Unitals which meet Baer subplanes in 1 moduloq points

We prove that a parabolic unitalU in a translation plane of orderq ² with kernel containing GF(q) is a Buekenhout-Metz unital if and only if certain Baer subplanes containing the translation line of meetU in 1 moduloq points. As a corollary we show that a unital 16-03 in PG(2,q ²) is classical if and only if it meets each Baer subplane of PG(2,q ²) in 1 moduloq points.     

Triads, Flocks of Conics and Q -(5,q)

We show that if an ovoid of Q (4,q),q even, admits a flock of conics then that flock must be linear. It follows that an ovoid of PG (3,q),q even, which admits a flock of conics must be an elliptic quadric. This latter result is used to give a characterisation of the classical example Q ^-(5,q) among the generalized quadrangles T ₃( ), where is an ovoid of PG (3q) and q is even, in terms of the geometric configuration of the centres of certain triads.     

A condition for the existence of ovals in PG(2, q), q even

A condition is found that determines whether a polynomial over GF(q) gives an oval in PG(2, q), q even. This shows that the set of all ovals of PG(2, q) corresponds to a certain variety of points of PG((q–4)/2, q). The condition improves upon that of Segre and Bartocci, who proved that all the terms of an oval polynomial have even degree. It is suitable for efficient computer searches.     

On a set of lines of PG(3, q) corresponding to a maximal cap contained in the Klein quadric of PG (5, q)

The problem is considered of constructing a maximal set of lines, with no three in a pencil, in the finite projective geometry PG(3, q) of three dimensions over GF(q). (A pencil is the set of q+1 lines in a plane and passing through a point.) It is found that an orbit of lines of a Singer cycle of PG(3, q) gives a set of size q ³ + q ² + q + 1 which is definitely maximal in the case of q odd. A (q ³ + q ² + q + 1)-cap contained in the hyperbolic (or Klein) quadric of PG(5, q) also comes from the construction. (A k-cap is a set of k points with no three in a line.) This is generalized to give direct constructions of caps in quadrics in PG(5, q). For q odd and greater than 3 these appear to be the largest caps known in PG(5, q). In particular it is shown how to construct directly a large cap contained in the Klein quadric, given an ovoid skew to an elliptic quadric of PG(3, q). Sometimes the cap is also contained in an elliptic quadric of PG(5, q) and this leads to a set of q ³ + q ² + q + 1 lines of PG(3,q ²) contained in the non-singular Hermitian surface such that no three lines pass through a point. These constructions can often be applied to real and complex spaces.     

Geometry of Cubic and Quartic Hypersurfaces over Finite Fields

Let YPⁿ be a cubic hypersurface defined over GF(q). Here, we study the Finite Field Nullstellensatz of order [q/3] for the set Y(q) of its GF(q)-points, the existence of linear subspaces of PG(n,q) contained in Y(q) and the possibility to join any two points of Y(q) by the union of two lines of PG(n,q) entirely contained in Y(q). We also study the existence of linear subspaces defined over GF(q) for the intersection of Y with s quadrics and for quartic hypersurfaces.     

On the non-existence of linear codes attaining the Griesmer bound

A lower bound on the size of a set K in PG(3, q) satisfying for any plane of PG(3, q), q4 is given. It induces the non-existence of linear [n,4,n + 1 – q ²]-codes over GF(q) attaining the Griesmer bound for .     

Characterizations of Buekenhout-Metz unitals

We give a characterization of the Buekenhout-Metz unitals in PG(2, q ²), in the cases that q is even or q=3, in terms of the secant lines through a single point of the unital. With the addition of extra conditions, we obtain further characterizations of Buekenhout-Metz unitals in PG(2, q ²), for all q. As an application, we show that the dual of a Buekenhout-Metz unital in PG(2, q ²) is a Buekenhout-Metz unital.     

