Quadratic Sets 4.5 Elliptic, Parabolic and Hyperbolic Quadratic Sets
|
|
- Madeleine Lewis
- 5 years ago
- Views:
Transcription
1 Quadratic Sets 4.5 Elliptic, Parabolic and Hyperbolic Quadratic Sets
2 Definitions Def: Let Q be a nondegenerate quadratic set in a d-dimensional projective space P (not assumed to be finite). If d is even and the index of Q is ½d then Q is called parabolic. If d is odd and the index of Q is ½(d-1) then Q is called elliptic. Finally, if d is odd and the index of Q is ½(d+1) then Q is called hyperbolic.
3 Examples a) Theorem can be reformulated as: Any nonempty nondegenerate quadratic set of a finite projective space is elliptic, parabolic or hyperbolic. b) The parabolic quadratic sets of a projective plane are the ovals. c) In a 3-dimensional projective space, the elliptic quadratic sets are the ovoids and the hyperbolic quadratic sets are the hyperboloids.
4 A More General Definition of a Cone Def: Let H be a hyperplane of a projective space P, and let V be a point outside of H. If Q* is a nondegenerate quadratic set of H the quadratic set Q = (VX), X Q* is called a cone with vertex V over Q*.
5 Parabolic Quadratic Sets Theorem 4.5.1: Let Q be a parabolic quadratic set in a 2tdimensional projective space P with t > 1. a) Let H = Q P be a tangent hyperplane. Then Q' = Q H is a cone over a parabolic quadratic set with vertex P. b) Let H* be a hyperplane that is not a tangent hyperplane. Then Q* = Q H* is an elliptic or hyperbolic quadratic set. Pf: a) Let W be a complement of P in H, and define Q'' = Q W. By Lemma 4.1.4, Q'' is a nondegenerate quadratic set. If U denotes a maximal Q-subspace through P, then U'' = U W has dimension dim(u) 1. By Theorem 4.2.4, U'' is a maximal Q''-subspace, therefore Q'' is parabolic. Since,by Lemma 4.1.3, the radical of Q' consists of just one point, namely P, Q' is a cone over Q''.
6 Parabolic Quadratic Sets Theorem 4.5.1: Let Q be a parabolic quadratic set in a 2tdimensional projective space P with t > 1. a) Let H = Q P be a tangent hyperplane. Then Q' = Q H is a cone over a parabolic quadratic set with vertex P. b) Let H* be a hyperplane that is not a tangent hyperplane. Then Q* = Q H* is an elliptic or hyperbolic quadratic set. Pf (cont): b) Since H* is not a tangent hyperplane, Q* is nondegenerate (see 4.1.4). A maximal Q-subspace (which is a subspace of dimension t-1) intersects H* in a subspace of dimension t-1 or t-2. This means that Q* is elliptic or hyperbolic.
7 Finite Parabolic Quadratic Sets Corollary 4.5.2: Let Q be a nonempty, nondegenerate quadratic set in P = PG(4,q). Then Q induces in any tangent hyperplane a cone, and in any other hyperplane an ovoid or a hyperboloid. Furthermore, Q consists of exactly q 3 + q 2 + q + 1 points, the number of hyperplanes in which Q induces an ovoid is ½q 2 (q 2-1), and the number of hyperplanes in which Q induces a hyperboloid is ½q 2 (q 2 +1). Pf: We know that Q is parabolic. From this the first assertion follows from the Theorem. In particular, we get that for each point P in Q, the quadratic set induced in Q P is a cone with vertex P. Thus the number a of Q-lines through P is q+1. So, by Thm we have Q = 1 + q 3 + aq = 1 + q 3 + (q+1)q.
8 Finite Parabolic Quadratic Sets Corollary 4.5.2: Let Q be a nonempty, nondegenerate quadratic set in P = PG(4,q). Then Q induces in any tangent hyperplane a cone, and in any other hyperplane an ovoid or a hyperboloid. Furthermore, Q consists of exactly q 3 + q 2 + q + 1 points, the number of hyperplanes in which Q induces an ovoid is ½q 2 (q 2-1), and the number of hyperplanes in which Q induces a hyperboloid is ½q 2 (q 2 +1). Pf(cont.): Let t = Q = q 3 + q 2 + q + 1, be the number of tangent hyperplanes, h the number of hyperboloid hyperplanes and e the number of ovoid hyperplanes. Clearly, t + h + e = q 4 + q 3 + q 2 + q + 1. So, h + e = q 4. Now a tangent hyperplane contains exactly q 2 +q+1 points of Q, while a hyperboloid hyperplane contains (q+1) 2 and an ovoid hyperplane contains q Any point of Q is on exactly q 3 + q 2 + q + 1 hyperplanes, thus we have:
9 Finite Parabolic Quadratic Sets Corollary 4.5.2: Let Q be a nonempty, nondegenerate quadratic set in P = PG(4,q). Then Q induces in any tangent hyperplane a cone, and in any other hyperplane an ovoid or a hyperboloid. Furthermore, Q consists of exactly q 3 + q 2 + q + 1 points, the number of hyperplanes in which Q induces an ovoid is ½q 2 (q 2-1), and the number of hyperplanes in which Q induces a hyperboloid is ½q 2 (q 2 +1). Pf(cont.): t(q 2 + q + 1) + h(q+1) 2 + e(q 2 + 1) = Q (q 3 + q 2 + q + 1). Which implies, h(q+1) 2 + e(q 2 + 1) = Q (q 3 + q 2 + q + 1) - t(q 2 + q + 1) = q 3 (q 3 + q 2 + q + 1). So, h(q+1) 2 + (q 4 - h)(q 2 + 1) = q 3 (q 3 + q 2 + q + 1) or 2qh = q 3 (q 2 + 1).
