Severi varieties and their varieties of reductions
Identifieur interne : 000B73 ( Istex/Checkpoint ); précédent : 000B72; suivant : 000B74Severi varieties and their varieties of reductions
Auteurs : A. Iliev ; L. ManivelSource :
- Journal für die reine und angewandte Mathematik [ 0075-4102 ] ; 2005-08-26.
Descripteurs français
- Wicri :
- geographic : Jordanie.
English descriptors
- KwdEn :
- Acts transitively, Adjoint, Adjoint variety, Algebra, Algebraic, Algebraic varieties, Ambient grassmannians, Automorphism, Automorphism group, Betti, Betti numbers, Characteristic polynomial, Chern classes, Chow ring, Closure, Codimension, Cubic form, Cubics, Determinant, Determinant hypersurface, Diagonal, Diagonal matrices, Discriminant, Discriminant hypersurface, Divisor, Dynkin, Dynkin diagram, Embedding, Exact sequence, Exceptional divisor, Explicit action, Explicit representative, Explicitely, Fano, Freudenthal, Freudenthal square, General line, General point, Generic, Global sections, Grassmannian, Grassmannians, Highest weight, Hilb, Hilbert, Hilbert scheme, Homogeneous vector bundle, Hyperplane, Hyperplane class, Hyperplane section, Hypersurface, Iliev, Intersection points, Irreducible, Isomorphic, Isomorphism, Isotropic, Isotropic subvariety, Isotropic varieties, Isotropy, Isotropy group, Jordan, Jordan algebra, Jordan algebras, Linear forms, Linear section, Linear sections, Linear subspace, Magic square, Manivel, Math, Matrix, Morphism, Nite, Nite group, Nite number, Normal bundle, Open orbit, Open subset, Other hand, Picard group, Polar hyperplane, Projective, Projective geometry, Projective plane, Projective representation, Projective section, Projective sections, Projective space, Projectivized, Quadratic form, Quadratic forms, Quotient, Ranestad, Rational projection, Reduction plane, Reduction planes, Second assertion, Second kind, Second variety, Severi, Severi varieties, Severi varieties proof, Severi variety, Simple computation, Simple contacts, Smooth quadric, Special lines, Standard representation, Straightforward computation, Subgroup, Subset, Subvariety, Surjective, Symmetric group, Tangent directions, Tangent line, Tangent space, Torus, Traceless, Traceless matrices, Transitively, Triality, Triality group, Triality subspaces, Triality varieties, Triality variety, Trisecant, Unique plane, Unique point, Vector bundle, Vector bundles, Veronese, Weighted dynkin diagram, Weighted dynkin diagrams.
- Teeft :
- Acts transitively, Adjoint, Adjoint variety, Algebra, Algebraic, Algebraic varieties, Ambient grassmannians, Automorphism, Automorphism group, Betti, Betti numbers, Characteristic polynomial, Chern classes, Chow ring, Closure, Codimension, Cubic form, Cubics, Determinant, Determinant hypersurface, Diagonal, Diagonal matrices, Discriminant, Discriminant hypersurface, Divisor, Dynkin, Dynkin diagram, Embedding, Exact sequence, Exceptional divisor, Explicit action, Explicit representative, Explicitely, Fano, Freudenthal, Freudenthal square, General line, General point, Generic, Global sections, Grassmannian, Grassmannians, Highest weight, Hilb, Hilbert, Hilbert scheme, Homogeneous vector bundle, Hyperplane, Hyperplane class, Hyperplane section, Hypersurface, Iliev, Intersection points, Irreducible, Isomorphic, Isomorphism, Isotropic, Isotropic subvariety, Isotropic varieties, Isotropy, Isotropy group, Jordan, Jordan algebra, Jordan algebras, Linear forms, Linear section, Linear sections, Linear subspace, Magic square, Manivel, Math, Matrix, Morphism, Nite, Nite group, Nite number, Normal bundle, Open orbit, Open subset, Other hand, Picard group, Polar hyperplane, Projective, Projective geometry, Projective plane, Projective representation, Projective section, Projective sections, Projective space, Projectivized, Quadratic form, Quadratic forms, Quotient, Ranestad, Rational projection, Reduction plane, Reduction planes, Second assertion, Second kind, Second variety, Severi, Severi varieties, Severi varieties proof, Severi variety, Simple computation, Simple contacts, Smooth quadric, Special lines, Standard representation, Straightforward computation, Subgroup, Subset, Subvariety, Surjective, Symmetric group, Tangent directions, Tangent line, Tangent space, Torus, Traceless, Traceless matrices, Transitively, Triality, Triality group, Triality subspaces, Triality varieties, Triality variety, Trisecant, Unique plane, Unique point, Vector bundle, Vector bundles, Veronese, Weighted dynkin diagram, Weighted dynkin diagrams.
