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.     

Complete Spans on Hermitian Varieties

Let L be a general linear complex in PG(3, q) for any prime power q. We show that when GF(q) is extended to GF(q ²), the extended lines of L cover a non-singular Hermitian surface H ? H(3, q ²) of PG(3, q ²). We prove that if Sis any symplectic spread PG(3, q), then the extended lines of this spread form a complete (q ² + 1)-span of H. Several other examples of complete spans of H for small values of q are also discussed. Finally, we discuss extensions to higher dimensions, showing in particular that a similar construction produces complete (q ³ + 1)-spans of the Hermitian variety H(5, q ²).     

Characterizations of Hermitian varieties by intersection numbers

In this paper, we give characterizations of the classical generalized quadrangles H(3, q ²) and H(4, q ²), embedded in PG(3, q ²) and PG(4, q ²), respectively. The intersection numbers with lines and planes characterize H(3, q ²), and H(4, q ²) is characterized by its intersection numbers with planes and solids. This result is then extended to characterize all Hermitian varieties in dimension at least 4 by their intersection numbers with planes and solids.      

Derivation techniques on the Hermitian surface

We discuss derivation‐like techniques for transforming one locally Hermitian partial ovoid of the Hermitian surface H(3,q²) into another one. These techniques correspond to replacing a regulus by its opposite in some naturally associated projective 3‐space PG(3,q) over a square root subfield. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 478–486, 2007     

Unitals in Finite Desarguesian Planes

We show that a suitable 2-dimensional linear system of Hermitian curves of PG(2,q ²) defines a model for the Desarguesian plane PG(2,q). Using this model we give the following group-theoretic characterization of the classical unitals. A unital in PG(2,q ²) is classical if and only if it is fixed by a linear collineation group of order 6(q + 1)² that fixes no point or line in PG(2,q ²).     

Subregular Spreads of Hermitian Unitals

In this paper, we consider the problem of constructing partitions of the points of a Hermitian unital into pairwise disjoint blocks, commonly known as spreads. We generalize a construction of Baker et al. (In Finite Geometry and Combinatorics, Vol. 191 of London Math. Soc. Lecture Not Ser., pages 17–30. Cambridge University Press, Cambridge, 1993.) to provide a new infinite family of spreads. Morover, we develop a structural connection between these new spreads of the Hermitian unital in PG(2, q²) and the subregular spreads of PG(3, q), allowing us to christen a new “subregular” family of spreads in the Hermitian unital in PG(2, q²).     

Transitive Arcs in Planes of Even Order

When one considers the hyperovals inPG(2,q),qeven,q>2, then the hyperoval inPG(2, 4) and the Lunelli-Sce hyperoval inPG(2, 16) are the only hyperovals stabilized by a transitive projective group [10]. In both cases, this group is an irreducible group fixing no triangle in the plane of the hyperoval, nor in a cubic extension of that plane. Using Hartley's classification of subgroups ofPGL₃(q),qeven [6], allk-arcs inPG(2,q) fixed by a transitive irreducible group, fixing no triangle inPG(2,q) or inPG(2,q³), are determined. This leads to new 18-, 36- and 72-arcs inPG(2,q),q=2^2h. The projective equivalences among the arcs are investigated and each section ends with a detailed study of the collineation groups of these arcs.     

Maximal partial spreads in PG (3, q)

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 over any field K and we apply the above representation to construct maximal partial spreads in PG(3, q). For q even we get new cardinalities for For q odd the cardinalities are partially known.     

Some p‐ranks related to a conic in PG(2, q)

Let A be the incidence matrix of lines and points of the classical projective plane PG(2, q) with q odd. With respect to a conic in PG(2, q), the matrix A is partitioned into 9 submatrices. The rank of each of these submatrices over F_q, the defining field of PG(2, q), is determined. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 224–236, 2010     

On the Twisted Cubic of PG(3, q)

In this paper we classify the lines of PG(3, q) whose points belong to imaginary chords of the twisted cubic of PG(3, q). Relying on this classification result, we obtain a complete classification of semiclassical spreads of the generalized hexagon H(q).     

