Hamid Abban

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All published works (21)

On K-moduli of quartic threefolds

2025-05-01

Hamid Abban , Ivan Cheltsov , Alexander Kasprzyk +2 more

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K-stability of pointless del Pezzo surfaces and Fano 3-folds

2024-11-01

Hamid Abban , Ivan Cheltsov , Takashi Kishimoto +1 more

We explore connections between existence of $\Bbbk$-rational points for Fano varieties defined over $\Bbbk$, a subfield of $\mathbb{C}$, and existence of K\"ahler-Einstein metrics on their geometric models. First, we show … We explore connections between existence of $\Bbbk$-rational points for Fano varieties defined over $\Bbbk$, a subfield of $\mathbb{C}$, and existence of K\"ahler-Einstein metrics on their geometric models. First, we show that geometric models of del Pezzo surfaces with at worst quotient singularities defined over $\Bbbk\subset\mathbb{C}$ admit (orbifold) K\"ahler--Einstein metrics if they do not have $\Bbbk$-rational points. Then we prove the same result for smooth Fano 3-folds with 8 exceptions. Consequently, we explicitly describe several families of pointless Fano 3-folds whose geometric models admit K\"ahler-Einstein metrics. In particular, we obtain new examples of prime Fano 3-folds of genus $12$ that admit K\"ahler--Einstein metrics. Our result can also be used to prove existence of rational points for certain Fano varieties, for example for any smooth Fano 3-fold over $\Bbbk\subset\mathbb{C}$ whose geometric model is strictly K-semistable.

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One-dimensional components in the K-moduli of smooth Fano 3-folds

2024-10-09

Hamid Abban , Ivan Cheltsov , Elena Denisova +5 more

By identifying K-polystable limits in 4 specific deformations families of smooth Fano 3-folds, we complete the classification of one-dimensional components in the K-moduli space of smoothable Fano 3-folds. By identifying K-polystable limits in 4 specific deformations families of smooth Fano 3-folds, we complete the classification of one-dimensional components in the K-moduli space of smoothable Fano 3-folds.

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Double Veronese cones with 28 nodes

2024-07-04

Hamid Abban , Ivan Cheltsov , Jihun Park +1 more

We study nodal del Pezzo 3 -folds of degree 1 (also known as double Veronese cones) with 28 singularities, which is the maximal possible number of singularities for such varieties. … We study nodal del Pezzo 3 -folds of degree 1 (also known as double Veronese cones) with 28 singularities, which is the maximal possible number of singularities for such varieties. We show that they are in one-to-one correspondence with smooth plane quartics and use this correspondence to study their automorphism groups. As an application, we find all G -birationally rigid varieties of this kind, and construct an infinite number of non-conjugate embeddings of the group \mathfrak{S}_{4} into the space Cremona group.

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Seshadri constants and K-stability of Fano manifolds

2023-03-24

Hamid Abban , Ziquan Zhuang

We give a lower bound of the δ-invariants of ample line bundles in terms of Seshadri constants. As applications, we prove the uniform K-stability of infinitely many families of Fano … We give a lower bound of the δ-invariants of ample line bundles in terms of Seshadri constants. As applications, we prove the uniform K-stability of infinitely many families of Fano hypersurfaces of arbitrarily large index, as well as the uniform K-stability of most families of smooth Fano threefolds of Picard number one.

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One-dimensional components in the K-moduli of smooth Fano 3-folds

2023-01-01

Hamid Abban , Ivan Cheltsov , Elena Denisova +5 more

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Stability of fibrations over one-dimensional bases

2022-06-09

Hamid Abban , Maksym Fedorchuk , Igor Krylov

We introduce and study a new notion of stability for varieties fibered over curves, motivated by Kollár’s stability for homogeneous polynomials with integral coefficients. We develop tools to study geometric … We introduce and study a new notion of stability for varieties fibered over curves, motivated by Kollár’s stability for homogeneous polynomials with integral coefficients. We develop tools to study geometric properties of stable birational models of fibrations whose fibers are complete intersections in weighted projective spaces. As an application, we prove the existence of standard models of threefold degree 1 del Pezzo fibrations, settling a conjecture of Corti.

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BIRATIONAL RIGIDITY OF ORBIFOLD DEGREE 2 DEL PEZZO FIBRATIONS

2022-06-02

Hamid Abban , Igor Krylov

Abstract Varieties fibered into del Pezzo surfaces form a class of possible outputs of the minimal model program. It is known that del Pezzo fibrations of degrees $1$ and $2$ … Abstract Varieties fibered into del Pezzo surfaces form a class of possible outputs of the minimal model program. It is known that del Pezzo fibrations of degrees $1$ and $2$ over the projective line with smooth total space satisfying the so-called $K^2$ -condition are birationally rigid: their Mori fiber space structure is unique. This implies that they are not birational to any Fano varieties, conic bundles, or other del Pezzo fibrations. In particular, they are irrational. The families of del Pezzo fibrations with smooth total space of degree $2$ are rather special, as for most families a general del Pezzo fibration has the simplest orbifold singularities. We prove that orbifold del Pezzo fibrations of degree $2$ over the projective line satisfying explicit generality conditions as well as a generalized $K^2$ -condition are birationally rigid.

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K-stability of Fano varieties via admissible flags

2022-01-01

Hamid Abban , Ziquan Zhuang

Abstract We develop a general approach to prove K-stability of Fano varieties. The new theory is used to (a) prove the existence of Kähler-Einstein metrics on all smooth Fano hypersurfaces … Abstract We develop a general approach to prove K-stability of Fano varieties. The new theory is used to (a) prove the existence of Kähler-Einstein metrics on all smooth Fano hypersurfaces of Fano index two, (b) compute the stability thresholds for hypersurfaces at generalised Eckardt points and for cubic surfaces at all points, and (c) provide a new algebraic proof of Tian’s criterion for K-stability, amongst other applications.

