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Severi varieties and their varieties of reductions

Identifieur interne : 000B73 ( Istex/Checkpoint ); précédent : 000B72; suivant : 000B74

Severi varieties and their varieties of reductions

Auteurs : A. Iliev ; L. Manivel

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RBID : ISTEX:47488442F0824CAB35FA35136B437217F71E5C6C

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English descriptors

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.

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DOI: 10.1515/crll.2005.2005.585.93


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ISTEX:47488442F0824CAB35FA35136B437217F71E5C6C

Le document en format XML

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<term>Determinant hypersurface</term>
<term>Diagonal</term>
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<term>Dynkin</term>
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<term>Irreducible</term>
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<term>Jordan algebra</term>
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<term>Reduction planes</term>
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<term>Severi varieties</term>
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<term>Special lines</term>
<term>Standard representation</term>
<term>Straightforward computation</term>
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<term>Subset</term>
<term>Subvariety</term>
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<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>
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<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>
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<term>Weighted dynkin diagrams</term>
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<term>Adjoint variety</term>
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<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>
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<term>Dynkin</term>
<term>Dynkin diagram</term>
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<term>Exact sequence</term>
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<term>Explicit action</term>
<term>Explicit representative</term>
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<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>
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<term>Hyperplane section</term>
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<term>Intersection points</term>
<term>Irreducible</term>
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<term>Isomorphism</term>
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<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>
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<term>Math</term>
<term>Matrix</term>
<term>Morphism</term>
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<term>Nite group</term>
<term>Nite number</term>
<term>Normal bundle</term>
<term>Open orbit</term>
<term>Open subset</term>
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<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>
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