Some Characterizations of Finite Hermitian Veroneseans

We characterize the finite Veronesean of all Hermitian varieties of PG(n,q²) as the unique representation of PG(n,q²) in PG(d,q), d n(n+2), where points and lines of PG(n,q²) are represented by points and ovoids of solids, respectively, of PG(d,q), with the only condition that the point set of PG(d,q) corresponding to the point set of PG(n,q²) generates PG(d,q). Using this result for n=2, we show that is characterized by the following properties: (1) ; (2) each hyperplane of PG(8,q) meets in q²+1, q³+1 or q³+q²+1 points; (3) each solid of PG(8,q) having at least q+3 points in common with shares exactly q²+1 points with it.51E24     

Disegni unitari nei piani di Hughes

Let H(q ²) be the Hughes plane on the regular nearfield of order q ₂ whose center is a field of order q. We construct in H(q ²) a unital of parameter q.

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Lavoro eseguito nell'ambito di attività di ricerca finanziate dal M.P.I. (fondi 60% e 40%).     

On pairs of permutable Hermitian surfaces

We investigate the intersection R of two permutable Hermitian surfaces of PG(3,q²), q odd. We show that R is a determinantal variety. From the combinatorial point of view R comprises a complete (q²+1)-span of the two corresponding Hermitian surfaces.     

Ovoids and spreads of finite classical polar spaces

LetP be a finite classical polar space of rankr, r2. An ovoidO ofP is a pointset ofP, which has exactly one point in common with every totally isotropic subspace of rankr. It is proved that the polar spaceW _n (q) arising from a symplectic polarity ofPG(n, q), n odd andn > 3, that the polar spaceQ(2n, q) arising from a non-singular quadric inPG(2n, q), n > 2 andq even, that the polar space Q^–(2n + 1,q) arising from a non-singular elliptic quadric inPG(2n + 1,q), n > 1, and that the polar spaceH(n,q ²) arising from a non-singular Hermitian variety inPG(n, q ²)n even andn > 2, have no ovoids.LetS be a generalized hexagon of ordern (1). IfV is a pointset of order n³ + 1 ofS, such that every two points are at distance 6, thenV is called an ovoid ofS. IfH(q) is the classical generalized hexagon arising fromG ₂ (q), then it is proved thatH(q) has an ovoid iffQ(6, q) has an ovoid. There follows thatQ(6, q), q=3^2h+1, has an ovoid, and thatH(q), q even, has no ovoid.A regular system of orderm onH(3,q ²) is a subsetK of the lineset ofH(3,q ²), such that through every point ofH(3,q ²) there arem (> 0) lines ofK. B. Segre shows that, ifK exists, thenm=q + 1 or (q + l)/2.If m=(q + l)/2,K is called a hemisystem. The last part of the paper gives a very short proof of Segre's result. Finally it is shown how to construct the 4-(11, 5, 1) design out of the hemisystem with 56 lines (q=3).     

Complete Systems of Lines on a Hermitian Surface over a Finite Field

The aim is to find the maximum size of a set of mutually ske lines on a nonsingular Hermitian surface in PG(3, q) for various values of q. For q = 9 such extremal sets are intricate combinatorial structures intimately connected ith hemisystems, subreguli, and commuting null polarities. It turns out they are also closely related to the classical quartic surface of Kummer. Some bounds and examples are also given in the general case.     