10 Hyperbolic Quadratic Sets Theorem 4.5.3: Let Q be a hyperbolic quadratic set of a (2t+1)- dimensional projective space P with t > 1. a) If H = Q P is a tangent hyperplane then Q' = Q H is a cone over a hyperbolic quadratic set with vertex P. b) If H* is a hyperplane which is not a tangent hyperplane then Q* = Q H* is a parabolic quadratic set. Pf: a) Let W be a complement of P in H, and define Q'' = Q W. By Lemma 4.1.4, Q'' is a nondegenerate quadratic set. If U denotes a maximal Q-subspace through P, then U'' = U W has dimension dim(u) 1. By Theorem 4.2.4, U'' is a maximal Q''-subspace, therefore Q'' is hyperbolic. Since,by Lemma 4.1.3, the radical of Q' consists of just one point, namely P, Q' is a cone over Q''. b) follows directly from and
11 A Finite Hyperbolic Quadratic Set Theorem 4.5.4: Let Q be a hyperbolic quadratic set in P = PG(5,q). Then a = (q+1) 2 and Q = q 4 + q 3 + 2q 2 + q + 1 = (q 2 + q + 1)(q 2 + 1). Pf: The number of Q lines through a point P ( = a) is the number of lines through the vertex P in Q P, which equals the number of points in the hyperbolic quadratic set of Theorem This hyperbolic quadratic set lies in a 3 dimensional space, and so, has (q+1) 2 points. By b) we obtain: Q = 1 + q 4 + (q+1) 2 q.
12 Equivalence Def: Let Q be a hyperbolic quadratic set of a 5-dimensional projective space P. We say that two Q-planes π 1 and π 2 are equivalent (written π 1 ~ π 2 ) if π 1 and π 2 are equal or intersect in precisely one point. Lemma 4.5.5: Let Q be a hyperbolic quadratic set of a 5- dimensional projective space P. Then the relation ~ is an equivalence relation. Pf: The relation is obviously reflexive and symmetric, so we need only show that it is transitive. Let π 1, π 2 and π 3 be three Q-planes such that π 1 and π 2 meet at a point P and π 2 and π 3 meet at a point R. Since all lines through P in π 1 and π 2 are tangent lines, it follows that Q P = <π 1,π 2 >. Similarly, Q R = <π 2,π 3 >.
13 Equivalence Lemma 4.5.5: Let Q be a hyperbolic quadratic set of a 5-dimensional projective space P. Then the relation ~ is an equivalence relation. Pf(cont.): Case I: P = R. We must show that π 1 and π 3 have no additional points in common. Let W be a complement of P in H = Q P and define Q' = Q W. Let g i = π i W (i = 1,2,3). By we know that Q' is a hyperboloid. Since g 1 and g 2 are skew lines, they belong to the same class of Q' (both in the regulus or both in the opposite regulus). Similarly, g 2 and g 3 are skew, so they are in the same class. Thus, both g 1 and g 3 are in the same class as g 2 and so are skew. Therefore, π 1 π 3 = P.
14 Equivalence Lemma 4.5.5: Let Q be a hyperbolic quadratic set of a 5-dimensional projective space P. Then the relation ~ is an equivalence relation. Pf(cont.): Case II: P R. In this case π 3 is not contained in Q P. Thus, π 3 intersects the hyperplane Q P in a line g 3. Let W be a complement of P in Q P containing g 3, and define Q' = Q W. Then Q' is a hyperboloid, and g 1 = π 1 W and g 2 = π 2 W are lines of the same class of Q'. Since g 3 intersects g 2 in the point R, g 3 belongs to the other class. This implies that g 3 and g 1 also intersect in some point S. Thus, π 1 π 3 = S.
15 Equivalence Classes Theorem 4.5.6: Let Q be a hyperbolic quadratic set of a 5- dimensional projective space P. Then the set of all Q-planes is partitioned into exactly two equivalence classes with respect to ~. Pf: Let π 1 and π 2 be two Q-planes that intersect in a line g (these exist by the proof of Theorem 4.2.3). These belong to different equivalence classes. We have to show that every other Q-plane π belongs to one of these two classes. The subspace V = <π 1,π 2 > has dimension 3 and the quadratic set induced by Q in V consists only of the points on these two planes (if it contained another point then V would be a Q-space... but its dimension is too large). Now let π be any Q-plane different from π 1 and π 2. The intersection of π with V must be contained in π 1 π 2 and by the dimension formula must be a point or a line since π is not contained in V (if empty the dimension of <π,v> would be 6).
