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.     

Ovoids and monomial ovals

This paper is a contribution to the classification of ovoids. We show, under some rather technical assumptions, that if an ovoid of PG(3, q) has a pencil of monomial ovals, then it is either an elliptic quadric or a Tits ovoid. Further, we show that if an ovoid of PG(3, q) has a bundle of translation ovals, again under some extra assumptions, then the ovoid is an elliptic quadric or a Tits ovoid.     

Note: A remark on a result of B. Segre concerning pseudoregular points of an elliptic quadric ofAG(2,q), q odd

In [1] S. ILKKA conjectured that pqeudoregular points of an elliptic quadric ofAG(2,q),q odd, only exist for small values ofq. In [3] B. SEGRE proves that an elliptic quadric ofAG(2,q),q odd, has pseudoregular points iffq=3 or 5. In [2], however, F. KáRTESZI shows that an elliptic quadric ofAG(2,7) has eight pseudoregular points. In this note we prove that part of B. Segre's proof is not correct, and that an elliptic quadric ofAG(2,q),q odd, has pseudoregular points iffq=3, 5 or 7.     

k-Arcs,Hyperovals, Partial Flocks and Flocks

Some recent results on k-arcs and hyperovals of PG(2,q),on partial flocks and flocks of quadratic cones of PG(3,q),and on line spreads in PG(3,q) are surveyed. Also,there is an appendix on how to use Veronese varieties as toolsin proving theorems.     

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).     

Irregular hyperovals inPG(2, 64)

Two irregular hyperovals in the Desarguesian projective planePG(2, 64) of order 64 are constructed. One has a collineation stabiliser of order 60, the other a stabiliser of order 15. It is a lso shown, with the aid of a computer, that there are no more (irregular) hyperova ls inPG(2, 64) stabilised by a collineation of order 5.     

Classification of ovoids inPG(3, 32)

A classification of the ovoids inPG(3, 32) is completed with the aid of a computer. The ovoids are examined in terms of which ovals can possibly appear as secant plane sections. A weak necessary condition for two ovals to appear together as plane sections of an ovoid surprisingly turns out to be sufficient to demonstrate that the only possible secant plane sections are translation ovals. A known result regarding ovoids with such plane sections then identifies the ovoids as either elliptic quadrics or Tits ovoids.     

Ovoids of PG(3,q), q Even, with a Conic Section

It is shown that if a plane of PG(3,q), q even, meets an ovoidin a conic, then the ovoid must be an elliptic quadric. Thisis proved by using the generalized quadrangles T₂(C) (C a conic),W(q) and the isomorphism between them to show that every secantplane section of the ovoid must be a conic. The result thenfollows from a well-known theorem of Barlotti.     

On the duals of certain d-dimensional dual hyperovals in

Let d2. A construction of d-dimensional dual hyperovals in PG(2d+1,2) using quadratic APN functions was discovered by Yoshiara in [S. Yoshiara, Dimensional dual hyperovals associated with quadratic APN functions, Innov. Incidence Geom., in press]. In this note, we prove that the duals of the d-dimensional dual hyperovals in PG(2d+1,2) constructed from quadratic APN functions are also d-dimensional dual hyperovals in PG(2d+1,2) if, and only if, d is even. Some examples are presented.     

Classification of hyperovals inPG(2,32)

In this paper, we describe an exhaustive computer search that demonstrates that there are precisely 6 isomorphism classes of hyperovals inPG(2,32). The six classes had previously been discovered, and it was known that any further hyperovals would have stabiliser groups of orders 1 or 2. As the techniques for finding hyperovals involved a mixture of group theory and computer search, an exhaustive search was regarded as the only feasible way to eliminate these final cases with small group.     

The determination of ovoids of PG(3,q) containing a pointed conic

It is shown that if a plane of PG(3,q),q even, meets an ovoid in a pointed conic, then eitherq=4 and the ovoid is an elliptic quadric, orq=8 and the ovoid is a Tits ovoid.     

On the intersection of ovoids sharing a polarity

In this article, an ovoidal fibration is used to show that any two ovoids of PG(3, q), q even, sharing a polarity, must meet in an odd number of points. This result was previously known only when one of the ovoids was an elliptic quadric or a Tits ovoid. It is also shown that an ovoid and an elliptic quadric of PG(3, q), sharing all of their tangents, must meet in 1 (mod 4) points.      

On d-Dimensional Dual Hyperovals

d-dimensional dual hyperovals in a projective space of dimension n are the natural generalization of dual hyperovals in a projective plane. After proving some general properties of them, we get the classification of two-dimensional dual hyperovals in projective spaces of order 2. A characterization of the only two-dimensional dual hyperoval which is known in PG(5,4) is also given. Finally the classification of 2-transitive two-dimensional dual hyperovals is reached.     

On the Sporadic Semifield Flock

We obtain the BLT-set associated with the sporadic semifield flock of the quadratic cone in PG(3,3⁵) as the complete intersection of the Payne–Thas and the Kantor–Knuth ovoids of the parabolic quadric Q(4,3⁵. Also, we give an alternative construction of the Penttila–Williams ovoid of Q(4,3⁵).     

On quadratic APN functions and dimensional dual hyperovals

In this paper we characterize the d-dimensional dual hyperovals in PG(2d + 1, 2) that can be obtained by Yoshiara’s construction (Innov Incid Geom 8:147–169, 2008) from quadratic APN functions and state a one-to-one correspondence between the extended affine equivalence classes of quadratic APN functions and the isomorphism classes of these dual hyperovals.     

A Geometric characterization of the perfect Suzuki-Tits ovoids

We characterize in a geometrical way those Suzuki-Tits ovoids which are defined over a perfect fieidK (or equivalently living inside a self polar symplectic quadrangle). We simplify our axioms in the particular cases that (1) the associated Suzuki group has exactly two orbits in the set of lines ofPG(3,K), and (2) the ovoid is finite.Senior Research Associate at the Belgian National Fund for Scientific Research     

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.     

On abelian difference set codes

In this paper we determine the ranks of the incidence matrices that belong to the following types of difference sets: Twin prime power difference sets, biquadratic residues and biquadratic residues with 0. We also prove a conjecture of Assmus and Key on the code generated by the hyperovals of PG(2, q).     

On Ovoids of Parabolic Quadrics

It is known that every ovoid of the parabolic quadric Q(4, q), q=p ^h , p prime, intersects every three-dimensional elliptic quadric in 1 mod p points. We present a new approach which gives us a second proof of this result and, in the case when p=2, allows us to prove that every ovoid of Q(4, q) either intersects all the three-dimensional elliptic quadrics in 1 mod 4 points or intersects all the three-dimensional elliptic quadrics in 3 mod 4 points. We also prove that every ovoid of Q(4, q), q prime, is an elliptic quadric. This theorem has several applications, one of which is the non-existence of ovoids of Q(6, q), q prime, q>3. We conclude with a 1 mod p result for ovoids of Q(6, q), q=p ^h , p prime.     

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).