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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... 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...
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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... 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 ...
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We classify uniruled compact Kähler threefolds whose groups of bimeromorphic selfmaps do not have the Jordan property.
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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 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... 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.
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We classify uniruled compact Kähler threefolds whose groups of bimeromorphic selfmaps do not have the Jordan property.
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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 $\mathbb{Q}$ have bounded orders. Further, we investigate which algebraic... 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 $\mathbb{Q}$ 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 $\Bbbk$ of characteristic zero.
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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 $\mathbb{Q}$ have bounded orders. Further, we investigate which algebraic... 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 $\mathbb{Q}$ 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 $\Bbbk$ of characteristic zero.
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This paper continues the author's study of extremal contractions in the sense of Mori from three-dimensional varieties onto surfaces. Such contractions occur in a natural way in the birational classification theory of... more
This paper continues the author's study of extremal contractions in the sense of Mori from three-dimensional varieties onto surfaces. Such contractions occur in a natural way in the birational classification theory of three-dimensional algebraic varieties. Reid's “general elephant” conjecture of the complementedness of the canonical divisor and also the conjecture about singularities of the base surface are discussed. The situation
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Without Abstract
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We prove the boundedness of complements modulo two conjectures: Borisov–Alexeev conjecture and effective adjunction for fibre spaces. We discuss the last conjecture and prove it in two particular cases.
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We consider KX-negative extremal contractions f:x→(Z,o), where X is an algebraic threefold with only ε-log terminal ℚ-factorial singularities and (Z,o) is a two (respectively, one)-dimensional germ. The main result is that KX is 1, 2, 3,... more
We consider KX-negative extremal contractions f:x→(Z,o), where X is an algebraic threefold with only ε-log terminal ℚ-factorial singularities and (Z,o) is a two (respectively, one)-dimensional germ. The main result is that KX is 1, 2, 3, 4 or 6-complementary or we have, so-called, exceptional case and then the singularity (Z∈o) is bounded (respectively, the multiplicity of the central fiber f-1(o) is bounded).
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We survey new results on finite groups of birational transformations of algebraic varieties. Primary 14E07; Secondary 14J50, 14J45, 14E30 Cremona group, birational transformation, Fano variety, Minimal Model Program
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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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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 −K X is f-ample. Assume that (X, C) contains a point of type (IIA).... more
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 −K X 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 paper.
The automorphism groups of the Fano–Mukai fourfold of genus 10 were studied in our previous paper [PZ18]. In particular, we found in [PZ18] the neutral components of these groups. In the present paper we finish the description of the... more
The automorphism groups of the Fano–Mukai fourfold of genus 10 were studied in our previous paper [PZ18]. In particular, we found in [PZ18] 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 with the automorphism groups GL2(k)⋊Z/2Z and (Ga×Gm)⋊Z/2Z, respectively. For any other Fano–Mukai fourfold V of genus 10 one has Aut(V ) = G m ⋊ Z/2Z, except for exactly one of them with Aut(V ) = G m ⋊ Z/6Z.
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We prove the boundedness of complements modulo two conjectures: Borisov-Alexeev conjecture and effective adjunction for fibre spaces. We discuss the last conjecture and prove it in two particular cases.
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We produce new families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of form Z × A 1 , where Z is a quasiprojective variety. The affine cones over such a fourfold admit effective G... more
We produce new families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of form Z × A 1 , where Z is a quasiprojective variety. The affine cones over such a fourfold admit effective G a-actions. Similar constructions of cylindrical Fano threefolds and fourfolds were done previously in [KPZ11, KPZ14, PZ15].
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We study Q-Fano threefolds of large Fano index. In particular, we prove that the maximum of Fano index is attained for the weighted projective space P(3,4,5,7).
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We discuss different generalizations of Zariski decomposition, relations between them and connections with finite generation of divisorial algebras.
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Let G be a nite group of order n and let k be a eld. Consider a rational (i.e., pure transcendental) extension K/k of transcendence degree n. We may assume that K = k({xg}), where g runs through all the elements of G. The group G... more
Let G be a nite group of order n and let k be a eld. Consider a rational (i.e., pure transcendental) extension K/k of transcendence degree n. We may assume that K = k({xg}), where g runs through all the elements of G. The group G naturally acts on K via h(xg) = xhg. E. Noether [Noe13] asked whether the eld of invariants K is rational over k or not. On the language of algebraic geometry, this is a question about the rationality of the quotient variety A/G. The most complete answer on this question is known for Abelian groups. Thus, if G is Abelian of exponent e, char k does not divide e and k contains a primitive eth roots of unity then A/G is rational [Fis15]. On the other hand, over an arbitrary eld k the rationality of A/G is related to some number theoretic questions. In this case A/G can be non-rational [Swa69] (see also [Vos73], [EM73], [Len74]). Noether's question can be generalized as follows.
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 ample. This paper continues our study of such germs... more
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 ample. This paper continues our study of such germs containing a point of type (IIA) started in [MP16].
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Smooth Fano threefolds were classified by Iskovskikh and Mori-Mukai. Ifa Fano threefold Xhas only terminal Gorenstein singularities, then it has asmoothing (see [Na]), i.e., such Xcan be considered as a degeneration of asmooth ones. One... more
Smooth Fano threefolds were classified by Iskovskikh and Mori-Mukai. Ifa Fano threefold Xhas only terminal Gorenstein singularities, then it has asmoothing (see [Na]), i.e., such Xcan be considered as a degeneration of asmooth ones. One can expect the same situation in the case of cDV singu-larities. In contrast, Fano threefolds with canonical non-cDV singularitiesare not necessarily smoothable:Examples 1.2. (i) (Weighted projective spaces.) All weighted projec-tive spaces P(a
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We prove that Q-Fano threefolds of Fano index ≥ 8 are rational.
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 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,... more
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.