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On K-moduli of quartic threefolds

2022-01-01

Hamid Abban , Ivan Cheltsov , Alexander Kasprzyk +1 more

The family of smooth Fano 3-folds with Picard rank 1 and anticanonical volume 4 consists of quartic 3-folds and of double covers of the 3-dimensional quadric branched along an octic … The family of smooth Fano 3-folds with Picard rank 1 and anticanonical volume 4 consists of quartic 3-folds and of double covers of the 3-dimensional quadric branched along an octic surface. They can all be parametrised as complete intersections of a quadric and a quartic in the weighted projective space $\mathbb{P}(1,1,1,1,1,2)$, denoted by $X_{2,4} \subset \mathbb{P}(1^5,2)$; all such smooth complete intersections are K-stable. With the aim of investigating the compactification of the moduli space of quartic 3-folds given by K-stability, we exhibit three phenomena: (i) there exist K-polystable complete intersection $X_{2,2,4} \subset \mathbb{P}(1^5,2^2)$ Fano 3-folds which deform to quartic 3-folds and are neither quartic 3-folds nor double covers of quadric 3-folds - in other words, the closure of the locus parametrising complete intersections $X_{2,4}\subset \mathbb{P}(1^5,2)$ in the K-moduli contains elements that are not of this type; (ii) any quasi-smooth $X_{2,2,4} \subset \mathbb{P}(1^5,2^2)$ is K-polystable; (iii) the closure in the K-moduli space of the locus parametrising complete intersections $X_{2,2,4} \subset \mathbb{P}(1^5,2^2)$ which are not complete intersections $X_{2,4} \subset \mathbb{P}(1^5,2)$ contains only points which correspond to complete intersections $X_{2,2,4} \subset \mathbb{P}(1^5,2^2)$.

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On Geometry of Fano Threefold Hypersurfaces

2021-01-01

Hamid Abban , Ivan Cheltsov , Jihun Park

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K-stability of Fano varieties via admissible flags

2020-01-01

Hamid Abban , Ziquan Zhuang

We develop a general approach to prove K-stability of Fano varieties. The new theory is used to (a) prove the existence of K\"ahler-Einstein metrics on all smooth Fano hypersurfaces of … We develop a general approach to prove K-stability of Fano varieties. The new theory is used to (a) prove the existence of K\"ahler-Einstein metrics on all smooth Fano hypersurfaces of Fano index two, (b) to compute the stability thresholds for hypersurfaces at generalized Eckardt points and for cubic surfaces at all points, and (c) to provide a new algebraic proof of Tian's criterion for K-stability, amongst other applications.

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Double Veronese cones with 28 nodes

2019-01-01

Hamid Abban , Ivan Cheltsov , Jihun Park +1 more

We study nodal del Pezzo 3-folds of degree $1$ (also known as double Veronese cones) with $28$ singularities, which is the maximal possible number of singularities for such varieties. We … We study nodal del Pezzo 3-folds of degree $1$ (also known as double Veronese cones) with $28$ singularities, which is the maximal possible number of singularities for such varieties. We show that they are in one-to-one correspondence with smooth plane quartics and use this correspondence to study their automorphism groups. As an application, we find all $G$-birationally rigid varieties of this kind, and construct an infinite number of non-conjugate embeddings of the group $\mathfrak{S}_4$ into the space Cremona group.

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Stability of fibrations over one-dimensional bases

2019-01-01

Hamid Abban , Maksym Fedorchuk , Igor B. Krylov

We introduce and study a new notion of stability for varieties fibered over curves, motivated by Koll\'ar's stability for homogeneous polynomials with integral coefficients. We develop tools to study geometric … We introduce and study a new notion of stability for varieties fibered over curves, motivated by Koll\'ar's stability for homogeneous polynomials with integral coefficients. We develop tools to study geometric properties of stable birational models of fibrations whose fibers are complete intersections in weighted projective spaces. As an application, we prove the existence of standard models of threefold degree one and two del Pezzo fibrations, settling a conjecture of Corti from 1996.

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Birational rigidity of orbifold degree 2 del Pezzo fibrations

2017-01-01

Hamid Abban , Igor Krylov

Varieties fibered into del Pezzo surfaces form a class of possible outputs of the minimal model program. It is known that del Pezzo fibrations of degrees $1$ and $2$ over … Varieties fibered into del Pezzo surfaces form a class of possible outputs of the minimal model program. It is known that del Pezzo fibrations of degrees $1$ and $2$ over the projective line with smooth total space satisfying the so-called $K^2$-condition are birationally rigid: their Mori fibre space structure is unique. This implies that they are not birational to any Fano varieties, conic bundles or other del Pezzo fibrations. In particular, they are irrational. The families of del Pezzo fibrations with smooth total space of degree $2$ are rather special, as for "most" families a general del Pezzo fibration has the simplest orbifold singularities. We prove that orbifold del Pezzo fibrations of degree $2$ over the projective line satisfying explicit generality conditions as well as a generalised $K^2$-condition are birationally rigid.

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On a conjecture of Tian

2015-01-01

Hamid Abban , Ivan Cheltsov , Josef Schicho

We study Tian's $\alpha$-invariant in comparison with the $\alpha_1$-invariant for pairs $(S_d,H)$ consisting of a smooth surface $S_d$ of degree $d$ in the projective three-dimensional space and a hyperplane section … We study Tian's $\alpha$-invariant in comparison with the $\alpha_1$-invariant for pairs $(S_d,H)$ consisting of a smooth surface $S_d$ of degree $d$ in the projective three-dimensional space and a hyperplane section $H$. A conjecture of Tian asserts that $\alpha(S_d,H)=\alpha_1(S_d,H)$. We show that this is indeed true for $d=4$ (the result is well known for $d\leqslant 3$), and we show that $\alpha(S_d,H)<\alpha_1(S_d,H)$ for $d\geqslant 8$ provided that $S_d$ is general enough. We also construct examples of $S_d$, for $d=6$ and $d=7$, for which Tian's conjecture fails. We provide a candidate counterexample for $S_5$.