Abstract
We study the varieties of reductions associated to the four Severi varieties, the first example of which is the Fano threefold of index 2 and degree 5 studied by Mukai and others. We prove that they are smooth but very special linear sections of Grassmann varieties, and rational Fano manifolds of dimension 3a and index a + 1, for a = 1, 2, 4, 8. We study their maximal linear spaces and prove that through the general point pass exactly three of them, a result we relate to Cartan’s triality principle. We also prove that they are compactifications of affine spaces.
Url:
DOI: 10.1515/crll.2005.2005.585.93
Affiliations:
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ISTEX:47488442F0824CAB35FA35136B437217F71E5C6CLe document en format XML
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<term>Adjoint variety</term>
<term>Algebra</term>
<term>Algebraic</term>
<term>Algebraic varieties</term>
<term>Ambient grassmannians</term>
<term>Automorphism</term>
<term>Automorphism group</term>
<term>Betti</term>
<term>Betti numbers</term>
<term>Characteristic polynomial</term>
<term>Chern classes</term>
<term>Chow ring</term>
<term>Closure</term>
<term>Codimension</term>
<term>Cubic form</term>
<term>Cubics</term>
<term>Determinant</term>
<term>Determinant hypersurface</term>
<term>Diagonal</term>
<term>Diagonal matrices</term>
<term>Discriminant</term>
<term>Discriminant hypersurface</term>
<term>Divisor</term>
<term>Dynkin</term>
<term>Dynkin diagram</term>
<term>Embedding</term>
<term>Exact sequence</term>
<term>Exceptional divisor</term>
<term>Explicit action</term>
<term>Explicit representative</term>
<term>Explicitely</term>
<term>Fano</term>
<term>Freudenthal</term>
<term>Freudenthal square</term>
<term>General line</term>
<term>General point</term>
<term>Generic</term>
<term>Global sections</term>
<term>Grassmannian</term>
<term>Grassmannians</term>
<term>Highest weight</term>
<term>Hilb</term>
<term>Hilbert</term>
<term>Hilbert scheme</term>
<term>Homogeneous vector bundle</term>
<term>Hyperplane</term>
<term>Hyperplane class</term>
<term>Hyperplane section</term>
<term>Hypersurface</term>
<term>Iliev</term>
<term>Intersection points</term>
<term>Irreducible</term>
<term>Isomorphic</term>
<term>Isomorphism</term>
<term>Isotropic</term>
<term>Isotropic subvariety</term>
<term>Isotropic varieties</term>
<term>Isotropy</term>
<term>Isotropy group</term>
<term>Jordan</term>
<term>Jordan algebra</term>
<term>Jordan algebras</term>
<term>Linear forms</term>
<term>Linear section</term>
<term>Linear sections</term>
<term>Linear subspace</term>
<term>Magic square</term>
<term>Manivel</term>
<term>Math</term>
<term>Matrix</term>
<term>Morphism</term>
<term>Nite</term>
<term>Nite group</term>
<term>Nite number</term>
<term>Normal bundle</term>
<term>Open orbit</term>
<term>Open subset</term>
<term>Other hand</term>
<term>Picard group</term>
<term>Polar hyperplane</term>
<term>Projective</term>
<term>Projective geometry</term>
<term>Projective plane</term>
<term>Projective representation</term>
<term>Projective section</term>
<term>Projective sections</term>
<term>Projective space</term>
<term>Projectivized</term>
<term>Quadratic form</term>
<term>Quadratic forms</term>
<term>Quotient</term>
<term>Ranestad</term>
<term>Rational projection</term>
<term>Reduction plane</term>
<term>Reduction planes</term>
<term>Second assertion</term>
<term>Second kind</term>
<term>Second variety</term>
<term>Severi</term>
<term>Severi varieties</term>
<term>Severi varieties proof</term>
<term>Severi variety</term>
<term>Simple computation</term>
<term>Simple contacts</term>
<term>Smooth quadric</term>
<term>Special lines</term>
<term>Standard representation</term>
<term>Straightforward computation</term>
<term>Subgroup</term>
<term>Subset</term>
<term>Subvariety</term>
<term>Surjective</term>
<term>Symmetric group</term>
<term>Tangent directions</term>
<term>Tangent line</term>
<term>Tangent space</term>
<term>Torus</term>
<term>Traceless</term>
<term>Traceless matrices</term>
<term>Transitively</term>
<term>Triality</term>
<term>Triality group</term>
<term>Triality subspaces</term>
<term>Triality varieties</term>
<term>Triality variety</term>
<term>Trisecant</term>
<term>Unique plane</term>
<term>Unique point</term>
<term>Vector bundle</term>
<term>Vector bundles</term>
<term>Veronese</term>
<term>Weighted dynkin diagram</term>
<term>Weighted dynkin diagrams</term>
