Angela Aguglia

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(q2), the extended lines of L cover a non-singular Hermitian surface H ≅ H(3, q2) of PG(3, q2). We prove that if Sis any symplectic spread PG(3, q), then the extended lines of this spread form a complete (q2 + 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 (q3 + 1)-spans of the Hermitian variety H(5, q2).

On pairs of permutable Hermitian surfaces

We investigate the intersection R of two permutable Hermitian surfaces of PG(3,q^2), q odd. We show that R is a determinantal variety. From the combinatorial point of view R comprises a complete (q^2+1)-span of the two corresponding Hermitian surfaces.

Algebraic curves and maximal arcs

AbstractA lower bound on the minimum degree of the plane algebraic curves containing every point in a large point-set

Intransitive collineation groups of ovals fixing a triangle

\mathcal{K}

Constructions of unitals in Desarguesian planes

of the Desarguesian plane PG(2,q) is obtained. The case where

Intersections of the Hermitian surface with irreducible quadrics in PG(3 ,q2), q odd

\mathcal{K}

Multiple blocking sets and multisets in Desarguesian planes

is a maximal (k,n)-arc is considered in greater depth.

An Algorithm for Constructing Some Maximal Arcs in PG(2, q 2)

We investigate collineation groups of a finite projective plane of odd order fixing an oval and having two orbits on it, one of which is assumed to be primitive. The situation in which there exists a fixed triangle off the oval is considered in detail. Our main result is the following. Theorem. Let π be a finite projective plane of odd order n containing an oval Ω. If a collineation group G of π satisfies the properties: (a) G fixes Ω and the action of G on Ω yields precisely two orbits Ω1 and Ω2, (b) G has even order and a faithful primitive action on Ω2, (c) G fixes neither points nor lines but fixes a triangle ABC in which the points A, B, C are not on the oval Ω, then n ∈ {7, 9, 27}, the orbit Ω2 has length 4 and G acts naturally on Ω2 as A4 or S4.Each order n ∈ {7, 9, 27} does furnish at least one example for the above situation; the determination of the planes and the groups which do occur is complete for n = 7, 9; the determination of the planes is still incomplete for n = 27.

Intersection sets, three-character multisets and associated codes

We present a new construction of non-classical unitals from a classical unital U in PG(2,q^2). The resulting non-classical unitals are B-M unitals. The idea is to find a non-standard model @P of PG(2,q^2) with the following three properties: (i)points of @P are those of PG(2,q^2); (ii)lines of @P are certain lines and conics of PG(2,q^2); (iii)the points in U form a non-classical B-M unital in @P. Our construction also works for the B-T unital, provided that conics are replaced by certain algebraic curves of higher degree.We present a new construction of non-classical unitals from a classical unital U in PG(2, q2). The resulting non-classical unitals are B-M unitals. The idea is to find a non-standard model Π of PG(2, q2) with the following three properties: (i) points of Π are those of PG(2, q2); (ii) lines of Π are certain lines and conics of PG(2, q2); (iii) the points in U form a non-classical B-M unital in Π. Our construction also works for the B-T unital, provided that conics are replaced by certain algebraic curves of higher degree.

Orthogonal arrays from Hermitian varieties

In PG(3,q^2), with q odd, we determine the possible intersection sizes of a Hermitian surface H and an irreducible quadric Q having the same tangent plane at a common point P@?Q@?H.