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Mori dream spaces and birational rigidity of Fano 3-folds

2014-01-01

Hamid Abban , Francesco Zucconi

We highlight a relation between the existence of Sarkisov links and the finite generation of (certain) Cox rings. We introduce explicit methods to use this relation in order to prove … We highlight a relation between the existence of Sarkisov links and the finite generation of (certain) Cox rings. We introduce explicit methods to use this relation in order to prove birational rigidity statements. To illustrate, we complete the birational rigidity results of Okada for Fano complete intersection 3-folds in singular weighted projective spaces.

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Singular del Pezzo fibrations and birational rigidity

2014-01-01

Hamid Abban

A known conjecture of Grinenko in birational geometry asserts that a Mori fibre space with the structure of del Pezzo fibration of low degree is birationally rigid if and only … A known conjecture of Grinenko in birational geometry asserts that a Mori fibre space with the structure of del Pezzo fibration of low degree is birationally rigid if and only if its anticanonical class is an interior point in the cone of mobile divisors. The conjecture is proved to be true for smooth models (with a generality assumption for degree 3). It is speculated that the conjecture holds for, at least, Gorenstein models in degree 1 and 2. In this article, I present a (Gorenstein) counterexample in degree 2 to this conjecture.

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On conjugacy classes of the Klein simple group in Cremona group

2013-01-01

Hamid Abban

We consider countably many three dimensional $\mathtt{PSL}_2(\mathbb{F}_7)$-del Pezzo surface fibrations over $\mathbb{P}^1$. Conjecturally they are all irrational except two families, one of which is the product of a del Pezzo … We consider countably many three dimensional $\mathtt{PSL}_2(\mathbb{F}_7)$-del Pezzo surface fibrations over $\mathbb{P}^1$. Conjecturally they are all irrational except two families, one of which is the product of a del Pezzo surface with $\mathbb{P}^1$. We show that the other model is $\mathtt{PSL}_2(\mathbb{F}_7)$-equivariantly birational to $\mathbb{P}^2\times\mathbb{P}^1$. Based on a result of Prokhorov, we show that they are non-conjugate as subgroups of the Cremona group $\mathtt{Cr}_3(\mathbb{C})$.

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An algebraic study of linkages with helical joints

2013-01-01

Hamid Abban , Zijia Li , Josef Schicho

Methods from algebra and algebraic geometry have been used in various ways to study linkages in kinematics. These methods have failed so far for the study of linkages with helical … Methods from algebra and algebraic geometry have been used in various ways to study linkages in kinematics. These methods have failed so far for the study of linkages with helical joints (joints with screw motion), because of the presence of some non-algebraic relations. In this article, we explore a delicate reduction of some analytic equations in kinematics to algebraic questions via a theorem of Ax. As an application, we give a classification of mobile closed 5-linkages with revolute, prismatic, and helical joints.

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Circle of Sarkisov links on a Fano $3$-fold

2013-01-01

Hamid Abban , Francesco Zucconi

For a general Fano $3$-fold of index $1$ in the weighted projective space $\mathbb{P}(1,1,1,1,2,2,3)$ we construct $2$ new birational models that are Mori fibre spaces, in the framework of the … For a general Fano $3$-fold of index $1$ in the weighted projective space $\mathbb{P}(1,1,1,1,2,2,3)$ we construct $2$ new birational models that are Mori fibre spaces, in the framework of the so-called Sarkisov program. We highlight a relation between the corresponding birational maps, as a circle of Sarkisov links, visualising the notion of relations (due to Kaloghiros) in Sarkisov program.

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Common Coauthors

Coauthor	Papers Together
Ivan Cheltsov	9
Igor Krylov	3
Ziquan Zhuang	3
Jihun Park	3
Maksym Fedorchuk	2
Dongchen Jiao	2
Alexander Kasprzyk	2
Anne-Sophie Kaloghiros	2
Francesco Zucconi	2
Josef Schicho	2
Andrea Petracci	2
Jesus Martinez‐Garcia	2
Erroxe Etxabarri-Alberdi	2
Elena Denisova	2
Theodoros Stylianos Papazachariou	2
Constantin Shramov	2
Frédéric Mangolte	1
Igor B. Krylov	1
Yuchen Liu	1
Takashi Kishimoto	1
Zijia Li	1

Commonly Cited References

K-STABILITY OF FANO MANIFOLDS WITH NOT SMALL ALPHA INVARIANTS

2017-03-30

Kento Fujita

We show that any $n$ -dimensional Fano manifold $X$ with $\unicode[STIX]{x1D6FC}(X)=n/(n+1)$ and $n\geqslant 2$ is K-stable, where $\unicode[STIX]{x1D6FC}(X)$ is the alpha invariant of $X$ introduced by Tian. In particular, any … We show that any $n$ -dimensional Fano manifold $X$ with $\unicode[STIX]{x1D6FC}(X)=n/(n+1)$ and $n\geqslant 2$ is K-stable, where $\unicode[STIX]{x1D6FC}(X)$ is the alpha invariant of $X$ introduced by Tian. In particular, any such $X$ admits Kähler–Einstein metrics and the holomorphic automorphism group $\operatorname{Aut}(X)$ of $X$ is finite.

