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Maximal exterior sets of hyperbolic quadrics: The complete classification

J.A Thas
1991 Journal of combinatorial theory. Series A  
MAXIMAL EXTERIOR SETS An exterior set with respect to the hyperbolic quadric Q+(2n -1, q) of PG(2n -1, q), IZ # 2, is a set X of points such that each line joining two distinct elements of X has no point  ...  An exterior set with respect to the hyperbolic quadric Q'(2n-1, q) of PG(2n -1, q), n 3 2, is a set X of points such that each line joining two distinct elements of X has no point in common with Q+(2n-  ...  MAXIMAL EXTERIOR SETS An exterior set with respect to the hyperbolic quadric Q+(2n -1, q) of PG(2n -1, q), IZ # 2, is a set X of points such that each line joining two distinct elements of X has no point  ... 
doi:10.1016/0097-3165(91)90040-n fatcat:i5veheewtngchgxyml2iwhracy

On the number of components of a complete intersection of real quadrics [article]

Alex Degtyarev, Ilia Itenberg, Viatcheslav Kharlamov
2012 arXiv   pre-print
We prove that the maximal number $B^0_2(N)$ of connected components that a regular complete intersection of three real quadrics in $\Bbb{P}^N$ can have differs at most by one from the maximal number of  ...  Our main results concern complete intersections of three real quadrics.  ...  In particular, the maximal number of connected components of a complete intersection of three real quadrics in P 5 R equals 10 = Hilb(6) + 1.  ... 
arXiv:0806.4077v3 fatcat:ocgzf6e4tvfnvd3stjglfdfa5m

Projection pencil of quadrics and Ivory theorem [article]

Ákos G.Horváth
2009 arXiv   pre-print
The "confocality property" of stachel can be get from our definition in the case when the projection transformation is regular, it is the identity transformation of the space.  ...  Our observation is that the generalized definition of confocality in stachel does not give back to the original definition of confocality of Euclidean conics.  ...  Definition 1 A maximal set of quadrics called by pencil of quadrics if the selfadjoint linear transformations corresponding their duals belong a two-dimensional subspace of the vector space L(V ).  ... 
arXiv:0905.4093v1 fatcat:cuscuu6slvcc7d3xhvw5rwekmm

Real algebraic curves of bidegree (5,5) on the quadric ellipsoid [article]

Matilde Manzaroli
2020 arXiv   pre-print
We complete the topological classification of real algebraic non-singular curves of bidegree (5, 5) on the quadric ellipsoid.  ...  We show in particular that previously known restrictions form a complete system for this bidegree. Therefore, the main part of the paper concerns the construction of real algebraic curves.  ...  The real part RX of X is the set of points fixed by σ.  ... 
arXiv:1809.03946v2 fatcat:uymqxzjinrbtjhjuo4xr6q7yjy

Floer cohomology and pencils of quadrics [article]

Ivan Smith
2011 arXiv   pre-print
We construct a derived equivalence between the Fukaya category of a curve and the nilpotent summand of the Fukaya category of the associated complete intersection of two quadrics.  ...  There is a classical relationship in algebraic geometry between a hyperelliptic curve and an associated pencil of quadric hypersurfaces.  ...  For completeness, we discuss quadrics of both even and odd dimension. Lemma 4.1.  ... 
arXiv:1006.1099v3 fatcat:2kfpvtwakjhytmgiwxcowmchai

On the Number of Components of a Complete Intersection of Real Quadrics [chapter]

Alex Degtyarev, Ilia Itenberg, Viatcheslav Kharlamov
2011 PROGRESS IN MATHEMATICS  
We prove that the maximal number B 0 2 (N ) of connected components that a regular complete intersection of three real quadrics in P N can have differs at most by one from the maximal number of ovals of  ...  Our main results concern complete intersections of three real quadrics.  ...  In particular, the maximal number of connected components of a complete intersection of three real quadrics in P 5 R equals 10 = Hilb(6) + 1.  ... 
doi:10.1007/978-0-8176-8277-4_5 fatcat:p6paec7tnrbptp72zdjpcgmwfm

Codes Associated with Nondegenerate Quadrics of a Symplectic Space of Even Order

N.S Narasimha Sastry, Peter Sin
2001 Journal of combinatorial theory. Series A  
This paper studies the incidence relation between the points and quadrics in the projective space of a symplectic vector space over a field of even order.  ...  The 2-rank of the incidence matrix is determined.  ...  ACKNOWLEDGMENT We take this opportunity to thank the referee for several suggestions which have improved the exposition.  ... 
doi:10.1006/jcta.2000.3118 fatcat:noourjw545h4viel2wa6azpa34

Bigraded Equivariant Cohomology of Real Quadrics [article]