16 Equivalence Classes Theorem 4.5.6: Let Q be a hyperbolic quadratic set of a 5-dimensional projective space P. Then the set of all Q-planes is partitioned into exactly two equivalence classes with respect to ~. Pf (cont.): Assume π intersects V in the points of g. Consider a point P on g. By 4.5.3, Q induces in any complement W of P in Q P a hyperboloid. However a point other than P in which π and g intersect is on three Q-planes, thus in W this point is on three distinct Q-lines, a contradiction. So π can not meet g in more than one point. Hence if the intersection is a line, π intersects either one of the planes π 1 or π 2 in a line and the other in a point (the point of intersection of the line with g). On the other hand, if the intersection is a point, the point could not be on g, since that implies that π is equivalent to both π 1 and π 2 (which are not equivalent to each other), so the intersection point is in just one of the planes. In either case π intersects only one of the two planes in a point and so is in the equivalence class of that plane.
17 Generalized Quadrangles Def: A generalized quadrangle is a rank 2 geometry consisting of points and lines such that a) Any two distinct points are on at most one line. b) All lines are incident with the same number of points; all points are incident with the same number of lines. c) If P is a point outside a line g, then there is precisely one line through P intersecting g.
18 Structure Theorem Theorem 4.5.8: The geometry consisting of the points, lines and planes of a hyperbolic quadratic set of a 5-dimensional projective space has the following diagram. Pf: The points, lines and planes of a hyperbolic quadratic set Q form a rank 3 geometry. The residue of a Q-plane (all Q-points and Q-lines incident with a plane) clearly form a projective plane, hence the first link. The residue of a Q-line consists of the points on the line and the planes containing the line, and since each of the former is incident with each of the latter, this is the trivial geometry. Finally, consider the residue of a Q-point, all Q-lines and Q-planes containing the point.
19 Structure Theorem Theorem 4.5.8: The geometry consisting of the points, lines and planes of a hyperbolic quadratic set of a 5-dimensional projective space has the following diagram. Pf: These lines and planes all lie in the tangent hyperplane at that point. Furthermore, by Theorem (a), the intersection of Q with this hyperplane is a cone over a hyperboloid in 3-dimensional space. The lines of this cone can be identified with the points of the hyperboloid and the planes with the lines on the hyperboloid. That is, the Q-planes, thought of as lines of the residue form a regulus and its opposite regulus. This structure is the 3-dimensional grid, a generalized quadrangle.
Analytical Geometry. 2.4 The Hyperbolic Quadric of PG(3,F)
Analytical Geometry 2.4 The Hyperbolic Quadric of PG(3,F) Transversals Def: A set of subspaces of a projective space are skew if no two of them have a point in common. A line is called a transversal of
More informationLax embeddings of the Hermitian Unital
Lax embeddings of the Hermitian Unital V. Pepe and H. Van Maldeghem Abstract In this paper, we prove that every lax generalized Veronesean embedding of the Hermitian unital U of PG(2, L), L a quadratic
More informationSUBSTRUCTURES OF FINITE CLASSICAL POLAR SPACES
In: Current Research Topics in Galois Geometry Editors: J. De Beule and L. Storme, pp. 33-59 ISBN 978-1-61209-523-3 c 2011 Nova Science Publishers, Inc. Chapter 2 SUBSTRUCTURES OF FINITE CLASSICAL POLAR
More informationFinite affine planes in projective spaces
Finite affine planes in projective spaces J. A.Thas H. Van Maldeghem Ghent University, Belgium {jat,hvm}@cage.ugent.be Abstract We classify all representations of an arbitrary affine plane A of order q
More informationHyperplanes of Hermitian dual polar spaces of rank 3 containing a quad
Hyperplanes of Hermitian dual polar spaces of rank 3 containing a quad Bart De Bruyn Ghent University, Department of Mathematics, Krijgslaan 281 (S22), B-9000 Gent, Belgium, E-mail: bdb@cage.ugent.be Abstract
More informationOn the structure of the directions not determined by a large affine point set
On the structure of the directions not determined by a large affine point set Jan De Beule, Peter Sziklai, and Marcella Takáts January 12, 2011 Abstract Given a point set U in an n-dimensional affine space
More informationDerivation Techniques on the Hermitian Surface
Derivation Techniques on the Hermitian Surface A. Cossidente, G. L. Ebert, and G. Marino August 25, 2006 Abstract We discuss derivation like techniques for transforming one locally Hermitian partial ovoid
More informationThis article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution
More informationLax Embeddings of Generalized Quadrangles in Finite Projective Spaces
Lax Embeddings of Generalized Quadrangles in Finite Projective Spaces J. A. Thas H. Van Maldeghem 1 Introduction Definition 1.1 A (finite) generalized quadrangle (GQ) S = (P, B, I) is a point-line incidence
More informationVertex opposition in spherical buildings
Vertex opposition in spherical buildings Anna Kasikova and Hendrik Van Maldeghem Abstract We study to which extent all pairs of opposite vertices of self-opposite type determine a given building. We provide