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A valuative criterion for uniform K-stability of ℚ-Fano varieties

2016-10-25

Kento Fujita

Abstract We give a simple necessary and sufficient condition for uniform K-stability of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℚ</m:mi> </m:math> \mathbb{Q} -Fano varieties. Abstract We give a simple necessary and sufficient condition for uniform K-stability of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ℚ</m:mi> </m:math> \mathbb{Q} -Fano varieties.

Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities

2014-03-28

Xiuxiong Chen , Simon Donaldson , Song Sun

This is the first of a series of three papers which prove the fact that a <italic>K</italic>-stable Fano manifold admits a Kähler-Einstein metric. The main result of this paper is … This is the first of a series of three papers which prove the fact that a <italic>K</italic>-stable Fano manifold admits a Kähler-Einstein metric. The main result of this paper is that a Kähler-Einstein metric with cone singularities along a divisor can be approximated by a sequence of smooth Kähler metrics with controlled geometry in the Gromov-Hausdorff sense.

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Vanishing sequences and Okounkov bodies

2014-08-28

Sébastien Boucksom , Alex Küronya , Catriona Maclean +1 more

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Essentials of the method of maximal singularities

2000-07-27

Александр Валентинович Пухликов

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

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Positivity in Algebraic Geometry I

2004-01-01

Robert Lazarsfeld

Stable rationality of orbifold Fano 3-fold hypersurfaces

2018-09-26

Takuzo Okada

We determine the rationality of very general quasi-smooth Fano<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"><mml:semantics><mml:mn>3</mml:mn><mml:annotation encoding="application/x-tex">3</mml:annotation></mml:semantics></mml:math></inline-formula>-fold weighted hypersurfaces completely and determine the stable rationality of them except for cubic<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"><mml:semantics><mml:mn>3</mml:mn><mml:annotation encoding="application/x-tex">3</mml:annotation></mml:semantics></mml:math></inline-formula>-folds. … We determine the rationality of very general quasi-smooth Fano<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"><mml:semantics><mml:mn>3</mml:mn><mml:annotation encoding="application/x-tex">3</mml:annotation></mml:semantics></mml:math></inline-formula>-fold weighted hypersurfaces completely and determine the stable rationality of them except for cubic<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"><mml:semantics><mml:mn>3</mml:mn><mml:annotation encoding="application/x-tex">3</mml:annotation></mml:semantics></mml:math></inline-formula>-folds. More precisely we prove that (i) very general Fano<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="3"><mml:semantics><mml:mn>3</mml:mn><mml:annotation encoding="application/x-tex">3</mml:annotation></mml:semantics></mml:math></inline-formula>-fold weighted hypersurfaces of index<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"><mml:semantics><mml:mn>1</mml:mn><mml:annotation encoding="application/x-tex">1</mml:annotation></mml:semantics></mml:math></inline-formula>or<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"><mml:semantics><mml:mn>2</mml:mn><mml:annotation encoding="application/x-tex">2</mml:annotation></mml:semantics></mml:math></inline-formula>are not stably rational except possibly for the cubic 3-folds, (ii) among the<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="27"><mml:semantics><mml:mn>27</mml:mn><mml:annotation encoding="application/x-tex">27</mml:annotation></mml:semantics></mml:math></inline-formula>families of Fano 3-fold weighted hypersurfaces of index greater than<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"><mml:semantics><mml:mn>2</mml:mn><mml:annotation encoding="application/x-tex">2</mml:annotation></mml:semantics></mml:math></inline-formula>, very general members of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="7"><mml:semantics><mml:mn>7</mml:mn><mml:annotation encoding="application/x-tex">7</mml:annotation></mml:semantics></mml:math></inline-formula>specific families are not stably rational, and the remaining<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="20"><mml:semantics><mml:mn>20</mml:mn><mml:annotation encoding="application/x-tex">20</mml:annotation></mml:semantics></mml:math></inline-formula>families consist of rational varieties.

Openness of K-semistability for Fano varieties

2022-01-01

Harold Blum , Yuchen Liu , Chenyang Xu

In this paper, we prove the openness of K-semistability in families of log Fano pairs by showing that the stability threshold is a constructible function on the fibers. We also … In this paper, we prove the openness of K-semistability in families of log Fano pairs by showing that the stability threshold is a constructible function on the fibers. We also prove that any special test configuration arises from a log canonical place of a bounded complement and establish properties of any minimizer of the stability threshold.

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Birational properties of pencils of del Pezzo surfaces of degrees 1 and 2

2000-06-30

Михаил Михайлович Гриненко

We study the birational rigidity problem for smooth Mori fibrations on del Pezzo surfaces of degree 1 and 2. For degree 1 we obtain a complete description of rigid and … We study the birational rigidity problem for smooth Mori fibrations on del Pezzo surfaces of degree 1 and 2. For degree 1 we obtain a complete description of rigid and non-rigid cases.

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K-semistability is equivariant volume minimization

2017-09-16

Chi Li

This is a continuation to the paper [Li15a] in which a problem of minimizing normalized volumes over Q-Gorenstein klt singularities was proposed.Here we consider its relation with K-semistability, which is … This is a continuation to the paper [Li15a] in which a problem of minimizing normalized volumes over Q-Gorenstein klt singularities was proposed.Here we consider its relation with K-semistability, which is an important concept in the study of Kähler-Einstein metrics on Fano varieties.In particular, we prove that for a Q-Fano variety V , the K-semistability of (V, -K V ) is equivalent to the condition that the normalized volume is minimized at the canonical valuation ord V among all C * -invariant valuations on the cone associated to any positive Cartier multiple of -K V .In this case, it's shown that ord V is the unique minimizer among all C * -invariant quasi-monomial valuations.These results allow us to give characterizations of K-semistability by using equivariant volume minimization, and also by using inequalities involving divisorial valuations over V .

