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Let H be a hyperbolic quadric in P G ( 3 , q ) , where q is a prime power. Let E (respectively, T ) denote the set of all lines of P G ( 3 , q ) which are external (respectively, tangent) to H . We characterize the minimum size blocking sets in P G ( 3 , q ) , q ≠ 2 , with respect to the line set E ∪ T . We also give an alternate proof characterizing the minimum size blocking sets in P G ( 3 , q ) with respect to the line set E for all odd q .

InźSharir and Solomon (2015), Sharir and Solomon showed that the number of incidences between m distinct points and n distinct lines in R 4 is (1) O ź m 2 ź 5 n 4 ź 5 + m 1 ź 2 n 1 ź 2 q 1 ź 4 + m 2 ź 3 n 1 ź 3 s 1 ź 3 + m + n , provided that no 2-flat contains more than s lines, and no hyperplane or quadric contains more than q lines, where the O ź hides a multiplicative factor of 2 c log m for some absolute constant c .In this paper we prove that, for integers m , n satisfying n 9 ź 8 < m < n 3 ź 2 , there exist m points and n lines on the quadratic hypersurface in R 4 { ( x 1 , x 2 , x 3 , x 4 ) ź R 4 ź x 1 = x 2 2 + x 3 2 - x 4 2 } , such that (i) at most s = O ( 1 ) lines lie on any 2-flat, (ii) at most q = O ( n ź m 1 ź 3 ) lines lie on any hyperplane, and (iii) the number of incidences between the points and the lines is ź ( m 2 ź 3 n 1 ź 2 ) , which is asymptotically larger than the upper bound inź(1), when n 9 ź 8 < m < n 3 ź 2 . This shows that the assumption that no quadric contains more than q lines (in the above mentioned theorem of Sharir and Solomon (2015)) is necessary in this regime of m and n .By a suitable projection from this quadratic hypersurface onto R 3 , we obtain m points and n lines in R 3 , with at most s = O ( 1 ) lines on a common plane, such that the number of incidences between the m points and the n lines is ź ( m 2 ź 3 n 1 ź 2 ) . It remains an interesting question to determine if this bound is also tight in general.

We give a combinatorial characterization of the family of lines of PG ( 3 , q ) , q ≥ 7 odd, which meet a hyperbolic quadric in two points (the so called secant lines) using their intersection properties with the points and planes of PG ( 3 , q ) .

Let E be a set of solids (hyperplanes) in PG ( 4 , q ) , q even, q > 2 , such that every point of PG ( 4 , q ) lies in either 0, 1 2 ( q 3 − q 2 ) or 1 2 q 3 solids of E , and every plane of PG ( 4 , q ) lies in either 0, 1 2 q or q solids of E . This article shows that E is either the set of solids that are disjoint from a hyperoval, or the set of solids that meet a non-singular quadric Q ( 4 , q ) in an elliptic quadric.

Let $U$ be a point set in the $n$-dimensional affine space ${\rm AG}(n,q)$ over the finite field of $q$ elements and $0\leq k\leq n-2$. In this paper we extend the definition of directions determined by $U$: a $k$-dimensional subspace $S_k$ at infinity is determined by $U$ if there is an affine $(k+1)$-dimensional subspace $T_{k+1}$ through $S_k$ such that $U\cap T_{k+1}$ spans $T_{k+1}$. We examine the extremal case $|U|=q^{n-1}$, and classify point sets NOT determining every $k$-subspace in certain cases., Comment: 7 pages. NOTICE: this is the author's version of a work that was accepted for publication in Discr. Math. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication

For q odd, it was shown in (J. Combin. Theory Ser. A 98 (2) (2002) 253) that the elliptic quadric Q^-(7,q) possesses a unique 1-system, the so-called classical 1-system. Here, the same result will be obtained for even q.