More informationNumber of points in the intersection of two quadrics defined over finite fields
Number of points in the intersection of two quadrics defined over finite fields Frédéric A. B. EDOUKOU e.mail:abfedoukou@ntu.edu.sg Nanyang Technological University SPMS-MAS Singapore Carleton Finite Fields
More informationLarge minimal covers of PG(3,q)
Large minimal covers of PG(3,q) Aiden A. Bruen Keldon Drudge Abstract A cover of Σ = PG(3,q) is a set of lines S such that each point of Σ is incident with at least one line of S. A cover is minimal if
More informationTwo-intersection sets with respect to lines on the Klein quadric
Two-intersection sets with respect to lines on the Klein quadric F. De Clerck N. De Feyter N. Durante Abstract We construct new examples of sets of points on the Klein quadric Q + (5, q), q even, having
More informationFunctional codes arising from quadric intersections with Hermitian varieties
Functional codes arising from quadric intersections with Hermitian varieties A. Hallez L. Storme June 16, 2010 Abstract We investigate the functional code C h (X) introduced by G. Lachaud [10] in the special
More informationCharacterizations of the finite quadric Veroneseans V 2n
Characterizations of the finite quadric Veroneseans V 2n n J. A. Thas H. Van Maldeghem Abstract We generalize and complete several characterizations of the finite quadric Veroneseans surveyed in [3]. Our
More informationOn small minimal blocking sets in classical generalized quadrangles
On small minimal blocking sets in classical generalized quadrangles Miroslava Cimráková a Jan De Beule b Veerle Fack a, a Research Group on Combinatorial Algorithms and Algorithmic Graph Theory, Department
More informationEuropean Journal of Combinatorics. Locally subquadrangular hyperplanes in symplectic and Hermitian dual polar spaces
European Journal of Combinatorics 31 (2010) 1586 1593 Contents lists available at ScienceDirect European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc Locally subquadrangular hyperplanes
More informationSynthetic Geometry. 1.4 Quotient Geometries
Synthetic Geometry 1.4 Quotient Geometries Quotient Geometries Def: Let Q be a point of P. The rank 2 geometry P/Q whose "points" are the lines of P through Q and whose "lines" are the hyperplanes of of
More informationTheorems of Erdős-Ko-Rado type in polar spaces
Theorems of Erdős-Ko-Rado type in polar spaces Valentina Pepe, Leo Storme, Frédéric Vanhove Department of Mathematics, Ghent University, Krijgslaan 28-S22, 9000 Ghent, Belgium Abstract We consider Erdős-Ko-Rado
More informationOn the geometry of regular hyperbolic fibrations
On the geometry of regular hyperbolic fibrations Matthew R. Brown Gary L. Ebert Deirdre Luyckx January 11, 2006 Abstract Hyperbolic fibrations of PG(3, q) were introduced by Baker, Dover, Ebert and Wantz
More informationThe geometry of secants in embedded polar spaces
The geometry of secants in embedded polar spaces Hans Cuypers Department of Mathematics Eindhoven University of Technology P.O. Box 513 5600 MB Eindhoven The Netherlands June 1, 2006 Abstract Consider
More informationCollineations of polar spaces with restricted displacements
Collineations of polar spaces with restricted displacements B. Temmermans J. A. Thas H. Van Maldeghem Department of Mathematics, Ghent University, Krijgslaan 281, S22, B 9000 Gent btemmerm@cage.ugent.be,
More informationShult Sets and Translation Ovoids of the Hermitian Surface
Shult Sets and Translation Ovoids of the Hermitian Surface A. Cossidente, G. L. Ebert, G. Marino, and A. Siciliano Abstract Starting with carefully chosen sets of points in the Desarguesian affine plane
More informationGeneralized Quadrangles Weakly Embedded in Finite Projective Space
Generalized Quadrangles Weakly Embedded in Finite Projective Space J. A. Thas H. Van Maldeghem Abstract We show that every weak embedding of any finite thick generalized quadrangle of order (s, t) in a
More informationBlocking sets of tangent and external lines to a hyperbolic quadric in P G(3, q), q even
Manuscript 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 Blocking sets of tangent and external lines to a hyperbolic quadric in P G(, q), q even Binod Kumar Sahoo Abstract Bikramaditya Sahu Let H be a fixed hyperbolic
More informationCodes from generalized hexagons
Codes from generalized hexagons A. De Wispelaere H. Van Maldeghem 1st March 2004 Abstract In this paper, we construct some codes that arise from generalized hexagons with small parameters. As our main
More informationA Quasi Curtis-Tits-Phan theorem for the symplectic group
A Quasi Curtis-Tits-Phan theorem for the symplectic group Rieuwert J. Blok 1 and Corneliu Hoffman 1,2 1 Department of Mathematics and Statistics Bowling Green State University Bowling Green, OH 43403-1874
More informationEmbeddings of Small Generalized Polygons
Embeddings of Small Generalized Polygons J. A. Thas 1 H. Van Maldeghem 2 1 Department of Pure Mathematics and Computer Algebra, Ghent University, Galglaan 2, B 9000 Ghent, jat@cage.rug.ac.be 2 Department
More informationOn the intersection of Hermitian surfaces
On the intersection of Hermitian surfaces Nicola Durante and Gary Ebert Abstract In [6] and [3] the authors determine the structure of the intersection of two Hermitian surfaces of PG(3, q 2 ) under the
