-Fano v...

Algebraic Geometry, 2021

We classify del Pezzo surfaces with Du Val singularities that have infinite automorphism groups, ... more We classify del Pezzo surfaces with Du Val singularities that have infinite automorphism groups, and describe the connected components of their automorphisms groups. Throughout this paper, we always assume that all varieties are projective and defined over an algebraically closed field k of characteristic 0. 2. Del Pezzo surfaces with Du Val singularities Let X be a Du Val del Pezzo surface with d := K 2 X. Then d is known as the degree of the surface X. Let µ : X → X be the minimal resolution of singularities. Then K X ∼ µ * K X , so that X is a weak del Pezzo surface, that is, the anticanonical divisor −K X is nef and big. By the Noether formula d = 10 − ρ(X) 9 and by the genus formula every irreducible curve on X with negative self-intersection number is either (−1

We give a brief review on recent developments in the three-dimensional minimal model program. In ... more We give a brief review on recent developments in the three-dimensional minimal model program. In this note we give a brief review on recent developments in the three-dimensional minimal model program (MMP for short). Certainly, this is not a complete survey of all advances in this area. For example, we do not discuss the minimal models of varieties of non-negative Kodaira dimension, as well as, applications to birational geometry and moduli spaces. The aim of the MMP is to find a good representative in a fixed birational equivalence class. Starting with an arbitrary smooth projective variety one can perform a finite number of certain elementary transformations, called divisorial contractions and flips, and at the end obtain a variety which is simpler in some sense. Most parts of the MMP are completed in arbitrary dimension. One of the basic remaining problems is the following: Describe all the intermediate steps and the outcome of the MMP. The MMP makes sense only in dimensions ≥ 2 ...

Izvestiya: Mathematics, 2020

We classify uniruled compact Kähler threefolds whose groups of bimeromorphic selfmaps do not have... more We classify uniruled compact Kähler threefolds whose groups of bimeromorphic selfmaps do not have the Jordan property.

Proceedings of the Steklov Institute of Mathematics, 2019

We classify some special classes of non-rational Fano threefolds with terminal singularities. In ... more We classify some special classes of non-rational Fano threefolds with terminal singularities. In particular, all such hyperelliptic and trigonal varieties are found.

We show that the Cremona group of rank 2 over a finite field is Jordan, and provide an upper boun... more 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.

Mathematical Notes, 2019

We study automorphism groups of Moishezon threefolds and show that such groups are always Jordan.

Mathematical Society of Japan Memoirs, 2001

Chapter 0. Introduction Chapter 1. Preliminary results 1.1. Singularities of pairs 1.2. Finite mo... more Chapter 0. Introduction Chapter 1. Preliminary results 1.1. Singularities of pairs 1.2. Finite morphisms and singularities of pairs 1.3. Log canonical covers Chapter 2. Inversion of adjunction 2.1. Two-dimensional toric singularities and log canonical singularities with a reduced boundary 2.2. Adjunction 2.3. Connectedness Lemma Chapter 3. Log terminal modifications 3.1. Log terminal modifications 3.2. Weighted blowups 3.3. Generalizations of Connectedness Lemma Chapter 4. Definition of complements and elementary properties 4.1. Introduction 4.2. Monotonicity 4.3. Birational properties of complements 4.4. Inductive properties of complements 4.5. Exceptionality Chapter 5. Log del Pezzo surfaces 5.1. Definitions and examples 5.2. Boundedness of log del Pezzos 5.3. On the existence of regular complements 5.4. Nonrational log del Pezzo surfaces Chapter 6. Birational contractions and two-dimensional log canonical singularities 6.1. Classification of two-dimensional log canonical singularities 6.2. Two-dimensional log terminal singularities as quotients ii CONTENTS iii Chapter 7. Contractions onto curves 7.1. Log conic bundles 7.2. Elliptic fibrations Chapter 8. Inductive complements 8.1. Examples 8.2. Nonrational case 8.3. The Main Inductive Theorem 8.4. Corollaries 8.5. Characterization of toric surfaces Chapter 9. Boundedness of exceptional complements 9.1. The main construction 9.2. Corollaries: Case of log Enriques surfaces 9.3. On the explicit bound of exceptional complements Chapter 10. On classification of exceptional complements: case δ ≥ 1 10.1. The inequlity δ ≤ 2 10.2. Case δ = 2 10.3. Examples Chapter 11. Appendix 11.1. Existence of complements 11.2. Minimal Model Program in dimension two Bibliography Index iv CONTENTS CHAPTER 0 * The proper transform is sometimes also called the birational or strict transform. † Note that our definition of ε-lt pairs is weaker than that given by Alexeev [A]: we do not claim that −d i > −1 + ε. * Shokurov pointed out that the stronger version of inequality (2.2) should be Weil(X)/Weil 0 ≥ d i − dim X, where Weil 0 ⊂ Weil(X) is the subgroup of all numerically trivial over Z (and Q-Cartier) divisors.

Sbornik: Mathematics, 2020

We classify compact complex surfaces whose groups of bimeromorphic selfmaps have bounded finite s... more 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 nonnegative Kodaira dimension, always has bounded finite subgroups. Bibliography: 23 titles.

Izvestiya: Mathematics, 2020

We classify uniruled compact Kähler threefolds whose groups of bimeromorphic selfmaps do not have... more We classify uniruled compact Kähler threefolds whose groups of bimeromorphic selfmaps do not have the Jordan property.

Compositio Mathematica, 2014

Assuming a particular case of the Borisov–Alexeev–Borisov conjecture, we prove that finite subgro... more Assuming a particular case of the Borisov–Alexeev–Borisov conjecture, we prove that finite subgroups of the automorphism group of a finitely generated field over mathbbQ\mathbb{Q}mathbbQ have bounded orders. Further, we investigate which algebraic varieties have groups of birational selfmaps satisfying the Jordan property. Unless explicitly stated otherwise, all varieties are assumed to be algebraic, geometrically irreducible and defined over an arbitrary field Bbbk\BbbkBbbk of characteristic zero.

Mathematical Notes, 2019

We study automorphism groups of Moishezon threefolds and show that such groups are always Jordan.

Sbornik Mathematics, 1995

String theories with branes can often be generalized by adding braneantibrane pairs. We explore t... more String theories with branes can often be generalized by adding braneantibrane pairs. We explore the cancellation of anomalies in this more general context, extending the familiar anomaly-cancelling mechanisms, both for ten-dimensional string theories with D-branes and for certain supersymmetric compactifications.

Tohoku Mathematical Journal, 2001

We verify a special case of V. V. Shokurov's conjecture about characterization of toric varieties... more We verify a special case of V. V. Shokurov's conjecture about characterization of toric varieties. More precisely, let (X, D = d i D i) be a threedimensional log variety such that K X + D is numerically trivial and (X, D) has only purely log terminal singularities. In this situation we prove the inequality d i ≤ rk Weil(X)/(algebraic equivalence) + dim(X). We describe such pairs for which the equality holds and show that all of them are toric.

Sbornik: Mathematics, 2007

Let U ⊂ P N be a projective variety which is not a cone and whose hyperplane sections are smooth ... more Let U ⊂ P N be a projective variety which is not a cone and whose hyperplane sections are smooth Enriques surfaces. We prove that the degree of such U is at most 32 and the bound is sharp.