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<keywords scheme="Teeft" xml:lang="en"><term>Acts transitively</term>
<term>Adjoint</term>
<term>Adjoint variety</term>
<term>Algebra</term>
<term>Algebraic</term>
<term>Algebraic varieties</term>
<term>Ambient grassmannians</term>
<term>Automorphism</term>
<term>Automorphism group</term>
<term>Betti</term>
<term>Betti numbers</term>
<term>Characteristic polynomial</term>
<term>Chern classes</term>
<term>Chow ring</term>
<term>Closure</term>
<term>Codimension</term>
<term>Cubic form</term>
<term>Cubics</term>
<term>Determinant</term>
<term>Determinant hypersurface</term>
<term>Diagonal</term>
<term>Diagonal matrices</term>
<term>Discriminant</term>
<term>Discriminant hypersurface</term>
<term>Divisor</term>
<term>Dynkin</term>
<term>Dynkin diagram</term>
<term>Embedding</term>
<term>Exact sequence</term>
<term>Exceptional divisor</term>
<term>Explicit action</term>
<term>Explicit representative</term>
<term>Explicitely</term>
<term>Fano</term>
<term>Freudenthal</term>
<term>Freudenthal square</term>
<term>General line</term>
<term>General point</term>
<term>Generic</term>
<term>Global sections</term>
<term>Grassmannian</term>
<term>Grassmannians</term>
<term>Highest weight</term>
<term>Hilb</term>
<term>Hilbert</term>
<term>Hilbert scheme</term>
<term>Homogeneous vector bundle</term>
<term>Hyperplane</term>
<term>Hyperplane class</term>
<term>Hyperplane section</term>
<term>Hypersurface</term>
<term>Iliev</term>
<term>Intersection points</term>
<term>Irreducible</term>
<term>Isomorphic</term>
<term>Isomorphism</term>
<term>Isotropic</term>
<term>Isotropic subvariety</term>
<term>Isotropic varieties</term>
<term>Isotropy</term>
<term>Isotropy group</term>
<term>Jordan</term>
<term>Jordan algebra</term>
<term>Jordan algebras</term>
<term>Linear forms</term>
<term>Linear section</term>
<term>Linear sections</term>
<term>Linear subspace</term>
<term>Magic square</term>
<term>Manivel</term>
<term>Math</term>
<term>Matrix</term>
<term>Morphism</term>
<term>Nite</term>
<term>Nite group</term>
<term>Nite number</term>
<term>Normal bundle</term>
<term>Open orbit</term>
<term>Open subset</term>
<term>Other hand</term>
<term>Picard group</term>
<term>Polar hyperplane</term>
<term>Projective</term>
<term>Projective geometry</term>
<term>Projective plane</term>
<term>Projective representation</term>
<term>Projective section</term>
<term>Projective sections</term>
<term>Projective space</term>
<term>Projectivized</term>
<term>Quadratic form</term>
<term>Quadratic forms</term>
<term>Quotient</term>
<term>Ranestad</term>
<term>Rational projection</term>
<term>Reduction plane</term>
<term>Reduction planes</term>
<term>Second assertion</term>
<term>Second kind</term>
<term>Second variety</term>
<term>Severi</term>
<term>Severi varieties</term>
<term>Severi varieties proof</term>
<term>Severi variety</term>
<term>Simple computation</term>
<term>Simple contacts</term>
<term>Smooth quadric</term>
<term>Special lines</term>
<term>Standard representation</term>
<term>Straightforward computation</term>
<term>Subgroup</term>
<term>Subset</term>
<term>Subvariety</term>
<term>Surjective</term>
<term>Symmetric group</term>
<term>Tangent directions</term>
<term>Tangent line</term>
<term>Tangent space</term>
<term>Torus</term>
<term>Traceless</term>
<term>Traceless matrices</term>
<term>Transitively</term>
<term>Triality</term>
<term>Triality group</term>
<term>Triality subspaces</term>
<term>Triality varieties</term>
<term>Triality variety</term>
<term>Trisecant</term>
<term>Unique plane</term>
<term>Unique point</term>
<term>Vector bundle</term>
<term>Vector bundles</term>
<term>Veronese</term>
<term>Weighted dynkin diagram</term>
<term>Weighted dynkin diagrams</term>
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<front><div type="abstract" xml:lang="en">We study the varieties of reductions associated to the four Severi varieties, the first example of which is the Fano threefold of index 2 and degree 5 studied by Mukai and others. We prove that they are smooth but very special linear sections of Grassmann varieties, and rational Fano manifolds of dimension 3a and index a + 1, for a = 1, 2, 4, 8. We study their maximal linear spaces and prove that through the general point pass exactly three of them, a result we relate to Cartan’s triality principle. We also prove that they are compactifications of affine spaces.</div>
</front>
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