More informationNeutral Geometry. October 25, c 2009 Charles Delman
Neutral Geometry October 25, 2009 c 2009 Charles Delman Taking Stock: where we have been; where we are going Set Theory & Logic Terms of Geometry: points, lines, incidence, betweenness, congruence. Incidence
More informationGeneralized quadrangles and the Axiom of Veblen
Geometry, Combinatorial Designs and Related Structures (ed. J. W. P. Hirschfeld), Cambridge University Press, London Math. Soc. Lecture Note Ser. 245 (1997), 241 -- 253 Generalized quadrangles and the
More informationSome Two Character Sets
Some Two Character Sets A. Cossidente Dipartimento di Matematica e Informatica Università degli Studi della Basilicata Contrada Macchia Romana 85100 Potenza (ITALY) E mail: cossidente@unibas.it Oliver
More informationA subset of the Hermitian surface
G page 1 / 11 A subset of the Hermitian surface Giorgio Donati Abstract Nicola Durante n this paper we define a ruled algebraic surface of PG(3, q 2 ), called a hyperbolic Q F -set and we prove that it
More informationSemifield flocks, eggs, and ovoids of Q(4, q)
Semifield flocks, eggs, and ovoids of Q(4, q) Michel Lavrauw Universita degli studi di Napoli Federico II Dipartimento di Matematica e Applicazioni R. Caccioppoli Via Cintia Complesso Monte S. Angelo 80126
More informationA characterization of the Split Cayley Generalized Hexagon H(q) using one subhexagon of order (1, q)
A characterization of the Split Cayley Generalized Hexagon H(q) using one subhexagon of order (1, q) Joris De Kaey and Hendrik Van Maldeghem Ghent University, Department of Pure Mathematics and Computer
More informationThe geometry of finite fields
UNIVERSITÀ DEGLI STUDI DI ROMA LA SAPIENZA The geometry of finite fields Simeon Ball Quaderni Elettronici del Seminario di Geometria Combinatoria 2E (Maggio 2001) http://www.mat.uniroma1.it/ combinat/quaderni
More informationUNIVERSITÀ DEGLI STUDI DI ROMA LA SAPIENZA. Semifield spreads
UNIVERSITÀ DEGLI STUDI DI ROMA LA SAPIENZA Semifield spreads Giuseppe Marino and Olga Polverino Quaderni Elettronici del Seminario di Geometria Combinatoria 24E (Dicembre 2007) http://www.mat.uniroma1.it/~combinat/quaderni
More informationA spectrum result on minimal blocking sets with respect to the planes of PG(3, q), q odd
A spectrum result on minimal blocking sets with respect to the planes of PG(3, q), q odd C. Rößing L. Storme January 12, 2010 Abstract This article presents a spectrum result on minimal blocking sets with
More informationChapter 3. Betweenness (ordering) A system satisfying the incidence and betweenness axioms is an ordered incidence plane (p. 118).
Chapter 3 Betweenness (ordering) Point B is between point A and point C is a fundamental, undefined concept. It is abbreviated A B C. A system satisfying the incidence and betweenness axioms is an ordered
More informationβ : V V k, (x, y) x yφ
CLASSICAL GROUPS 21 6. Forms and polar spaces In this section V is a vector space over a field k. 6.1. Sesquilinear forms. A sesquilinear form on V is a function β : V V k for which there exists σ Aut(k)
More informationTight Sets and m-ovoids of Quadrics 1
Tight Sets and m-ovoids of Quadrics 1 Qing Xiang Department of Mathematical Sciences University of Delaware Newark, DE 19716 USA xiang@mathudeledu Joint work with Tao Feng and Koji Momihara 1 T Feng, K
More informationIdentifying codes in vertex-transitive graphs and strongly regular graphs
Identifying codes in vertex-transitive graphs and strongly regular graphs Sylvain Gravier Institut Fourier University of Grenoble Grenoble, France sylvain.gravier@ujf-grenoble.fr Aline Parreau LIRIS University
More informationA characterization of the finite Veronesean by intersection properties
A characterization of the finite Veronesean by intersection properties J. Schillewaert, J.A. Thas and H. Van Maldeghem AMS subject classification: 51E0, 51A45 Abstract. A combinatorial characterization
More informationThe L 3 (4) near octagon
The L 3 (4) near octagon A. Bishnoi and B. De Bruyn October 8, 206 Abstract In recent work we constructed two new near octagons, one related to the finite simple group G 2 (4) and another one as a sub-near-octagon
More informationContents. Index... 15
Contents Filter Bases and Nets................................................................................ 5 Filter Bases and Ultrafilters: A Brief Overview.........................................................
More informationOn the intersection of Hermitian surfaces
J. Geom. 85 (2006) 49 60 0047 2468/06/020049 12 Birkhäuser Verlag, Basel, 2006 DOI 10.1007/s00022-006-0042-4 On the intersection of Hermitian surfaces Luca Giuzzi Abstract. We provide a description of
More informationDense near octagons with four points on each line, III
Dense near octagons with four points on each line, III Bart De Bruyn Ghent University, Department of Pure Mathematics and Computer Algebra, Krijgslaan 281 (S22), B-9000 Gent, Belgium, E-mail: bdb@cage.ugent.be
More informationarxiv: v6 [math.mg] 9 May 2014
arxiv:1311.0131v6 [math.mg] 9 May 2014 A Clifford algebraic Approach to Line Geometry Daniel Klawitter Abstract. In this paper we combine methods from projective geometry, Klein s model, and Clifford algebra.
