A Characterisation of the Lines Externaltoan Oval Cone in PG(3,q), q Even

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Consider the planes through m. Any plane through m contains V , and so is not a secant plane. Also, any plane through m has at least two black points P and V , so it is not a V-plane. Thus all planes through m are 0-planes. Since there are no lines of L in any 0-plane, there can be no lines of L meeting m. Thus m must consist entirely of black points. So a line through V either has V as its only black point, or consists entirely of black points.

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Now let m be a line not through V and suppose m is not a line of L. Let Bm be the number of black points on m. We will count the total number of all black points by considering the black points in the planes about m. Now exactly one plane passes through both m and V. If this plane were a V-plane, then m would be a line of L as noted in the proof of Lemma 2. The other q planes 26 S. Hence a line not through V is either a line of L or contains exactly 2 black points.

Then m cannot contain the point V , since no secant plane passes through V. Hence, by Lemma 2.

The set of black points C is a hyperoval cone and L is its set of external lines. Now each of these lines passes through V and contains at least two black points, so by Lemma 2. Thus, C is the set of points on the lines joining V to the points of a hyperoval O.

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That is, C is a hyperoval cone. It remains to show that each of these cones has L as its set of external lines. Each of the oval cones contained in C has L as its set of external lines. Di Gennaro, N. Durante and D. Olanda, A characterization of the family of lines external to a hyperbolic quadric of PG 3, q , J. Olanda, A characterization of the family of secant or external lines of an ovoid of PG 3, q , Bull. Simon Stevin 12 1—4. Hell On Dirt Oval. Read more. Crompton Characterisation of Polymers. Even In The Darkness. Even in the Grave.

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Even in the Best of Families. Even in Death. Characterisation of Porous Solids V. A bilinear approach to cone multipliers I. A bilinear approach to cone multipliers II. Recommend Documents. Crompton 2 C ontents 1. Pyrolysis — Gas Chromatography Now, let any four collinear points, in the schema of H, and any other point together with the lines incident with them, be neglected, the result is an configuration. To illustrate the method, upon neglecting the four colinear points incident with f 12 and any other point, say 3, together with the five lines incident with them, we obtain the configuration 83 , which is described by the following schema d 9 d 8 E7 La5 E4.

Figure 2. We thus have the following simple result: Proposition 2. For this purpose, select any two points not both incident with any line of the configuration , say 17 and Thus we have shown: Proposition 3. The symmetric net S 4 of order 4 may be partitioned into two disjoint configurations Consequently, both AG 2, 4 and PG 2, 4 contain pairs of disjoint 83's. Of course, we may also describe PG 2, 4 by using homogeneous coordinates over GF 4 see e.

Pickett [8].

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After discarding the five points 0, 1, 0 , 1, 0, i , 1, 1, 1 , 1, co, 1 , 1, co2, 1 , which are incident with the line [1, 0, 1], we obtain the following two 83 configurations: co2, co, 1 0,o, 1 co, 1,1 0, co, l O,l, co 0,1,1 coz 1, 1 1, 1, coz co, 0, 1 1, 0, co 1,1, co 1,0,0 1, co, co2 1,1,0 o ,1,0 1, co, 0 Summarizing our results, we have proved Theorem 3,2.

In PG 2, 4 , there exist: 1 2 two disjoint configurations of the type 83; and a chain of four simple quadrangles each is inscribed in the next. In Section 5, we shall show that any 83 contained in PG 2, 4 extends to an affine subplane of order 3 in PG 2, 4 ; then it will be easy to see that the following holds: Proposition 3.

PG 2, 4 contains exactly configurations 83 and exactly affine subplanes of order 3. In this section we briefly consider the existence of pairwise disjoint 83's in projective and affine spaces over GF 4 ; this will be a simple application of Proposition 3. We first note: Proposition 4. AG n, 4 can be partitioned into 2 2n-3 pairwise disjoint 83'S. By Proposition 3.

But each of these affine planes can be partitioned into two disjoint 83's. Assume 9 c H; then the assertion follows by induction, observing that H is isomorphic to PG n - 1, 4. We now consider the general question of embedding 83 into the Pappian projective plane PG 2, F over some commutative field F. The coordinates are now as shown in Figure 4. In all other cases, x has to be a primitive 6'th root of unity. Thus we have the following Theorem 5. Let F be a commutative field. Thus consider the 83 of Figure 4, embedded into PG 2, F.

We have shown: Theorem 5. For example, the two disjoint 83's within PG 2, 4 given in section 3 extend to the two affine planes 14 17 18 12 0 15 6 20 19 and 1 16 13 5 0 7 10 2 8 intersecting in the point O. We do not know whether or not PG 2, 4 contains two disjoint affine subplanes of order 3. We now specialize the results of section 5 to the finite case. PG 2, q contains 83 if and only if q is a power of 3 or 4 or q is congruent to 1 modulo 6.

Any 83 contained in PG 2, q extends to an affine subplane of order 3 of PG 2, q. Our proof in fact allows us to determine the number of 83's contained in PG 2, q. O n the other hand, each 83 contains two quadrangles and thus is counted twice. Hirschfeld [5]. Hence we have: Proposition 6. Since each 83 contained in PG 2, q extends to a unique AG 2, 3 and since AG 2, 3 contains 9 copies of 83, this also implies the following: M. Corollary 6. This still leaves the question, however, whether "large parts" of AG 2, r for example, BAG 2, r can be c o n t a i n e d in PG 2, q for other values of q.

The third author would like to thank the University of Kuwait for its hospitality during the time of this research. References [1] A.