A NEW METHOD OF CONSTRUCTING MAXIMAL PARTIAL SPREADS OF PG(3, q), MAPPING PG(3, q) OVER A NON-SINGULAR QUADRIC Q(4, q) OF PG(4, q)
نویسندگان
چکیده
We transfer the whole geometry of PG (3, q) over a non-singular quadric Q(4, q) of PG (4, q) mapping suitably PG (3, q) over Q(4, q). More precisely the points of PG (3, q) are the lines of Q(4, q); the lines of PG (3, q) are the tangent cones of Q(4, q) and the reguli of the hyperbolic quadrics hyperplane section of Q(4, q). A plane of PG (3, q) is the set of lines of Q(4, q) meeting a fixed line of Q(4, q). We remark that this representation is valid also for a projective space P3,K over any field K and we apply the above representation to construct maximal partial spreads F in PG(3, q). For q even we get new cardinalities for F. For q odd the cardinalities are partially known.
منابع مشابه
Symplectic spreads from twisted fields
A ,yml'/eclic 'l'J"wd of PG(2n + l,q) is a spread of the symplectic polar space ~V(2n + l,q) defined by a nonsingular alternating bilinear form on a (2n+2)dimensional vector space over GF(q), i.e., a set of q"+l + 1 pairwise disjoint maximal totally isotropic subspaces. Note that a symplectic spread of PG(3, q) is equivalent, under the Klein correspondence, to an ovoid of the quadric Q( 4, q). ...
متن کاملA spectrum result on minimal blocking sets with respect to the planes of PG(3, q), q odd
This article presents a spectrum result on minimal blocking sets with respect to the planes of PG(3, q) , q odd. We prove that for every integer k in an interval of, roughly, size [q 2 /4, 3q 2 /4] , there exists such a minimal blocking set of size k in PG(3, q) , q odd. A similar result on the spectrum of minimal blocking sets with respect to the planes of PG(3, q) , q even, was presented in [...
متن کاملOn the non-existence of a maximal partial spread of size 76 in PG(3, 9)
We prove the non-existence of maximal partial spreads of size 76 in PG(3, 9). Relying on the classification of the minimal blocking sets of size 15 in PG(2, 9) [22], we show that there are only two possibilities for the set of holes of such a maximal partial spread. The weight argument of Blokhuis and Metsch [3] then shows that these sets cannot be the set of holes of a maximal partial spread o...
متن کاملMaximal partial line spreads of non-singular quadrics
For n ≥ 9, we construct maximal partial line spreads for non-singular quadrics of PG(n, q) for every size between approximately (cn + d)(qn−3 + qn−5) log 2q and qn−2, for some small constants c and d. These results are similar to spectrum results on maximal partial line spreads in finite projective spaces by Heden, and by Gács and Szőnyi. These results also extend spectrum results on maximal pa...
متن کاملA spectrum result on maximal partial ovoids of the generalized quadrangle Q(4, q), q even
In this article, we prove a spectrum result on maximal partial ovoids of the generalized quadrangle Q(4, q), q odd, i.e. for every integer k in the interval [a, b], where a ≈ 3 5 q and b ≈ 9 10 q, there exists a maximal partial ovoid of Q(4, q), q odd, of size k. Since the generalized quadrangle W(q) defined by a symplectic polarity of PG(3, q) is isomorphic to the dual of the generalized quadr...
متن کامل