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Valuation spaces and multiplier ideals on singular varieties

2015-01-07

Sébastien Boucksom , Tommaso de Fernex , C. Favre +1 more

We generalize to all normal complex algebraic varieties the valuative characterization of multiplier ideals due to Boucksom-Favre-Jonsson in the smooth case. To that end, we extend the log discrepancy function … We generalize to all normal complex algebraic varieties the valuative characterization of multiplier ideals due to Boucksom-Favre-Jonsson in the smooth case. To that end, we extend the log discrepancy function to the space of all real valuations, and prove that it satisfies an adequate properness property, building upon previous work by Jonsson and Mustaţă. We next give an alternative definition of the concept of numerically Cartier divisors previously introduced by the first three authors, and prove that numerically ℚ-Cartier divisors coincide with ℚ-Cartier divisors for rational singularities. These ideas naturally lead to the notion of numerically ℚ-Gorenstein varieties, for which our valuative characterization of multiplier ideals takes a particularly simple form.

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On del Pezzo fibrations that are not birationally rigid

2012-03-09

Hamid Ahmadinezhad

We consider ℙ(1, 1, 1, 2) bundles over ℙ1 and construct hypersurfaces of these bundles which form a degree 2 del Pezzo fibration over ℙ1 as a Mori fibre space. … We consider ℙ(1, 1, 1, 2) bundles over ℙ1 and construct hypersurfaces of these bundles which form a degree 2 del Pezzo fibration over ℙ1 as a Mori fibre space. We classify all such hypersurfaces whose type III or IV Sarkisov link, inherited from the ambient space, pass to a different Mori fibre space. A similar result for cubic surface fibrations over ℙ2 is also presented.

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K-stability of cubic threefolds

2019-07-02

Yuchen Liu , Chenyang Xu

We prove the K-moduli space of cubic threefolds is identical to their GIT moduli. More precisely, the K-(semi,poly)-stability of cubic threefolds coincide to the corresponding GIT stabilities, which could be … We prove the K-moduli space of cubic threefolds is identical to their GIT moduli. More precisely, the K-(semi,poly)-stability of cubic threefolds coincide to the corresponding GIT stabilities, which could be explicitly calculated. In particular, this implies that all smooth cubic threefolds admit K\"ahler-Einstein metric as well as provides a precise list of singular KE ones. To achieve this, our main new contribution is an estimate in dimension three of the volumes of kawamata log terminal singularities introduced by Chi Li. This is obtained via a detailed study of the classification of three dimensional canonical and terminal singularities, which was established during the study of the explicit three dimensional minimal model program.

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Convex bodies associated to linear series

2009-01-01

Robert Lazarsfeld , Mircea Mustaţǎ

Dans son travail sur la log-concavité des multiplicités, Okounkov montre au passage que l'on peut associer un corps convexe à un système linéaire sur une variété projective, puis utiliser la … Dans son travail sur la log-concavité des multiplicités, Okounkov montre au passage que l'on peut associer un corps convexe à un système linéaire sur une variété projective, puis utiliser la géométrie convexe pour étudier ces systèmes linéaires. Bien qu'Okounkov travaille essentiellement dans le cadre classique des fibrés en droites amples, il se trouve que sa construction s'étend au cas d'un grand diviseur arbitraire. De plus, ce point de vue permet de rendre transparentes de nombreuses propriétés de base des invariants asymptotiques des systèmes linéaires, et ouvre la porte à de nombreuses extensions. Le but de cet article est d'initier un développement systématique de la théorie et de donner quelques applications et exemples.

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Birational properties of pencils of del Pezzo surfaces of degrees 1 and 2. II

2003-06-30

Михаил Михайлович Гриненко

The problem of birational rigidity is investigated for smooth Mori fibre spaces representable as a pencil of del Pezzo surfaces of degree 1 or 2. The problem of birational rigidity is investigated for smooth Mori fibre spaces representable as a pencil of del Pezzo surfaces of degree 1 or 2.

Singularities of linear systems and 3-fold birational geometry

2000-07-27

Alessio Corti

Mori structures on a Fano threefold of index 2 and degree 1

2004-01-01

Михаил Михайлович Гриненко

Valuations and asymptotic invariants for sequences of ideals

2012-01-01

Mattias Jönsson , Mircea Mustaţǎ

We study asymptotic jumping numbers for graded sequences of ideals, and show that every such invariant is computed by a suitable real valuation of the function field. We conjecture that … We study asymptotic jumping numbers for graded sequences of ideals, and show that every such invariant is computed by a suitable real valuation of the function field. We conjecture that every valuation that computes an asymptotic jumping number is necessarily quasi-monomial. This conjecture holds in dimension two. In general, we reduce it to the case of affine space and to graded sequences of valuation ideals. Along the way, we study the structure of a suitable valuation space.

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Uniqueness of K-polystable degenerations of Fano varieties

2019-08-22

Harold Blum , Chenyang Xu

We prove that K-polystable degenerations of $\mathbb{Q}$-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable $\mathbb{Q}$-Fano varieties is separated. Together with recently proven boundedness and openness … We prove that K-polystable degenerations of $\mathbb{Q}$-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable $\mathbb{Q}$-Fano varieties is separated. Together with recently proven boundedness and openness statements, the latter result yields a separated Deligne-Mumford stack parametrizing all uniformly K-stable $\mathbb{Q}$-Fano varieties of fixed dimension and volume. The result also implies that the automorphism group of a K-stable $\mathbb{Q}$-Fano variety is finite.