More informationConstructing the Tits Ovoid from an Elliptic Quadric. Bill Cherowitzo UCDHSC-DDC July 1, 2006 Combinatorics 2006
Constructing the Tits Ovoid from an Elliptic Quadric Bill Cherowitzo UCDHSC-DDC July 1, 2006 Combinatorics 2006 Ovoids An ovoid in PG(3,q) is a set of q 2 +1 points, no three of which are collinear. Ovoids
More information1.4 Equivalence Relations and Partitions
24 CHAPTER 1. REVIEW 1.4 Equivalence Relations and Partitions 1.4.1 Equivalence Relations Definition 1.4.1 (Relation) A binary relation or a relation on a set S is a set R of ordered pairs. This is a very
More informationExterior powers and Clifford algebras
10 Exterior powers and Clifford algebras In this chapter, various algebraic constructions (exterior products and Clifford algebras) are used to embed some geometries related to projective and polar spaces
More informationCodewords of small weight in the (dual) code of points and k-spaces of P G(n, q)
Codewords of small weight in the (dual) code of points and k-spaces of P G(n, q) M. Lavrauw L. Storme G. Van de Voorde October 4, 2007 Abstract In this paper, we study the p-ary linear code C k (n, q),
More informationPlanar and Affine Spaces
Planar and Affine Spaces Pýnar Anapa İbrahim Günaltılı Hendrik Van Maldeghem Abstract In this note, we characterize finite 3-dimensional affine spaces as the only linear spaces endowed with set Ω of proper
More information10. Noether Normalization and Hilbert s Nullstellensatz
10. Noether Normalization and Hilbert s Nullstellensatz 91 10. Noether Normalization and Hilbert s Nullstellensatz In the last chapter we have gained much understanding for integral and finite ring extensions.
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013
18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013 Throughout this lecture k denotes an algebraically closed field. 17.1 Tangent spaces and hypersurfaces For any polynomial f k[x
More informationarxiv: v1 [math.mg] 4 Jan 2013
On the boundary of closed convex sets in E n arxiv:1301.0688v1 [math.mg] 4 Jan 2013 January 7, 2013 M. Beltagy Faculty of Science, Tanta University, Tanta, Egypt E-mail: beltagy50@yahoo.com. S. Shenawy
More informationRecursive constructions for large caps
Recursive constructions for large caps Yves Edel Jürgen Bierbrauer Abstract We introduce several recursive constructions for caps in projective spaces. These generalize the known constructions in an essential
More informationOrthogonal Arrays & Codes
Orthogonal Arrays & Codes Orthogonal Arrays - Redux An orthogonal array of strength t, a t-(v,k,λ)-oa, is a λv t x k array of v symbols, such that in any t columns of the array every one of the possible
More informationTwo-character sets as subsets of parabolic quadrics
Two-character sets as subsets of parabolic uadrics Bart De Bruyn Ghent University Department of Mathematics Krijgslaan 81 (S) B-9000 Gent Belgium E-mail: bdb@cageugentbe Abstract A two-character set is
More informationOn Exceptional Lie Geometries
On Exceptional Lie Geometries Anneleen De Schepper Jeroen Schillewaert Hendrik Van Maldeghem Magali Victoor Dedicated to the memory of Ernie Shult Abstract Parapolar spaces are point-line geometries introduced
More informationSPG systems and semipartial geometries
Adv. Geom. 1 (2001), 229±244 Advances in Geometry ( de Gruyter 2001 SPG systems and semipartial geometries (Communicated by H. Van Maldeghem) Abstract. First, the paper contains new necessary and su½cient
More informationGeneralized Quadrangles with a Spread of Symmetry
Europ. J. Combinatorics (999) 20, 759 77 Article No. eujc.999.0342 Available online at http://www.idealibrary.com on Generalized Quadrangles with a Spread of Symmetry BART DE BRUYN We present a common
More informationA characterization of the set of lines either external to or secant to an ovoid in PG(3,q)
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 49 (011), Pages 159 163 A characterization of the set of lines either external to or secant to an ovoid in PG(3,q) Stefano Innamorati Dipartimento di Ingegneria
More informationCHAPTER 7. Connectedness
CHAPTER 7 Connectedness 7.1. Connected topological spaces Definition 7.1. A topological space (X, T X ) is said to be connected if there is no continuous surjection f : X {0, 1} where the two point set
More informationOn the nucleus of the Grassmann embedding of the symplectic dual polar space DSp(2n, F), char(f) = 2
On the nucleus of the Grassmann embedding of the symplectic dual polar space DSp(2n, F), char(f) = 2 Rieuwert J. Blok Department of Mathematics and Statistics Bowling Green State University Bowling Green,
More informationGroup theoretic characterizations of Buekenhout Metz unitals in PG(2,q 2 )
J Algebr Comb (2011) 33: 401 407 DOI 10.1007/s10801-010-0250-8 Group theoretic characterizations of Buekenhout Metz unitals in PG(2,q 2 ) Giorgio Donati Nicola Durante Received: 1 March 2010 / Accepted:
More informationIntransitive geometries
Intransitive geometries Ralf Gramlich Hendrik Van Maldeghem April 10, 006 Abstract A lemma of Tits establishes a connection between the simple connectivity of an incidence geometry and the universal completion