Commuting polarities and maximal partial ovoids of H(4,q2)

In PG(4,q²), q odd, let Q(4,q²) be a non‐singular quadric commuting with a non‐singular Hermitian variety H(4,q²). Then these varieties intersect in the set of points covered by the extended generators of a non‐singular quadric Q₀ in a Baer subgeometry Σ₀ of PG(4,q²). It is proved that any maximal partial ovoid of H(4,q²) intersecting Q₀ in an ovoid has size at least 2(q²+1). Further, given an ovoid O of Q₀, we construct maximal partial ovoids of H(4,q²) of size q³+1 whose set of points lies on the hyperbolic lines 〈P,X〉 where P is a fixed point of O and X varies in O\{P}. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 307–313, 2009     

On Distinguished Solutions of Truncated Matricial Hamburger Moment Problems

We study two slightly different versions of the truncated matricial Hamburger moment problem. A central topic is the construction and investigation of distinguished solutions of both moment problems under consideration. These solutions turn out to be nonnegative Hermitian q × q Borel measures on the real axis which are concentrated on a finite number of points. These points and the corresponding masses will be explicitly described in terms of the given data. Furthermore, we investigate a particular class of sequences (s_j)^∞_{j = 0} of complex q × q matrices for which the corresponding infinite matricial Hamburger moment problem has a unique solution. Our approach is mainly algebraic. It is based on the use of particular matrix polynomials constructed from a nonnegative Hermitian block Hankel matrix. These matrix polynomials are immediate generalizations of the monic orthogonal matrix polynomials associated with a positive Hermitian block Hankel matrix. We generalize a classical theorem due to Kronecker on infinite Hankel matrices of finite rank to block Hankel matrices and discuss its consequences for the nonnegative Hermitian case.     

The Multiplicity Free Permutation Representations of the Ree Groups 2 G 2(q), the Suzuki Groups 2 B 2(q), and Their Automorphism Groups

Abstract

Let G a simple group of type ² B ₂(q) or ² G ₂(q), where q is an odd power of 2 or 3, respectively. The main goal of this paper is to determine the multiplicity free permutation representations of G and A ≤ Aut(G) where A is a subgroup containing a copy of G. Let B be a Borel subgroup of G. If G = ² B ₂(q) we show that there is only one non-trivial multiplicity free permutation representation, namely the representation of G associated to the action on G/B. If G = ² G ₂(q) we show that there are exactly two such non-trivial representations, namely the representations of G associated to the action on G/B and the action on G/M, where M = UC with U the maximal unipotent subgroup of B and C the unique subgroup of index 2 in the maximal split torus of B. The multiplicity free permutation representations of A correspond to the actions on A/H where H is isomorphic to a subgroup containing B if G = ² B ₂(q), and containing M if G = ² G ₂(q). The problem of determining the multiplicity free representations of the finite simple groups is important, for example, in the classification of distance-transitive graphs.     

Complete ( q 2?+ q +?8)/2-caps in the spaces PG (3, q ), q ≡ 2 (mod 3) an odd prime, and a complete 20-cap in PG (3, 5)

An infinite family of complete (q ² + q + 8)/2-caps is constructed in PG(3, q) where q is an odd prime ≡ 2 (mod 3), q ≥ 11. This yields a new lower bound on the second largest size of complete caps. A variant of our construction also produces one of the two previously known complete 20-caps in PG(3, 5). The associated code weight distribution and other combinatorial properties of the new (q ² + q + 8)/2-caps and the 20-cap in PG(3, 5) are investigated. The updated table of the known sizes of the complete caps in PG(3, q) is given. As a byproduct, we have found that the unique complete 14-arc in PG(2, 17) contains 10 points on a conic. Actually, this shows that an earlier general result dating back to the Seventies fails for q = 17.      

Fischer Matrices of the Affine Groups

Let GL_n(q) be the general linear group and let H_n ; V_n(q) · GL_n(q) denote the affine group of V_n(q). In [1] and [4], we determined Fischer matrices for the conjugacy classes of GL_n(q) where n = 2, 3, 4 and we obtained the number of conjugacy classes and irreducible characters of H₂, H₃, and H₄. In this paper, we find the Fischer matrices of the affine group H_n for arbitrary n.AMS Subject Classification Primary 20C15 Secondary 20C33