The quadric Veronesean V 2 4 in PG ( 5 , q ) is characterized as a ( q 2 + q + 1 ) -cap of class [ 0 , 1 , 2 , 3 , q + 1 ] 2 and type ( 1 , q + 1 , 2 q + 1 ) 4 of PG ( 5 , q ) by Ferri (q odd) and Thas and Van Maldeghem (q even). In this note we generalize this result slightly by proving that for a ( q 2 + q + 1 ) -cap of class [ 0 , 1 , 2 , 3 , q + 1 ] 2 and type ( 1 , m , n ) 4 of PG ( 5 , q ) , the parameters m and n are uniquely determined and equal q + 1 and 2 q + 1 , respectively.

A flock of a cone in PG(5,q) with a line as vertex and a hyperbolic quadric Q^+(3,q) as base is associated with every locally hermitian 1-system of Q^+(7,q) and conversely, so that the two objects are equivalent. We construct an example of such a flock, starting from a Segre variety S"1";"2, and study the corresponding 1-system of Q^+(7,q). Locally hermitian semiclassical 1-systems of Q^+(7,q), which are not contained in a hyperplane of PG(7,q), are characterized in terms of their flock. Finally, the previously known locally hermitian semiclassical 1-systems of Q^+(7,q) are investigated and it seems that many new examples can be found.

In the linear representation of the desarguesian plane PG(2,q2) in PG(5,q), the classical unital (the hermitian curve) is represented by an elliptic quadric ruled by lines of a normal spread. In this paper, we prove that a Buekenhout–Metz unital U in PG(2,q2), arising from an elliptic quadric, is represented in PG(5,q) by an algebraic hypersurface of degree four minus the complement of a line in a three-dimensional subspace. Also, such a hypersurface is reducible if and only if U is classical.

This article discusses two problems on classical generalized quadrangles. It is known that the generalized quadrangle Q(4,q) arising from the parabolic quadric in PG(4,q) has a spread if and only if q is even. Hence, for q odd, the problem arises of the cardinality of the smallest set of lines of Q(4,q) covering all points of Q(4,q). We show in this paper that this set of lines must contain more than q2+1+(q−1)/3 lines. We also show that Q(4,q), q even, does not contain minimal covers of sizes q2+1+r when q⩾32 and 0

In an r-dimensional projective Galois space, PG(r,q), of order q, let K be a k-set of class [0,1,m,n]1, with respect to the lines. We prove that: if r=2s−1(s⩾2 and q=2,q=4 or q odd if s=2), k=θ2s−1 and there exists a point V of K through which exactly q2(s−1) 1-secant lines pass and through any other point of K pass q2s−3 1-secants, then K is a quadric cone projecting from V a non-singular quadric of a PG(2(s−1),q) skew with V; if r=2(s−1) (s⩾3),k=θ2s−3+qs−1 and there exists a point V of K through which exactly q2s−3−qs−2 1-secant lines pass and through any other point of K q2(s−2)−qs−2 1-secants pass, then K is a quadric cone projecting from V a hyperbolic quadric of a PG(2s−3,q) skew with V.

Generalized Q-sets in a locally projective planar space S of order n are defined. If S contains a non-degenerate generalized Q-set Q and the order n (>4) of S is even, we prove that S is embeddable in PG(3,n) and Q is isomorphic to an ovoid or a hyperbolic quadric of PG(3,n).

We discuss various configurations of planes in PG(5,2). The supporting point-sets of most of the configurations, or else the complements of these point-sets, have cubic equations and are of elliptic or hyperbolic type. In our discussion of conclaves of eight planes in PG(5,2) we introduce a second addition upon the space V(6,2)=X⊕X ∗ , where X ∗ is the dual of X . We show that the 8 planes of a conclave lying on a hyperbolic quadric for one addition mutate into the 8 Conwell heptads lying off a hyperbolic quadric for the other addition. In our discussion of double-seven of planes in PG(5,2) we also consider their relation with the double-seven of planes in PG(8,2) which is supported by the Segre variety S 2,2 over GF(2).