More informationDefinition. Example: In Z 13
Difference Sets Definition Suppose that G = (G,+) is a finite group of order v with identity 0 written additively but not necessarily abelian. A (v,k,λ)-difference set in G is a subset D of G of size k
More informationDomesticity in projective spaces
Innovations in Incidence Geometry Volume 12 (2011), Pages 141 149 ISSN 1781-6475 ACADEMIA PRESS Domesticity in projective spaces Beukje Temmermans Joseph A. Thas Hendrik Van Maldeghem Abstract Let J be
More informationA Lemma on the Minimal Counter-example of Frankl s Conjecture
A Lemma on the Minimal Counter-example of Frankl s Conjecture Ankush Hore email: ankushore@gmail.com Abstract Frankl s Conjecture, from 1979, states that any finite union-closed family, containing at least
More informationThe Symmetric Space for SL n (R)
The Symmetric Space for SL n (R) Rich Schwartz November 27, 2013 The purpose of these notes is to discuss the symmetric space X on which SL n (R) acts. Here, as usual, SL n (R) denotes the group of n n
More informationComputational Methods in Finite Geometry
Computational Methods in Finite Geometry Anton Betten Colorado State University Summer School, Brighton, 2017 Topic # 4 Cubic Surfaces Cubic Surfaces with 27 Lines A cubic surface in PG(, q) is defined
More informationGeneralized Veronesean embeddings of projective spaces, Part II. The lax case.
Generalized Veronesean embeddings of projective spaces, Part II. The lax case. Z. Akça A. Bayar S. Ekmekçi R. Kaya J. A. Thas H. Van Maldeghem Abstract We classify all embeddings θ : PG(n, K) PG(d, F),
More informationThe maximum size of a partial spread in H(4n + 1,q 2 ) is q 2n+1 + 1
The maximum size of a partial spread in H(4n +, 2 ) is 2n+ + Frédéric Vanhove Dept. of Pure Mathematics and Computer Algebra, Ghent University Krijgslaan 28 S22 B-9000 Ghent, Belgium fvanhove@cage.ugent.be
More information13. Forms and polar spaces
58 NICK GILL In this section V is a vector space over a field k. 13. Forms and polar spaces 13.1. Sesquilinear forms. A sesquilinear form on V is a function β : V V k for which there exists σ Aut(k) such
More informationCharacterizations of Segre Varieties
Characterizations of Segre Varieties J. A Thas H. Van Maldeghem Abstract In this paper several characterizations of Segre varieties and their projections are given. The first two characterization theorems
More informationUnextendible Mutually Unbiased Bases (after Mandayam, Bandyopadhyay, Grassl and Wootters)
Article Unextendible Mutually Unbiased Bases (after Mandayam, Bandyopadhyay, Grassl and Wootters) Koen Thas Department of Mathematics, Ghent University, Ghent 9000, Belgium; koen.thas@gmail.com Academic
More informationON THE STRUCTURE OF (t mod q)-arcs IN FINITE PROJECTIVE GEOMETRIES
GODIXNIK NA SOFI SKI UNIVERSITET SV. KLIMENT OHRIDSKI FAKULTET PO MATEMATIKA I INFORMATIKA Tom 103 ANNUAL OF SOFIA UNIVERSITY ST. KLIMENT OHRIDSKI FACULTY OF MATHEMATICS AND INFORMATICS Volume 103 ON THE
More informationOn bisectors in Minkowski normed space.
On bisectors in Minkowski normed space. Á.G.Horváth Department of Geometry, Technical University of Budapest, H-1521 Budapest, Hungary November 6, 1997 Abstract In this paper we discuss the concept of
More informationCollineation groups of translation planes admitting hyperbolic Buekenhout or parabolic Buekenhout Metz unitals
Journal of Combinatorial Theory, Series A 114 2007 658 680 www.elsevier.com/locate/jcta Collineation groups of translation planes admitting hyperbolic Buekenhout or parabolic Buekenhout Metz unitals Norman
More informationRepresentation Theory and Orbital Varieties. Thomas Pietraho Bowdoin College
Representation Theory and Orbital Varieties Thomas Pietraho Bowdoin College 1 Unitary Dual Problem Definition. A representation (π, V ) of a group G is a homomorphism ρ from G to GL(V ), the set of invertible
More informationLINKAGE CLASSES OF GRADE 3 PERFECT IDEALS
LINKAGE CLASSES OF GRADE 3 PERFECT IDEALS LARS WINTHER CHRISTENSEN, OANA VELICHE, AND JERZY WEYMAN Abstract. While every grade 2 perfect ideal in a regular local ring is linked to a complete intersection
More informationGroups of Prime Power Order with Derived Subgroup of Prime Order
Journal of Algebra 219, 625 657 (1999) Article ID jabr.1998.7909, available online at http://www.idealibrary.com on Groups of Prime Power Order with Derived Subgroup of Prime Order Simon R. Blackburn*
More information6.4 BASIS AND DIMENSION (Review) DEF 1 Vectors v 1, v 2,, v k in a vector space V are said to form a basis for V if. (a) v 1,, v k span V and
6.4 BASIS AND DIMENSION (Review) DEF 1 Vectors v 1, v 2,, v k in a vector space V are said to form a basis for V if (a) v 1,, v k span V and (b) v 1,, v k are linearly independent. HMHsueh 1 Natural Basis
More informationHyperbolic lines in generalized polygons
J. van Bon, H. Cuypers and H. Van Maldeghem August 29, 2013 Abstract In this paper we develop the theory of hyperbolic lines in generalized polygons. In particular, we investigate the extremal situation
More informationA Geometric Characterization of the Perfect Ree-Tits Generalized Octagons.
