1,355,200 research outputs found
Intransitive collineation groups of ovals fixing a triangle
No full textWe investigate collineation groups of a finite projective plane of odd order fixing an oval and having two orbits on it, one of which is assumed to be primitive. The situation in which there exists a fixed triangle off the oval is considered in detail. Our main result is the following. Theorem. Let P be a finite projective plane of odd order n containing an oval O. If a collineation group G of P satisfies the properties: (a) G fixes O and the action of G on O yields precisely two orbits O_1 and O_2, (b) G has even order and a faithful primitive action on O_2, (c) G fixes neither points nor lines but fixes a triangle ABC in which the points A, B, C are not on the oval O, then n is an element of {7, 9, 27}, the orbit O_2 has length 4 and G acts naturally on O_2 as A_4 or S_4. Each order n in {7, 9, 27} does furnish at least one example for the above situation; the determination of the planes and the groups which do occur is complete for n = 7, 9; the determination of the planes is still incomplete for n = 27
On quasi-Hermitian varieties in even characteristic and related orthogonal arrays
No full textIn this paper we study the BM quasi-Hermitian varieties introduced in [A.
Aguglia, A. Cossidente, G. Korchm\`aros, On quasi-Hermitian Varieties, J.
Combin. Des. 20 (2012) 433-447.] in characteristc and dimension . After
a brief investigation of their combinatorial properties, we first show that all
of these varieties are projectively equivalent in non-zero even characteristic,
exhibiting a behavior which is strikingly different from what happens in odd
characteristic, see [A. Aguglia, L. Giuzzi, On the equivalence of certain
quasi-Hermitian varieties, J. Combin. Des. 1-15 (2022)]. This completes the
classification project started in that paper. Next, by using previous results,
we explicitly determine and investigate the structure of the full collineation
group stabilizing these varieties. Finally, as a byproduct of our
investigation, we also construct and a family of simple orthogonal arrays
with entries in where is a power of .Comment: 22 pages/submitted version. Work supported by INDAM. A. Aguglia and
V. Siconolfi have been partially supported by the EU under the Italian NRRP
of NextGenerationEU, partnership on "Telecommunications of the Future"
(PE00000001 - program RESTART, CUP: D93C22000910001
Irreducible collineation groups with two orbits forming an oval
No full textLet G be a collineation group of a finite projective plane P of odd order fixing an oval Ω. We investigate the case in which G has even order, has two orbits Ω_0 and Ω_1 on Ω, and the action of G on Ω_0 is primitive.We show that if G is irreducible, then P has a G-invariant desarguesian subplane P_0 and Ω_0 is a conic of P_0
On the non-existence of a projective plane of order 15 with an A_4-invariant oval
No full textLet P be a projective plane of order 15 with an oval O. Assume P admits a collineation group G fixing O such that G is isomorphic to A_4 and the action of G on O yields precisely two orbits O(1) and O(2) with |O(2)|= 4. We prove that the Buekenhout oval arising from O cannot exist
Blocking sets of nonsecant lines to a conic in PG(2,q), q odd
No full textIn a previous paper [1], all point sets of minimum size in PG(2,q), blocking all external lines to a given irreducible conic C, have been determined for every odd q. Here we obtain a similar classification for those point sets of minimum size, which meet every external and tangent line to C
Blocking sets of external lines to a conic in PG(2,q), q odd
No full textWe determine all point-sets of minimum size in , odd that meet every external line to a conic in . The proof uses a result on the linear system of polynomials vanishing at every internal point to the conic and a corollary to the classification theorem of all subgroups of
On algebraic curves over a finite field with many rational points
No full textIn [12], a new upper bound for the number of F-q-rational points on an absolutely irreducible algebraic plane curve defined over a finite field F-q of degree d q - 2 was obtained. The present paper is a continuation of [12] and the main result is a similar upper bound for the case d = rootq - 2
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