On (q 2 + q + 2, q + 2)-arcs in the Projective Plane PG(2, q)

A (k,n)-arc in PG(2,q) is usually defined to be a set of k points in the plane such that some line meets in n points but such that no line meets in more than n points. There is an extensive literature on the topic of (k,n)-arcs. Here we keep the same definition but allow to be a multiset, that is, permit to contain multiple points. The case k=q ²+q+2 is of interest because it is the first value of k for which a (k,n)-arc must be a multiset. The problem of classifying (q ²+q+2,q+2)-arcs is of importance in coding theory, since it is equivalent to classifying 3-dimensional q-ary error-correcting codes of length q ²+q+2 and minimum distance q ². Indeed, it was the coding theory problem which provided the initial motivation for our study. It turns out that such arcs are surprisingly rich in geometric structure. Here we construct several families of (q ²+q+2,q+2)-arcs as well as obtain some bounds and non-existence results. A complete classification of such arcs seems to be a difficult problem.     

On translation planes of orderq 3 that admit a collineation group of orderq 3

Let II be a translation plane of orderq ³, with kernel GF(q) forq a prime power, that admits a collineation groupG of orderq ³ in the linear translation complement. Moreover, assume thatG fixes a point at infinity, acts transitively on the remaining points at infinity andG/E is an abelian group of orderq ², whereE is the elation group ofG.In this article, we determined all such translation planes. They are (i) elusive planes of type I or II or (ii) desirable planes.Furthermore, we completely determined the translation planes of orderp ³, forp a prime, admitting a collineation groupG of orderp ³ in the translation complement such thatG fixes a point at infinity and acts transitively on the remaining points at infinity. They are (i) semifield planes of orderp ³ or (ii) the Sherk plane of order 27.     

Some remarks on embeddings of the flag geometries of projective planes in finite projective spaces

The flag geometry of a finite projective plane II of orders is the generalized hexagon of order (s, 1) obtained from II by putting equal to the set of all flags of II, by putting equal to the set of all points and lines of II and where I is the natural incidence relation (inverse containment), that is, is the dual of the double of II in the sense of [8]. Then we say that is fully (and weakly) embedded in the finite projective space PG(d, q) if is a subgeometry of the natural point-line geometry associated with PG(d, q), if s=q, if the set of points of generates PG(d, q) (and if the set of points of not opposite any given point of does not generate PG(d, q)). We have classified all such embeddings in [3, 4, 5, 6]. In the present paper, we weaken the hypotheses in some special cases, and we give an alternative formulation of the classification.     

On the Embedding of (k,p)-Arcs is Maximal Arcs

Our main result is that a (k,p)-arc in PG (2,q),q = p ^h , with k qp - q + p - can be extended to a maximal arc. Combining this result with the recent Ball, Blokhuis, Mazzocca theorem about the non-existence of maximal arcs for p > 2, it gives an upper bound for the size of a (k,p)-arc. The method can be regarded as a generalization of B. Segre's method for proving similar embeddability theorems for k-arcs (that is when n= 2). It is based on associating an algebraic envelope containing the short lines to the (k,p)-arc. However, the construction of the envelope is independent of Segre's method using the generalization of Menelaus' theorem.     

Tubes of even order and flat π.C 2 geometries

Atube of even orderq=2 ^d is a setT={L, } ofq+3 pairwise skew lines in PG(3,q) such that every plane onL meets the lines of in a hyperoval. Thequadric tube is obtained as follows. Take a hyperbolic quadricQ=Q ₃ ⁺ (q) in PG(3,q); letL be an exterior line, and let consist of the polar line ofL together with a regulus onQ.In this paper we show the existence of tubes of even order other than the quadric one, and we prove that the subgroup of PL(4,q) fixing a tube {L, } cannot act transitively on . As pointed out by a construction due to Pasini, this implies new results for the existence of flat .C ₂ geometries whoseC ₂-residues are nonclassical generalized quadrangles different from nets. We also give the results of some computations on the existence and uniqueness of tubes in PG(3,q) for smallq. Further, we define tubes for oddq (replacing hyperoval by conic in the definition), and consider briefly a related extremal problem.Dedicated to luigi antonio rosati on the occasion of his 70th birthday     

Partial t-Spreads in PG(2t+1,q)