A Geometric Characterization of the Perfect Ree-Tits Generalized Octagons. H. Van Maldeghem Dedicated to Prof. J. Tits on the occasion of his 65th birthday 1 Introduction The world of Tits-buildings, created
More informationOn the chromatic number of q-kneser graphs
On the chromatic number of q-kneser graphs A. Blokhuis & A. E. Brouwer Dept. of Mathematics, Eindhoven University of Technology, P.O. Box 53, 5600 MB Eindhoven, The Netherlands aartb@win.tue.nl, aeb@cwi.nl
More informationCompact hyperbolic Coxeter n-polytopes with n + 3 facets
Compact hyperbolic Coxeter n-polytopes with n + 3 facets Pavel Tumarkin Independent University of Moscow B. Vlassievskii 11, 11900 Moscow, Russia pasha@mccme.ru Submitted: Apr 3, 007; Accepted: Sep 30,
More informationDistance-j Ovoids and Related Structures in Generalized Polygons
Distance-j Ovoids and Related Structures in Generalized Polygons Alan Offer and Hendrik Van Maldeghem Department of Pure Mathematics and Computer Algebra, Ghent University, Galglaan 2, B 9000 Ghent, Belgium
More informationBase subsets of polar Grassmannians
Journal of Combinatorial Theory, Series A 114 (2007) 1394 1406 www.elsevier.com/locate/jcta Base subsets of polar Grassmannians Mark Pankov Department of Mathematics and Information Technology, University
More informationProjection pencils of quadrics and Ivory s theorem
Projection pencils of quadrics and Ivory s theorem Á.G. Horváth Abstract. Using selfadjoint regular endomorphisms, the authors of [7] defined, for an indefinite inner product, a variant of the notion of
More informationCourse Notes for EE227C (Spring 2018): Convex Optimization and Approximation
Course Notes for EE227C (Spring 2018): Convex Optimization and Approximation Instructor: Moritz Hardt Email: hardt+ee227c@berkeley.edu Graduate Instructor: Max Simchowitz Email: msimchow+ee227c@berkeley.edu
More informationOn Cameron-Liebler line classes with large parameter
On with large parameter J. De Beule (joint work with Jeroen Demeyer, Klaus Metsch and Morgan Rodgers) Department of Mathematics Ghent University Department of Mathematics Vrije Universiteit Brussel June
More informationGeneralized polygons in projective spaces
Generalized polygons in projective spaces Hendrik Van Maldeghem Department of Pure Mathematics and Computer Algebra, Ghent University, Galglaan 2, B-9000 Gent, Belgium, e-mail: hvm@cage.rug.ac.be 1 Introduction
More informationThe Flag-Transitive C3-Geometries of Finite Order
Journal of Algebraic Combinatorics 5 (1996), 251-284 1996 Kluwer Academic Publishers. Manufactured in The Netherlands. The Flag-Transitive C3-Geometries of Finite Order SATOSHI YOSHIARA yoshiara@cc.osaka-kyoiku.ac.jp
More informationTransitive Partial Hyperbolic Flocks of Deficiency One
Note di Matematica Note Mat. 29 2009), n. 1, 89-98 ISSN 1123-2536, e-issn 1590-0932 DOI 10.1285/i15900932v29n1p89 Note http://siba-ese.unisalento.it, di Matematica 29, n. 2009 1, 2009, Università 89 98.
More informationAbout Maximal Partial 2-Spreads in PG(3m 1, q)
Innovations in Incidence Geometry Volume 00 (XXXX), Pages 000 000 ISSN 1781-6475 About Maximal Partial 2-Spreads in PG(3m 1, q) Sz. L. Fancsali and P. Sziklai Abstract In this article we construct maximal
More informationNAME: Mathematics 205A, Fall 2008, Final Examination. Answer Key
NAME: Mathematics 205A, Fall 2008, Final Examination Answer Key 1 1. [25 points] Let X be a set with 2 or more elements. Show that there are topologies U and V on X such that the identity map J : (X, U)
More information1 Some loose ends from last time
Cornell University, Fall 2010 CS 6820: Algorithms Lecture notes: Kruskal s and Borůvka s MST algorithms September 20, 2010 1 Some loose ends from last time 1.1 A lemma concerning greedy algorithms and
More informationSPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS
SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS ROYA BEHESHTI AND N. MOHAN KUMAR Abstract. We prove that the space of smooth rational curves of degree e in a general complete intersection of multidegree
More information