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# Documents  14J35 | enregistrements trouvés : 10

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## Automorphisms of hyperkähler manifolds​ - Lecture 1 Sarti, Alessandra | CIRM H

Post-edited

Research talks;Algebraic and Complex Geometry

In the 80's Beauville generalized several foundational results of Nikulin on automorphism groups of K3 surfaces to hyperkähler manifolds. Since then the study of automorphism groups of hyperkähler manifolds and in particular of hyperkähler fourfolds got very much attention. I will present some classification results for automorphisms on hyperkähler fourfolds that are deformation equivalent to the Hilbert scheme of two points on a K3 surface and describe some explicit examples. I will give particular attention to double EPW sextics, that admit in a natural way a non-symplectic involution. Time permitting I will show how the rich geometry of double EPW sextics has an important connection to a classical question of U. Morin (1930). In the 80's Beauville generalized several foundational results of Nikulin on automorphism groups of K3 surfaces to hyperkähler manifolds. Since then the study of automorphism groups of hyperkähler manifolds and in particular of hyperkähler fourfolds got very much attention. I will present some classification results for automorphisms on hyperkähler fourfolds that are deformation equivalent to the Hilbert scheme of two points on a K3 surface and ...

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## Special rational fibrations in Fano 4-folds Casagrande, Cinzia | CIRM H

Multi angle

Research talks

Smooth, complex Fano 4-folds are not classified, and we still lack a good understanding of their general properties. We focus on Fano 4-folds with large second Betti number $b_{2}$, studied via birational geometry and the detailed analysis of their contractions and rational contractions (we recall that a contraction is a morphism with connected fibers onto a normal projective variety, and a rational contraction is given by a sequence of flips followed by a contraction). The main result that we want to present is the following: let $X$ be a Fano 4-fold having a nonconstant rational contraction $X --> Y$ of fiber type. Then either $b_{2}(X)$ is at most 18, with equality only for a product of surfaces, or $Y$ is $\mathbb{P}^{1}$ or $\mathbb{P}^{2}$. The proof is achieved by reducing to the case of "special" rational contractions of fiber type. We will explain this notion and give an idea of the techniques that are used. Smooth, complex Fano 4-folds are not classified, and we still lack a good understanding of their general properties. We focus on Fano 4-folds with large second Betti number $b_{2}$, studied via birational geometry and the detailed analysis of their contractions and rational contractions (we recall that a contraction is a morphism with connected fibers onto a normal projective variety, and a rational contraction is given by a sequence of flips ...

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## Gushel-Mukai varieties and their periods Debarre, Olivier | CIRM H

Multi angle

Research talks

Gushel-Mukai varieties are defined as the intersection of the Grassmannian Gr(2, 5) in its Plücker embedding, with a quadric and a linear space. They occur in dimension 6 (with a slighty modified construction), 5, 4, 3, 2 (where they are just K3 surfaces of degree 10), and 1 (where they are just genus 6 curves). Their theory parallels that of another important class of Fano varieties, cubic fourfolds, with many common features such as the presence of a canonically attached hyperkähler fourfold: the variety of lines for a cubic is replaced here with a double EPW sextic.
There is a big difference though: in dimension at least 3, GM varieties attached to a given EPW sextic form a family of positive dimension. However, we prove that the Hodge structure of any of these GM varieties can be reconstructed from that of the EPW sextic or of an associated surface of general type, depending on the parity of the dimension (for cubic fourfolds, the corresponding statement was proved in 1985 by Beauville and Donagi). This is joint work with Alexander Kuznetsov.
Gushel-Mukai varieties are defined as the intersection of the Grassmannian Gr(2, 5) in its Plücker embedding, with a quadric and a linear space. They occur in dimension 6 (with a slighty modified construction), 5, 4, 3, 2 (where they are just K3 surfaces of degree 10), and 1 (where they are just genus 6 curves). Their theory parallels that of another important class of Fano varieties, cubic fourfolds, with many common features such as the ...

