Embedding (r, 1)-designs in finite projective planes

It is known that any non-trivial (r,1)-design on υ varieties (υ ? (r? 1)² ? 1) is extendible; this fact implies the existence of a projective plane of order r ? 1. In this paper it is shown that any non-trivial (r, 1)-design on (r ? 1)² ? α varieties, where r and α are appropriately bounded, is extendible; hence this fact implies the existence of a projective plane of order r ? 1. We also show that, for υ ? (r ? 1)² ? 2, any non-trivial (r, 1)-design on υ varieties is extendible.     

Steiner triple systems of order 19 associated with a certain type of projective plane of order 10

It is shown that the existence of a Steiner triple system of order 19 satisfying certain very restrictive conditions would lead to the completion of a large portion of the incidence matrix of a projective plane of order 10.     

On tangentially transitive projective planes

The existence of Baer collineations in a projective plane is related to the existence of desargues-like configurations. The plane of order four is characterized as the only finite plane that possesses a Baer subplane partition into tangentially transitive Baer subplanes which is preserved by each of the tangentially transitive groups. It is shown that a finite projective plane has either no or one tangentially transitive Baer subplane or is partially transitive of Hughes type (4, m), (5, m) or (6, m) for some m. The Lenz-Barlotti classes which contain a finite plane which is not a translation plane nor its dual and which possesses a tangentially transitive Baer subplane are shown to be classes I.1 and II.1.     

On the original Steiner Systems

It is shown that in every n-colouring ((n ? 1)-colouring) of a projective plane (affine plane) of odd order n at least one line has three points of the same colour.     

A theorem on equidistant codes

It is shown that a nontrivial code consisting of k²+k+2 words at mutual distance 2k (k > 1) exists if and only if there exists a projective plane of order k.     

Co-calibrated G 2 structure from cuspidal cubics

We establish a twistor correspondence between a cuspidal cubic curve in a complex projective plane, and a co-calibrated homogeneous G ₂ structure on the seven-dimensional parameter space of such cubics. Imposing the Riemannian reality conditions leads to an explicit co-calibrated G ₂ structure on SU(2, 1)/U(1). This is an example of an SO(3) structure in seven dimensions. Cuspidal cubics and their higher degree analogues with constant projective curvature are characterised as integral curves of certain seventh order ODEs. Projective orbits of such curves are shown to be analytic continuations of Aloff?CWallach manifolds, and it is shown that only cubics lift to a complete family of contact rational curves in a projectivised cotangent bundle to a projective plane.     

A fake projective plane with an order 7 automorphism

A fake projective plane is a compact complex surface (a compact complex manifold of dimension 2) with the same Betti numbers as the complex projective plane, but not isomorphic to the complex projective plane. As was shown by Mumford, there exists at least one such surface.In this paper we prove the existence of a fake projective plane which is birational to a cyclic cover of degree 7 of a Dolgachev surface.     

Sklyanin algebras and Hilbert schemes of points

We construct projective moduli spaces for torsion-free sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective plane P².The generic noncommutative plane corresponds to the Sklyanin algebra S=Skl(E,σ) constructed from an automorphism σ of infinite order on an elliptic curve E⊂P². In this case, the fine moduli space of line bundles over S with first Chern class zero and Euler characteristic 1−n provides a symplectic variety that is a deformation of the Hilbert scheme of n points on P²?E.     

A projective plane of order 16

A projective plane of order 16 is constructed. It is a translation plane and appears to be new. The representation of the collineation group on the axis of the plane has a normal subgroup isomorphic to L₃ (2) with factor group isomorphic to S₃. The orbits of this representation have lengths 14 and 3. If two points in the latter orbit are chosen to define a sharply doubly transitive set of permutations, the permutations from the multiplicative loop generate a group isomorphic to A₇. The plane is of Lenz-Barlotti class IVa.1.     

On small {;k; q};-arcs in planes of order q2

In this paper we examine some properties of complete {;k; q};-arcs in projective planes of order q². In particular, we derive a lower bound for k, and we exhibit a family of arcs having low values of k which exist in every such plane having a Baer subplane. In addition we resolve the existence problem for complete {;k; 3};-arcs in PG(2, 9).     

Hadamard matrices and projective planes

The construction of a Hadamard matrix of order n² from a projective plane of order n, n even, is given. Alternative constructions, specialized to the case n = 10, from sets of mutually orthogonal Latin squares are also given. Special properties of the Hadamard matrices are discussed and a partial example is given in the case n = 10.     