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A minimizing valuation is quasi-monomial

2020-04-15

Chenyang Xu

We prove a version of Jonsson-Mustaţǎ's Conjecture, which says for any graded sequence of ideals, there exists a quasi-monomial valuation computing its log canonical threshold. As a corollary, we confirm … We prove a version of Jonsson-Mustaţǎ's Conjecture, which says for any graded sequence of ideals, there exists a quasi-monomial valuation computing its log canonical threshold. As a corollary, we confirm Chi Li's conjecture that a minimizer of the normalized volume function is always quasi-monomial. Applying our techniques to a family of klt singularities, we show that the volume of klt singularities is a constructible function. As a corollary, we prove that in a family of klt log Fano pairs, the K-semistable ones form a Zariski open set. Together with previous works by many people, we conclude that all K-semistable klt Fano varieties with a fixed dimension and volume are parametrized by an Artin stack of finite type, which then admits a separated good moduli space, whose geometric points parametrize K-polystable klt Fano varieties.

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On pliability of del Pezzo fibrations and Cox rings

2014-10-28

Hamid Ahmadinezhad

We develop some concrete methods to build Sarkisov links, starting from Mori fibre spaces. This is done by studying low rank Cox rings and their properties. As part of this … We develop some concrete methods to build Sarkisov links, starting from Mori fibre spaces. This is done by studying low rank Cox rings and their properties. As part of this development, we give an algorithm to construct explicitly the coarse moduli space of a toric Deligne-Mumford stack. This can be viewed as the generalisation of the notion of well-formedness for weighted projective spaces to homogeneous coordinate ring of toric varieties. As an illustration, we apply these methods to study birational transformations of certain fibrations of del Pezzo surfaces over $\mathbb{P}^1$, into other Mori fibre spaces, using Cox rings and variation of geometric invariant theory. We show that the pliability of these Mori fibre spaces is at least three and they are not rational.

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Uniform K-stability and plt blowups of log Fano pairs

2019-04-19

Kento Fujita

We show relationships between uniform K-stability and plt blowups of log Fano pairs. We see that it is enough to evaluate certain invariants defined by volume functions for all plt … We show relationships between uniform K-stability and plt blowups of log Fano pairs. We see that it is enough to evaluate certain invariants defined by volume functions for all plt blowups in order to test uniform K-stability of log Fano pairs. We also discuss the uniform K-stability of two log Fano pairs under crepant finite covers. Moreover, we give another proof of K-semistability of the projective plane.

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Finite generation for valuations computing stability thresholds and applications to K-stability

2022-06-29

Yuchen Liu , Chenyang Xu , Ziquan Zhuang

We prove that on any log Fano pair of dimension $n$ whose stability threshold is less than $\frac{n+1}{n}$, any valuation computing the stability threshold has a finitely generated associated graded … We prove that on any log Fano pair of dimension $n$ whose stability threshold is less than $\frac{n+1}{n}$, any valuation computing the stability threshold has a finitely generated associated graded ring. Together with earlier works, this implies that (a) a log Fano pair is uniformly K-stable (resp. reduced uniformly K-stable) if and only if it is K-stable (resp. K-polystable); (b) the K-moduli spaces are proper and projective; and combining with the previously known equivalence between the existence of Kähler-Einstein metric and reduced uniform K-stability proved by the variational approach, (c) the Yau-Tian-Donaldson conjecture holds for general (possibly singular) log Fano pairs.

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K-stability of Fano varieties: an algebro-geometric approach

2021-08-31

Chenyang Xu

We give a survey of the recent progress on the study of K-stability of Fano varieties by an algebro-geometric approach. We give a survey of the recent progress on the study of K-stability of Fano varieties by an algebro-geometric approach.

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Optimal destabilizing centers and equivariant K-stability

2021-04-15

Ziquan Zhuang

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Kähler-Einstein metrics with positive scalar curvature

1997-09-03

Gang Tian

Symmetries, quotients and Kähler-Einstein metrics

2006-01-24

Claudio Arezzo , Alessandro Ghigi , Gian Pietro Pirola

We consider Fano manifolds M that admit a collection of finite automorphism groups G1, …, Gk, such that the quotients M/Gi are smooth Fano manifolds possessing a Kähler-Einstein metric. Under … We consider Fano manifolds M that admit a collection of finite automorphism groups G1, …, Gk, such that the quotients M/Gi are smooth Fano manifolds possessing a Kähler-Einstein metric. Under some numerical and smoothness assumptions on the ramification divisors, we prove that M admits a Kähler-Einstein metric too.

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Restricted volumes and base loci of linear series

2009-06-01

Lawrence Ein , Robert Lazarsfeld , Mircea Mustaţǎ +2 more

We introduce and study the restricted volume of a divisor along a subvariety. Our main result is a description of the irreducible components of the augmented base locus by the … We introduce and study the restricted volume of a divisor along a subvariety. Our main result is a description of the irreducible components of the augmented base locus by the vanishing of the restricted volume.

K-stability of Fano varieties via admissible flags

2022-01-01

Hamid Abban , Ziquan Zhuang

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Thresholds, valuations, and K-stability

2020-02-20

Harold Blum , Mattias Jönsson

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Mori dream spaces and birational rigidity of Fano 3-folds

2016-02-10

Hamid Ahmadinezhad , Francesco Zucconi

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K‐Stability and Kähler‐Einstein Metrics

2015-04-30

Gang Tian

We prove that if a Fano manifold M is K‐stable, then it admits a Kähler‐Einstein metric. It affirms a longstanding conjecture for Fano manifolds. © 2015 Wiley Periodicals, Inc. We prove that if a Fano manifold M is K‐stable, then it admits a Kähler‐Einstein metric. It affirms a longstanding conjecture for Fano manifolds. © 2015 Wiley Periodicals, Inc.

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Stable rationality of orbifold Fano threefold hypersurfaces

2016-01-01

Takuzo Okada

We determine the rationality of very general quasismooth Fano 3-fold weighted hypersurfaces completely and determine the stable rationality of them except for cubic 3-folds. More precisely we prove that (i) … We determine the rationality of very general quasismooth Fano 3-fold weighted hypersurfaces completely and determine the stable rationality of them except for cubic 3-folds. More precisely we prove that (i) very general Fano 3-fold weighted hypersurfaces of index 1 or 2 are not stably rational except possibly for the cubic threefolds, (ii) among the 27 families of Fano 3-fold weighted hypersurfaces of index greater than 2, very general members of specific 7 families are not stably rational and the remaining 20 families consists of rational varieties.