Pedro F. dos Santos, Paulo Lima-Filho
2006 arXiv   pre-print
We give a complete description of the bigraded Bredon cohomology ring of smooth projective real quadrics, with coefficients in the constant Mackey functor Z .  ...  Some of the results and techniques introduced can be applied to other geometrically cellular real varieties.  ...  Left to the reader. Isotropic quadrics Consider a real quadratic form q of rank n + 2 that can be written as q = q ′ + h, where h is a hyperbolic factor.  ... 
arXiv:math/0605711v1 fatcat:lpjlvgssffeyffhslh7wktkqg4

Birational geometry of quadrics in characteristic $2$

Burt Totaro
2008 Journal of Algebraic Geometry  
The theory of quadratic forms can be regarded as studying an important special case of the general problem of birational classification of algebraic varieties.  ...  As in other advances on quadratic forms over the past decade, the proofs use the Chow groups of algebraic cycles on quadrics and products of quadrics.  ...  We want to compute the total index (maximal dimension of an isotropic subspace) of s over the function field E/F of the corresponding projective quadric S.  ... 
doi:10.1090/s1056-3911-08-00472-4 fatcat:h7wp2hdp4fbt7p6stsf6oxwymu

Bianchi's Bäcklund transformation for higher dimensional quadrics [article]

Ion I. Dinca
2008 arXiv   pre-print
The starting point of our investigation is the higher dimensional (infinitesimal) version of Bianchi's main four theorems on the theory of deformations of quadrics and Bianchi's treatment of the Bäcklund  ...  We provide a generalization of Bianchi's Bäcklund transformation from 2-dimensional quadrics to higher dimensional quadrics.  ...  on the preservation under the Ivory affinity of lengths of segments between confocal quadrics were already known to other authors; for example Henrici's construction of the articulated hyperbolic paraboloid  ... 
arXiv:0808.2007v2 fatcat:qxqkg7balbgbzb2y3ojp7lsumm

Intersection of quadrics in C^n, moment-angle manifolds, complex manifolds and convex polytopes [article]

Alberto Verjovsky
2018 arXiv   pre-print
in an open set in complex projective space mathbb P_ C^n-1  ...  It is an overview of different properties of a class of non-Kähler compact complex manifolds called LVMB manifolds, obtained as the Hausdorff space of leaves of systems of commuting complex linear equations  ...  different aspects of the subject of these notes with me during several years.  ... 
arXiv:1807.00838v1 fatcat:hwllfoso2fabtmpq7vknxvubje

The Method of Archimedes in the geometry of quadrics [article]

Ion I. Dinca
2006 arXiv   pre-print
letter and may provide the key to generalizations in other settings.  ...  Certain metric-projective properties of confocal quadrics (most of them established in the first half of the XIX^th century) carry out (stick and transfer) by rolling to and influence surfaces applicable  ...  A •−1 (Ao + B) = 0, Q(o) = 0, equivalent to o = −A •−1 B + B T A •−1 B−C 2v T B v (o Complete a maximal set of orthonormal eigenvectors of A to an orthonormal basis of C n and consider the vectors of this  ... 
arXiv:math/0612375v1 fatcat:qyt6legcg5bo7p4mnnvbffvty4

Stabilizers of subspaces under similitudes of the Klein quadric, and automorphisms of Heisenberg algebras

Michael Gulde, Markus Stroppel
2012 Linear Algebra and its Applications  
We determine the groups of automorphisms and their orbits for nilpotent Lie algebras of class 2 and small dimension, over arbitrary fields (including the characteristic 2 case).  ...  Acknowledgements The authors owe Norbert Knarr at least one cup of coffee for the simple computational arguments in 3.6 (replacing much deeper arguments involving Witt cancellation and the Skolem-Noether  ...  The second author was supported by SFB 478 "Geometrische Strukturen in der Mathematik", Münster, Germany; a substantial part of this paper was written during a stay at Münster in 2009.  ... 
doi:10.1016/j.laa.2012.03.018 fatcat:g6mqabg2j5da7bcphaidjracuq

Page 4316 of Mathematical Reviews Vol. , Issue 87h [page]

1987 Mathematical Reviews  
By definition a maximal exterior set is a set of g? +q+1 points such that any line joining two elements of the set is exterior to the quadric.  ...  (B-GHNT-G) Exterior sets with respect to the hyperbolic quadric in PG(2n-—1,q). Finite geometries (Winnipeg, Man., 1984), 83-89, Lecture Notes in Pure and Appl. Math., 103, Dekker, New York, 1985.  ... 

Page 3337 of Mathematical Reviews Vol. , Issue 94f [page]

1994 Mathematical Reviews  
This article also discusses some new results on partial flocks and maximal exterior sets of hyperbolic quadrics in PG(2m — 1, q). {For the entire collection see MR 93m:00039.} Gary L.  ...  the hyperbolic quadric in PG(3, q).  ... 
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