A Note on Embedding the 4-arcs of Fano Plane with Quadric and Cubic Veroneseans to Projective Spaces (original) (raw)
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On The Fano Planes in (7n-1)-dimensional Projective Spaces
Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi
In this study, we obtain Fano planes whose points are (n-1)-dimensional subspaces and lines are (3n-1)-dimensional subspaces of P in (7n-1)-dimensional projective space P ,using (n;k)-SCID. We give examplesof Fano planes whose the lines of Fano configuration are planes and the points are points ofPG(6,2)and thelines of Fano configuration are 5-spaces and the points are lines ofPG(13,2).
A Note on Fano Configurations in the Projective Space PG(5,2)
Konuralp Journal of Mathematics (KJM), 2021
Let n ≥ 2 and let U j | j ∈ J, with |J| = n 2 + n + 1, be a set of disjoint subspaces (of the same dimension) of some finite projective space PG(N, q) with the property that the number of such subspaces in the span of any two such subspaces is always n + 1 and the intersection of any two different such spans is always a subspace U j (so we have a projective plane of order n with point set U j | j ∈ J.) In this work we search for Fano configurations in PG(5,2) whose lines are 3-spaces and points are lines.
Journal of Combinatorial Theory, Series A, 2000
The flag geometry 1=(P, L, I) of a finite projective plane 6 of order s is the generalized hexagon of order (s, 1) obtained from 6 by putting P equal to the set of all flags of 6, by putting L equal to the set of all points and lines of 6, and where I is the natural incidence relation (inverse containment), i.e., 1 is the dual of the double of 6 in the sense of H. Van Maldeghem (1998,``Generalized Polygons,'' Birkha user Verlag, Basel). Then we say that 1 is fully and weakly embedded in the finite projective space PG(d, q) if 1 is a subgeometry of the natural point-line geometry associated with PG(d, q), if s=q, if the set of points of 1 generates PG(d, q), and if the set of points of 1 not opposite any given point of 1 does not generate PG(d, q). In two earlier papers we have shown that the dimension d of the projective space belongs to [6, 7, 8], that the projective plane 6 is Desarguesian, and we have classified the full and weak embeddings of 1 (1 as above) in the case that there are two opposite lines L, M of 1 with the property that the subspace U L, M of PG(d, q) generated by all lines of 1 meeting either L or M has dimension 6 (which is automatically satisfied if d=6). In the present paper, we partly handle the case d=7; more precisely, we consider for d=7 the case where for all pairs (L, M) of opposite lines of 1, the subspace U L, M has dimension 7 and where there exist four lines concurrent with L contained in a 4-dimensional subspace of PG(7, q).
On the incidence structures of polar spaces and quadrics
Advances in Geometry, 2000
In 1969 Buekenhout characterized non-singular quadrics of finite-dimensional projective spaces as polar spaces spanning the whole space and containing every line of the projective space which they intersect in at least three points (see ). Using suitable synthetic properties of the pairs of non-collinear disjoint lines, in this paper I present a new characterization of polar spaces and a combinatorial characterization of non-singular quadrics of a projective space of arbitrary dimension. Moreover, the extension of the theorem of Buekenhout even to the infinitedimensional case is given.
A note on Nk configurations and theorems in projective space
Bulletin of The Australian Mathematical Society, 2007
A method of embedding nk configurations into projective space of k-1 dimensions is given. It breaks into the easy problem of finding a rooted spanning tree of the associated Levi graph. Also it is shown how to obtain a "complementary" n n-k "theorem" about projective space (over a field or skew-field F) from any n* theorem over F. Some elementary matroid theory is used, but with an explanation suitable for most people. Various examples are mentioned, including the planar configurations: Fano 73, Pappus 93, Desargues IO3 (also in 3d-space), Mobius 84 (in 3d-space), and the resulting 7t in 3d-space, 96 in 5d-space, and IO7 in 6d-space. (The Mobius configuration is self-complementary.) There are some n/t configurations that are not embeddable in certain projective spaces, and these will be taken to similarly not embeddable configurations by complementation. Finally, there is a list of open questions.
The Reverse construction of complete (k, n)-arcs in three-dimensional projective space PG(3,4)
2020
In this work, the complete (k, n) arcs in PG(3,4) over Galois field GF(4) can be creat ed by removing some points from the complete arcs of degree m, where m = n + 1, 3 n q2 + q is used . In addition, where k ≤ 85, we geometrically prove that the minimum complete (k, n)-arc in PG(3,4) is (5,3)-arc. A(k, n)-arcs is a set of k points no n+1 of which collinear. A(k, n ) -arcs is complete unless it is embedded in an arc (k+1,n).
On regular {v, n}-arcs in finite projective spaces
Journal of Combinatorial Designs, 1993
A regular {v,n}-arc of a projective space P of order q is a set S of Y points such that each line of P has exactly 0 , l or n points in common with S and such that there exists a line of P intersecting S in exactly n points. Our main results are as follows: (1) If P is a projective plane of order q and if S is a regular {v, n}-arc with n 2 f i + 1, then S is a set of n collinear points, a Baer subplane, a unital, or a maximal arc. (2) If P is a projective space of order q and if S is a regular {v,n}-arc with n 2 f i + 1 spanning a subspace U of dimension at least 3, then S is a Baer subspace of U, an affine space of order q in U, or S equals the point Set Of u.
Triple System and Fano Plane Structure in
2020
A triple system is an absolutely fascinating concept in projective geometry. This paper is an extension of previously done work on triple systems, specifically the triples that fit into a Fano plane and the (i, j, k) triples of the quaternion group. Here, we have explored and determined the existence of triple systems in for n = p, n = pq and n = 2p with m εN, p, qεP, and p > q, where N is the set of natural numbers, P is the set of primes and is the set of units in Zn. A triple system in has been denoted by (k1,k2,k3) where there exists ki > 1, i = 1,2,3, such that ki 2 ≡ 1(mod n) with k1k2 ≡ k3(mod n), k1k3 ≡ k2 (mod n) and k2k3 ≡ k1 (mod n). We have also investigated the number of triples in and determined the general formula for getting the triples. Further, we have fitted the triples into Fano planes and established the projective geometry structure for the above defined . AMS Subject Classification: 20B05
On the construction of arcs using quadrics
Australas. J Comb., 1994
A new method of constructing arcs in projective space is given. It is a generalisation of the fact that a normal rational curve can be given by the complete intersection of a set of quadrics. The non-classicallO-arc of PG(4, 9) together with its special point is the set of derived points of a. cubic primal. This property is shared with the normal rational curve of this spaceo L INTRODUCTION AND NOTATION This paper shows that quadrics (quadratic or hypersurfaces in finite projective space) are related to k-arcs and their associated curves in a fundamental way. The methods of classical algebraic geometry are necessary for much of the discussion; see [1,26]. Before delving into these connections we shall give the reader a gentle introduction to the known constructions and theory of arcs. The geometries which are the object for this discussion are, in the main, the finite projective spaces PG(n,q) of dimension n over GF(q), for n 2: 2, and q = ph, P prime. However, many of the results a...
Note on Embedding a Class of Finite Planar Spaces into 3-Dimensional Projective Spaces
Results in Mathematics, 2009
Let S = (P, L, H) be a finite planar space such that any two planes intersect in a line. Such planar spaces are interesting, since they provide examples for a class of finite geometries which is conjectured to be embeddable in a finite projective geometry (see e.g. ). In this paper, a new embedding condition for a class of these finite planar spaces is given.