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On the K-stability of Fano varieties and anticanonical divisors

2018-12-01

Kento Fujita , Yuji Odaka

We apply a recent theorem of Li and the first author to give some criteria for the K-stability of Fano varieties in terms of anticanonical $\mathbb{Q}$-divisors. First, we propose a … We apply a recent theorem of Li and the first author to give some criteria for the K-stability of Fano varieties in terms of anticanonical $\mathbb{Q}$-divisors. First, we propose a condition in terms of certain anti-canonical $\mathbb{Q}$-divisors of given Fano variety, which we conjecture to be equivalent to the K-stability. We prove that it is at least a sufficient condition and also related to the Berman-Gibbs stability. We also give another algebraic proof of the K-stability of Fano varieties which satisfy Tian's alpha invariants condition.

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Birationally rigid Pfaffian Fano 3-folds

2018-03-01

Hamid Ahmadinezhad

We classify birationally rigid orbifold Fano 3-folds of index 1 defined by 5 × 5 Pfaffian varieties.We give a sharp criterion for the birational rigidity of these families based on … We classify birationally rigid orbifold Fano 3-folds of index 1 defined by 5 × 5 Pfaffian varieties.We give a sharp criterion for the birational rigidity of these families based on the type of singularities that the varieties admit.Various conjectures are born out of our study, highlighting a possible approach to the classification of terminal Fano 3-folds.The birationally rigid cases are the first known rigid examples of Fano varieties that are not (weighted) complete intersections.

Alpha invariant and K-stability of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi mathvariant="double-struck">Q</mml:mi></mml:math>-Fano varieties

2012-02-10

Yuji Odaka , Yuji Sano

On K-stability of finite covers

2016-06-28

Ruadhaí Dervan

We show that certain Galois covers of K-semistable Fano varieties are K-stable. We use this to give some new examples of Fano manifolds admitting Kähler–Einstein metrics, including hypersurfaces, double solids … We show that certain Galois covers of K-semistable Fano varieties are K-stable. We use this to give some new examples of Fano manifolds admitting Kähler–Einstein metrics, including hypersurfaces, double solids and threefolds.

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Del Pezzo Surfaces Over Dedekind Schemes

1996-11-01

Alessio Corti

Let S be a Dedekind scheme with fraction field K. We study the following problem: given a Del Pezzo surface X, defined over K, construct a distinguished integral model of … Let S be a Dedekind scheme with fraction field K. We study the following problem: given a Del Pezzo surface X, defined over K, construct a distinguished integral model of X, defined over all of S. We provide a satisfactory answer if S is a smooth complex curve, and a conjectural answer if X is a cubic Del Pezzo surface over (nearly) arbitrary S.

BIRATIONAL AUTOMORPHISMS OF CONIC BUNDLES

1981-02-28

V G Sarkisov

This paper studies birational automorphisms of algebraic threefolds representable in the form of conic bundles. It is shown that, under certain conditions on the degeneration curve of a conic bundle, … This paper studies birational automorphisms of algebraic threefolds representable in the form of conic bundles. It is shown that, under certain conditions on the degeneration curve of a conic bundle, the automorphisms preserve the conic bundle structure.

Uniform stability of twisted constant scalar curvature K\"ahler metrics

2014-12-01

Ruadhaí Dervan

We introduce a norm on the space of test configurations, which we call the minimum norm. We conjecture that uniform K-stability with respect to this norm is equivalent to the … We introduce a norm on the space of test configurations, which we call the minimum norm. We conjecture that uniform K-stability with respect to this norm is equivalent to the existence of a constant scalar curvature Kahler metric. This notion of uniform K-stability is analogous to coercivity of the Mabuchi functional. We characterise the triviality of test configurations, by showing that a test configuration has zero minimum norm if and only if it has zero $L^2$-norm, if and only if it is almost trivial. We prove that the existence of a twisted constant scalar curvature Kahler metric implies uniform twisted K-stability with respect to the minimum norm, when the twisting is ample. We give algebro-geometric proofs of uniform K-stability in the general type and Calabi-Yau cases, as well as in the Fano case under an alpha invariant condition. Our results hold for line bundles sufficiently close to the (anti)-canonical line bundle, and also in the twisted setting. We show that log K-stability implies twisted K-stability, and also that twisted K-semistability of a variety implies that the variety has mild singularities.

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Multiplicities and log canonical threshold

2004-09-01

Tommaso de Fernex , Lawrence Ein , Mircea Mustaţǎ

Given an $n$-dimensional local ring $R$ of a smooth variety, and a zero-dimensional ideal $I\subset R$, we prove the following inequality involving the Samuel multiplicity and the log canonical threshold: … Given an $n$-dimensional local ring $R$ of a smooth variety, and a zero-dimensional ideal $I\subset R$, we prove the following inequality involving the Samuel multiplicity and the log canonical threshold: $e(I)\geq n^n/\operatorname {lc}(I)^n$. Moreover, equality holds if and only if the integral closure of $I$ is a power of the maximal ideal in $R$. When $n=2$, we give a similar inequality for an arbitrary ideal $I$.

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Über Determinanten und ihre Anwendung in der Geometrie, insbesondere auf Curven vierter Ordnung.