This article first of all discusses the problem of the cardinality of maximal partial spreads in PG(3,q), q square, q>4. Let r be an integer such that 2rq+1 and such that every blocking set of PG(2,q) with at most q+r points contains a Baer subplane. If S is a maximal partial spread of PG(3,q) with q ²-1-r lines, then r=s( +1) for an integer s2 and the set of points of PG(3,q) not covered byS is the disjoint union of s Baer subgeometriesPG(3, ). We also discuss maximal partial spreads in PG(3,p ³), p=p ₀ ^h , p ₀ prime, p ₀ 5, h 1, p 5. We show that if p is non-square, then the minimal possible deficiency of such a spread is equal to p ²+p+1, and that if such a maximal partial spread exists, then the set of points of PG(3,p ³) not covered by the lines of the spread is a projected subgeometryPG(5,p) in PG(3,p ³). In PG(3,p ³),p square, for maximal partial spreads of deficiency p ²+p+1, the combined results from the preceding two cases occur. In the final section, we discuss t-spreads in PG(2t+1,q), q square or q a non-square cube power. In the former case, we show that for small deficiencies , the set of holes is a disjoint union of subgeometries PG(2t+1, ), which implies that 0 (mod +1) and, when (2t+1)( -1) <q-1, that 2( +1). In the latter case, the set of holes is the disjoint union of projected subgeometries PG(3t+2, ) and this implies 0 (mod q ^2/3+q ^1/3+1). A more general result is also presented.     

A group theoretic characterization of classical unitals

Let G be the group of projectivities stabilizing a unital $\mathcal{U}$ in PG(2,q ²). In?this paper, we prove that $\mathcal{U}$ is a classical unital if and only if there are two points in $\mathcal{U}$ such that the stabilizer of these two points in G has order?q ²?1.     

On nets of orderq 2 and degreeq+1 admitting GL(2,q)

In this paper we consider finite nets of orderq ² and degreeq + 1 which admit GL(2,q). Our main result says that if a net of orderq ² and degreeq + 1 admits a collineation group with a point-regular normal subgroupT such that /T GL(2,q), then is isomorphic to a regulus net, a twisted regulus net, a Hering net, or . Except in the last one, each of them corresponds to a surface in PG(3,q) obtained from a homogeneous polynomial in two variables.     

Nuclei of point sets of sizeq+1 contained in the union of two lines inPG(2,q)

We give a complete classification for pairs ( (),) where () is the set of all nuclei of a set ofq+1 not collinear points contained in the union of two lines in a desarguesian planePG(2,q) of orderq. We also obtain some new results concerning blocking sets of Rédei type and certain point-sets of type [0,1,m,n] inPG(2, q).     

A Combinatorial Characterization of Hermitian Curves

A unital U with parameter q is a 2 – (q ³ + 1, q + 1, 1) design. If a point set U in PG(2, q ²) together with its (q + 1)-secants forms a unital, then U is called a Hermitian arc. Through each point p of a Hermitian arc H there is exactly one line L having with H only the point p in common; this line L is called the tangent of H at p. For any prime power q, the absolute points and nonabsolute lines of a unitary polarity of PG(2, q ²) form a unital that is called the classical unital. The points of a classical unital are the points of a Hermitian curve in PG(2, q ²).Let H be a Hermitian arc in the projective plane PG(2, q ²). If tangents of H at collinear points of H are concurrent, then H is a Hermitian curve. This result proves a well known conjecture on Hermitian arcs.     

Multiple Blocking Sets in PG(n, q), n > 3

This article discusses minimal s-fold blocking sets B in PG (n, q), q = p^h, p prime, q > 661, n > 3, of size |B| > sq + c ^p q ^2/3 - (s - 1) (s - 2)/2 (s > min (c ^p q ^1/6, q ^1/4/2)). It is shown that these s-fold blocking sets contain the disjoint union of a collection of s lines and/or Baer subplanes. To obtain these results, we extend results of Blokhuis–Storme–Szönyi on s-fold blocking sets in PG(2, q) to s-fold blocking sets having points to which a multiplicity is given. Then the results in PG(n, q), n 3, are obtained using projection arguments. The results of this article also improve results of Hamada and Helleseth on codes meeting the Griesmer bound.     

Blocking sets and partial spreads in finite projective spaces

A t-blocking set in the finite projective space PG(d, q) with dt+1 is a set of points such that any (d–t)-dimensional subspace is incident with a point of and no t-dimensional subspace is contained in . It is shown that | |q ^t +...+1+q ^t–1q and the examples of minimal cardinality are characterized. Using this result it is possible to prove upper and lower bounds for the cardinality of partial t-spreads in PG(d, q). Finally, examples of blocking sets and maximal partial spreads are given.