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## Automorphisms of hyperkähler manifolds​ - Lecture 3 Sarti, Alessandra | CIRM H

Multi angle

Research talks;Algebraic and Complex Geometry

In the 80's Beauville generalized several foundational results of Nikulin on automorphism groups of K3 surfaces to hyperkähler manifolds. Since then the study of automorphism groups of hyperkähler manifolds and in particular of hyperkähler fourfolds got very much attention. I will present some classification results for automorphisms on hyperkähler fourfolds that are deformation equivalent to the Hilbert scheme of two points on a K3 surface and describe some explicit examples. I will give particular attention to double EPW sextics, that admit in a natural way a non-symplectic involution. Time permitting I will show how the rich geometry of double EPW sextics has an important connection to a classical question of U. Morin (1930). In the 80's Beauville generalized several foundational results of Nikulin on automorphism groups of K3 surfaces to hyperkähler manifolds. Since then the study of automorphism groups of hyperkähler manifolds and in particular of hyperkähler fourfolds got very much attention. I will present some classification results for automorphisms on hyperkähler fourfolds that are deformation equivalent to the Hilbert scheme of two points on a K3 surface and ...

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## Automorphisms of hyperkähler manifolds​ - Lecture 2 Sarti, Alessandra | CIRM H

Multi angle

Research talks;Algebraic and Complex Geometry

In the 80's Beauville generalized several foundational results of Nikulin on automorphism groups of K3 surfaces to hyperkähler manifolds. Since then the study of automorphism groups of hyperkähler manifolds and in particular of hyperkähler fourfolds got very much attention. I will present some classification results for automorphisms on hyperkähler fourfolds that are deformation equivalent to the Hilbert scheme of two points on a K3 surface and describe some explicit examples. I will give particular attention to double EPW sextics, that admit in a natural way a non-symplectic involution. Time permitting I will show how the rich geometry of double EPW sextics has an important connection to a classical question of U. Morin (1930). In the 80's Beauville generalized several foundational results of Nikulin on automorphism groups of K3 surfaces to hyperkähler manifolds. Since then the study of automorphism groups of hyperkähler manifolds and in particular of hyperkähler fourfolds got very much attention. I will present some classification results for automorphisms on hyperkähler fourfolds that are deformation equivalent to the Hilbert scheme of two points on a K3 surface and ...

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## Segre's reflexivity and an inductive characterization of hyperquadrics Kachi, Yasuyuki ; Sato, Eiichi | American Mathematical Society 2002

Ouvrage

- 116 p.
ISBN 978-0-8218-3225-7

Memoirs of the american mathematical society , 0763

Localisation : Collection 1er étage

variété # cycle algèbrique # théorie de l'intersection # distribution de cycle # principe de Lefschetz généralisé # réflexivité de Segre # dégénérescence # variété de Severi # pseudo-quotient universel # variété grassmannienne # espace de Kuranishi # déformation de fibre vectoriel # fibré universel

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## Flips for 3-folds and 4-folds Corti, Alessio | Oxford University Press 2007

Ouvrage

- 189 p.
ISBN 978-0-19-857061-5

Oxford lecture series in mathematics and its applications , 0035

Localisation : Ouvrage RdC (Flip)

géométrie algébrique # variétés à 3 dimension # variétés à 4 dimensions

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## Cohomological and geometric approaches to rationality problems.New perspectives Bogomolov, Fedor ; Tschinkel, Yuri | Birkhäuser 2010

Ouvrage

- ix; 311 p.
ISBN 978-0-8176-4933-3

Progress in mathematics , 0282

Localisation : Collection 1er étage

rationalité # cohomologie # géométrie algébrique # invariant cohomologique # groupe fini # groupe de Lie # espace de modules # point rationel # variétés algébriques

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## Brauer groups and obstruction problems:moduli spaces and arithmetic Auel, Asher ; Hassett, Brendan ; Varilly-Alvarado, Anthony ; Viray, Bianca | Birkhäuser 2017

Ouvrage

- ix; 247 p.
ISBN 978-3-319-46851-8

Progress in mathematics , 0320

Localisation : Collection 1er étage

théorie des groupes # géométrie algébrique # groupe de Brauer # catégorie dérivée des faisceaux cohérents # isogénie # point de torsion # courbe modulaire # surface K3

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## Untersuchung einiger singulärer Hyperflächen des vier-und fünf-dimensionalen projektiven Raumes Gläser, Michael | Mathematische Naturwissenschaften Fakultat Universitat 1982

Thèse

- 115 p.

Bonner mathematische schriften , 0142

Localisation : Publication 1er étage

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