A (line) spread in PG(3, q ) is any set of q 2 + 1 disjoint lines in PG(3, q ). The spread S is called symplectic if all lines of S are totally isotropic for some symplectic polarity ς of PG(3, q ). An ovoid of PG(3, q ), q > 2, is a set of q 2 + 1 points, no three of which are collinear; an ovoid of PG(3,2) is the same as an elliptic quadric. An ovoid of the nonsingular quadric Q (4, q ) of PG(4, q ) is any set O of points of Q (4, q ) which has exactly one point in common with each line of Q (4, q ). Symplectic spreads of PG(3, q ) and ovoids of Q (4, q ) are equivalent objects, and, for q even, also ovoids of Q (4, q ) and ovoids of PG(3, q ) are equivalent objects. With each symplectic spread of PG(3, q ) there corresponds a plane of order q 2 . A 3 − ( q 2 + 1, q + 1, 1) design is called an inversive plane of order q . With each ovoid of PG(3, q ) there corresponds an inversive plane of order q . Here we survey some important characterizations of finite inversive planes, the inversive planes of small order, the planes of small order arising from symplectic spreads of PG(3, q ), and all known classes of ovoids of PG(3, q ) and Q (4, q ). In Appendix A we also survey the ovoids of the nonsingular quadric Q (2 n , q ), n ⩾ 3, of PG(2 n , q ). In second Appendix B we discuss an interesting relationship between flocks of the quadratic cone K of PG(3, q ) and ovoids of Q (4, q ).

In 1983 Barahona defined the class of cut polytopes; recently Padberg defined the class of Boolean quadric polytopes. We show that every Boolean quadric polytope is the image of a cut polytope under a bijective linear transformation, and so studying Boolean quadric polytopes reduces to studying special cut polytopes. Our proof uses an idea introduced in 1965 by Hammer.

Bruen proved that if A is a set of points in AG(n, q) which intersects every hyperplane in at least t points, then |A| ≥ (n+t-1)(q-1) + 1, leaving as an open question how good such bound is. Here we prove that, up to a trivial case, if t > ((n - 1)(q - 1) + 1)/2, then Bruen's bound can be improved. If t is equal to the integer part of ((n - 1)(q - 1) + 1)/2, then there are some examples which attain such a lower bound. Somehow, this suggests the following combinatorial characterization: if a set S of points in PG(3, q) meets every affine plane in at least q - 1 points and is of minimum size with respect to this property, then S is a hyperbolic quadric.

We will classify, up to linear representations, all geometries fully embedded in an affine space with the property that for every antiflag {p,L} of the geometry there are either 0, @a, or q lines through p intersecting L. An example of such a geometry with @a=2 is the following well known geometry HT"n. Let Q"n"+"1 be a nonsingular quadric in a finite projective space PG(n+1,q), n>=3, q even. We project Q"n"+"1 from a point [email protected]?Q"n"+"1, distinct from its nucleus if n+1 is even, on a hyperplane PG(n,q) not through r. This yields a partial linear space HT"n whose points are the points p of PG(n,q), such that the line is a secant to Q"n"+"1, and whose lines are the lines of PG(n,q) which contain q such points. This geometry is fully embedded in an affine subspace of PG(n,q) and satisfies the antiflag property mentioned. As a result of our classification theorem we will give a new characterization theorem of this geometry.

For n ⩾ 2, the boolean quadric polytope Pn is the convex hull in d:=(n+12) dimensions of the binary solutions xixj = yij, for all i < j in N ≔ 1,2,. …,n. The polytope is naturally modeled by a somewhat larger polytope; namely, Ln the solution set of uij ⩽ xij, yij ⩽ xj, xi + xj ⩽ 1 + yij, yij ⩾ 0, for all i, j in N. In a first step toward seeing how well Ln approximates Pn we establish that the d-dimensional volume of Ln is 22n−dn!/(2n)!. Using a well-known connection between Pn and the ‘cut polytope’ of a complete graph n + 1 vertices, we also establish the volume of a relaxation of this cut polytope.