Fixed points of projectivities of prime order

This work begins with a review of the classical results for fixed points of projectivities in a projective plane over a general commutative field. The second section of this work features all the material necessary to prove the main result, which is presented in Theorem 2.8. It is shown that, in a finite projective plane of order q, there exists a projectivity g? of prime order p?>?3 if and only if p divides exactly one of the integers q ? 1, q, q?+?1, q ² + q + 1. Theorem 2.8 establishes a correspondence between the possible structures of points fixed by g?, as presented in Theorem 1.3, and the integer that is divisible by p. The special case of p = 2 is handled in Sect. 2.1, where it is shown that every involution is a harmonic homology for q odd and an elation for q even. The special case of p?=?3 is handled in Sect. 2.2, and Theorem 2.8 is adapted for p?=?3 and presented as Theorem 2.15. An application of Theorems 2.8 and 2.15 is determining the sizes of (n, r)-arcs that are stabilized by projectivities of prime order p in the finite projective plane of order q; in Sect. 3, this application is presented in Propositions 3.2 and 3.3.     

Harmonic conjugation in harmonic matroids

We study a generalization of the concept of harmonic conjugation from projective geometry and full algebraic matroids to a larger class of matroids called harmonic matroids. We use harmonic conjugation to construct a projective plane of prime order in harmonic matroids without using the axioms of projective geometry. As a particular case we have a combinatorial construction of a projective plane of prime order in full algebraic matroids.     

Regular Hjelmslev planes

A projective Hjelmslev plane is called regular iff it admits an Abelian collineation group that is regular on both the points and lines of the plane and that splits into a summand regular on the elements of any given neighborhood and another summand permuting the points and lines of the projective image plane regularly. Regular Hjelmslev planes are shown to correspond to so-called special difference sets. We construct regular Hjelmslev planes with parameters (qⁿ, q) for any prime power q and any natural number n as well as for infinitely many series of parameters (t, q), where t is not a power of q. Our construction also yields series of parameters for which the existence of a Hjelmslev plane was not known up to now as well as the first information on the existence of nontrivial collineations in the case of parameters (t, q) with t not a power of q.     

Cyclic projective planes and binary,extended cyclic self-dual codes

If P is a cyclic projective plane of order n, we give number theoretic conditions on n² + n + 1 so that the binary code of P is contained in a binary cyclic code C whose extension is self-dual. When this containment occurs C does not contain any ovals of P. As a corollary to these conditions we obtain that the extended binary code of a cyclic projective plane of order 2^s is contained in a binary extended cyclic self-dual code if and only if s is odd.     

Proof of a conjecture of Segre and Bartocci on monomial hyperovals in projective planes

The existence of certain monomial hyperovals D(x ^k ) in the finite Desarguesian projective plane PG(2, q), q even, is related to the existence of points on certain projective plane curves g _k (x, y, z). Segre showed that some values of k (k?=?6 and 2 ⁱ ) give rise to hyperovals in PG(2, q) for infinitely many q. Segre and Bartocci conjectured that these are the only values of k with this property. We prove this conjecture through the absolute irreducibility of the curves g _k .     

On the complement of the set of n-collinear points in PG(3, n)

Let ${S = (\mathcal{P}, \mathcal{L}, \mathcal{H})}$ be the finite planar space obtained from the 3-dimensional projective space PG(3, n) of order n by deleting a set of n-collinear points. Then, for every point ${p\in S}$ , the quotient geometry S/p is either a projective plane or a punctured projective plane, and every line of S has size n or n + 1. In this paper, we prove that a finite planar space with lines of size n + 1 ? s and n + 1, (s ≥ 1), and such that for every point ${p\in S}$ , the quotient geometry S/p is either a projective plane of order n or a punctured projective plane of order n, is obtained from PG(3, n) by deleting either a point, or a line or a set of n-collinear points.     

A class of unitals of order q which can be embedded in two different planes of order q2

By deriving the desarguesian plane of order q² for every prime power q a unital of order q is constructed which can be embedded in both the Hall plane and the dual of the Hall plane of order q² which are non-isomorphic projective planes. The representation of translation planes in the fourdimensional projective space of J. André and F. Buekenhouts construction of unitals in these planes are used. It is shown that the full automorphism groups of these unitals are just the collineation groups inherited from the classical unitals.     

Embeddings of SL(2; ?) into the cremona group

Geometric and dynamic properties of embeddings of SL(2; ℤ) into the Cremona group are studied. Infinitely many nonconjugate embeddings that preserve the type (i.e., that send elliptic, parabolic and hyperbolic elements onto elements of the same type) are provided. The existence of infinitely many nonconjugate elliptic, parabolic and hyperbolic embeddings is also shown. In particular, a group G of automorphisms of a smooth surface S obtained by blowing up 10 points of the complex projective plane is given. The group G is isomorphic to SL(2; ℤ), preserves an elliptic curve and all its elements of infinite order are hyperbolic.