1855-01-01

O. Hesse

Article Über Determinanten und ihre Anwendung in der Geometrie, insbesondere auf Curven vierter Ordnung. was published on January 1, 1855 in the journal Journal für die reine und angewandte Mathematik … Article Über Determinanten und ihre Anwendung in der Geometrie, insbesondere auf Curven vierter Ordnung. was published on January 1, 1855 in the journal Journal für die reine und angewandte Mathematik (volume 1855, issue 49).

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Stable rationality and conic bundles

2015-09-16

Brendan Hassett , Andrew Kresch , Yuri Tschinkel

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Local positivity of ample line bundles

1995-01-01

Lawrence Ein , Oliver Küchle , Robert Lazarsfeld

Let $L$ be a nef line bundle on a smooth complex projective variety $X$ of dimension $n$. Demailly has introduced a very interesting invariant --- the Seshadri constant $\epsilon(L,x)$ --- … Let $L$ be a nef line bundle on a smooth complex projective variety $X$ of dimension $n$. Demailly has introduced a very interesting invariant --- the Seshadri constant $\epsilon(L,x)$ --- which in effect measures how positive $L$ is locally near a given point $x \in X$. For instance, Seshadri's criterion for ampleness may be phrased as stating that $L$ is ample if and only if there exists a positive number $e > 0$ such that $\epsilon(L,x) > e$ for all $x \in X$, and if $L$ is VERY ample, then $\epsilon(L,x) \ge 1$ for every $x$. We prove the somewhat surprising result that in each dimension $n$ there is a uniform lower bound on the Seshadri constant of an ample line bundle $L$ at a very general point of $X$. Specifically, $\epsilon(L,x) \ge (1/n) $ for all $x \in X$ outside the union of countably many proper subvarieties of $X$. Examples of Miranda show that there cannot exist a bound (independent of $X$ and $L$) that holds at every point. The proof draws inspiration from two sources: first, the arguments used to prove boundedness of Fano manifolds of Picard number one; and secondly some of the geometric ideas involving zero-estimates appearing in the work of Faltings and others on Diophantine approximation and transcendence theory. We give some elementary applications of the main theorem to adjoint and pluricanonical linear series.

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Canonical Metrics in Kähler Geometry

2000-01-01

Gang Tian

1 Introduction to Kahler manifolds.- 1.1 Kahler metrics.- 1.2 Curvature of Kahler metrics.- 2 Extremal Kahler metrics.- 2.1 The space of Kahler metrics.- 2.2 A brief review of Chern classes.- … 1 Introduction to Kahler manifolds.- 1.1 Kahler metrics.- 1.2 Curvature of Kahler metrics.- 2 Extremal Kahler metrics.- 2.1 The space of Kahler metrics.- 2.2 A brief review of Chern classes.- 2.3 Uniformization of Kahler-Einstein manifolds.- 3 Calabi-Futaki invariants.- 3.1 Definition of Calabi-Futaki invariants.- 3.2 Localization formula for Calabi-Futaki invariants.- 4 Scalar curvature as a moment map.- 5 Kahler-Einstein metrics with non-positive scalar curvature.- 5.1 The Calabi-Yau Theorem.- 5.2 Kahler-Einstein metrics for manifolds with c1(M) < 0.- 6 Kahler-Einstein metrics with positive scalar curvature.- 6.1 A variational approach.- 6.2 Existence of Kahler-Einstein metrics.- 6.3 Examples.- 7 Applications and generalizations.- 7.1 A manifold without Kahler-Einstein metric.- 7.2 K-energy and metrics of constant scalar curvature.- 7.3 Relation to stability.

Fibrations into del Pezzo surfaces

2006-04-30

Михаил Михайлович Гриненко

The problem of birational rigidity for three-dimensional algebraic varieties fibered over rational curves into del Pezzo surfaces of degree 1 is discussed. A criterion for the rigidity of such fibrations … The problem of birational rigidity for three-dimensional algebraic varieties fibered over rational curves into del Pezzo surfaces of degree 1 is discussed. A criterion for the rigidity of such fibrations in the Mori category is suggested and the inverse implication is proved (Theorem 3.3). Surgeries on fibers in fibrations of this type, which turn out to be closely related to the rigidity problem, are considered. In particular, an important result on the uniqueness of a smooth model in a class of maps over a base is stated (Corollary 4.5).

A generalization of Ross-Thomas' slope theory

2009-01-01

Yuji Odaka

We give a formula of the Donaldson-Futaki invariants for certain type of semi test configurations, which essentially generalizes Ross-Thomas' slope theory. The positivity (resp. non-negativity) of those "a priori special" … We give a formula of the Donaldson-Futaki invariants for certain type of semi test configurations, which essentially generalizes Ross-Thomas' slope theory. The positivity (resp. non-negativity) of those "a priori special" Donaldson-Futaki invariants implies K-stability (resp. K-semistability). We show its applicability by proving K-(semi)stability of certain polarized varieties with semi-log-canonical singularities, generalizing some results by Ross-Thomas.

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On Calabi's conjecture for complex surfaces with positive first Chern class

1990-12-01

Gang Tian

Cayley-Bacharach theorems and conjectures

1996-01-01

David Eisenbud , Mark Green , Joe Harris

A theorem of Pappus of Alexandria, proved in the fourth century A.D., began a long development in algebraic geometry. In its changing expressions one can see reflected the changing concerns … A theorem of Pappus of Alexandria, proved in the fourth century A.D., began a long development in algebraic geometry. In its changing expressions one can see reflected the changing concerns of the field, from synthetic geometry to projective plane curves to Riemann surfaces to the modern development of schemes and duality. We survey this development historically and use it to motivate a brief treatment of a part of duality theory. We then explain one of the modern developments arising from it, a series of conjectures about the linear conditions imposed by a set of points in projective space on the forms that vanish on them. We give a proof of the conjectures in a new special case.

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Extremal metrics on del Pezzo threefolds

2009-04-01

Ivan Cheltsov , Constantin Shramov

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