Michel Lavrauw

On linear sets on a projective line

Linear sets generalise the concept of subgeometries in a projective space. They have many applications in finite geometry. In this paper we address two problems for linear sets: the equivalence problem and the intersection problem. We consider linear sets as quotient geometries and determine the exact conditions for two linear sets to be equivalent. This is then used to determine in which cases all linear sets of rank 3 of the same size on a projective line are (projectively) equivalent. In (Donati and Durante, Des Codes Cryptogr, 46:261–267), the intersection problem for subgeometries of PG(n, q) is solved. The intersection of linear sets is much more difficult. We determine the intersection of a subline PG(1, q) with a linear set in PG(1, qh) and investigate the existence of irregular sublines, contained in a linear set. We also derive an upper bound, which is sharp for odd q, on the size of the intersection of two different linear sets of rank 3 in PG(1, qh).

Scattered Spaces with Respect to a Spread in PG(n,q)

A scattered subspace of PG(n-1,q) with respect to a (t-1)-spread S is a subspace intersecting every spread element in at most a point. Upper and lower bounds for the dimension of a maximum scattered space are given. In the case of a normal spread new classes of two intersection sets with respect to hyperplanes in a projective space are obtained using scattered spaces.

On the classification of semifield flocks

Abstract A classical lemma of Weil is used to characterise quadratic polynomials f with coefficients GF ( q n ), q odd, with the property that f ( x ) is a non-zero square for all x ∈ GF ( q ). This characterisation is used to prove the main theorem which states that there are no subplanes of order q contained in the set of internal points of a conic in PG (2, q n ) for q ⩾4 n 2 −8 n +2. As a corollary to this theorem it then follows that the only semifield flocks of the quadratic cone of PG (3, q n ) for those q exceeding this bound are the linear flocks and the Kantor–Knuth semifield flocks.

On embeddings of minimum dimension of PG(n,q) × PG(n,q)

A construction is given of an embedding of \({\mathrm{PG}}(n-1,q)\times {\mathrm{PG}}(n-1,q)\) into \({\mathrm{PG}}(2n-1,q)\), i.e. of minimum dimension, and it is shown that the image is a nonsingular hypersurface of degree \(n\). The construction arises from a scattered subspace with respect to a Desarguesian spread in \({\mathrm{PG}}(2n-1,q)\). By construction there are two systems of maximum subspaces (in this case \((n-1)\)-dimensional) which cover this hypersurface. However, unlike the standard Segre embedding, the minimum embedding constructed here allows another \(n-2\) systems of maximum subspaces which cover this embedding. We describe these systems and study the stabiliser of these embeddings. The results can be considered as a generalization of the properties of the hyperbolic quadric \(Q^+(3,q)\).

Subgeometries and linear sets on a projective line

We define the splash of a subgeometry on a projective line, extending the definition of 1] to general dimension and prove that a splash is always a linear set. We also prove the converse: each linear set on a projective line is the splash of some subgeometry. Therefore an alternative description of linear sets on a projective line is obtained. We introduce the notion of a club of rank r, generalizing the definition from 4], and show that clubs correspond to tangent splashes. We obtain a condition for a splash to be a scattered linear set and give a characterization of clubs, or equivalently of tangent splashes. We also investigate the equivalence problem for tangent splashes and determine a necessary and sufficient condition for two tangent splashes to be (projectively) equivalent.

On the isotopism classes of finite semifields

A projective plane is called a translation plane if there exists a line L such that the group of elations with axis L acts transitively on the points not on L. A translation plane whose dual plane is also a translation plane is called a semifield plane. The ternary ring corresponding to a semifield plane can be made into a non-associative algebra called a semifield, and two semifield planes are isomorphic if and only if the corresponding semifields are isotopic. In [S. Ball, G. Ebert, M. Lavrauw, A geometric construction of finite semifields, J. Algebra 311 (1) (2007) 117-129] it was shown that each finite semifield gives rise to a particular configuration of two subspaces with respect to a Desarguesian spread, called a BEL-configuration, and vice versa that each BEL-configuration gives rise to a semifield. In this manuscript we investigate the question when two BEL-configurations determine isotopic semifields. We show that there is a one-to-one correspondence between the isotopism classes of finite semifields and the orbits of the action a subgroup of index two of the automorphism group of a Segre variety on subspaces of maximum dimension skew to a determinantal hypersurface.

Symplectic Spreads

We construct an infinite family of symplectic spreads in spaces of odd rank and characteristic.

Finite semifields and nonsingular tensors

In this article, we give an overview of the classification results in the theory of finite semifields (note that this is not intended as a survey of finite semifields including a complete state of the art (see also Remark 1.10)) and elaborate on the approach using nonsingular tensors based on Liebler (Geom Dedicata 11(4):455–464, 1981).

Fq-pseudoreguli of PG(3,q3) and scattered semifields of order q6

In this paper, we study rank two semifields of order q^6 that are of scattered type. The known examples of such semifields are some Knuth semifields, some Generalized Twisted Fields and the semifields recently constructed in Marino et al. (in press) [12] for q=1(mod3). Here, we construct new infinite families of rank two scattered semifields for any q odd prime power, with q=1(mod3); for any q=2^2^h, such that h=1(mod3) and for any q=3^h with h@?0(mod3). Both the construction and the proof that these semifields are new, rely on the structure of the linear set and the so-called pseudoregulus associated to these semifields.

On Eggs and Translation Generalised Quadrangles

We study eggs in PG(4n?1, q). A new model for eggs is presented in which all known examples are given. We calculate the general form of the dual egg for eggs arising from a semifield flock. Applying this to the egg obtained by L. Bader et al. (1999, J. Combin. Theory Ser. A86, 49?62) from the Penttila?Williams ovoid, we obtain the dual egg, which is not isomorphic to any of the previous known examples. Furthermore we give a new proof of a conjecture of J. A. Thas (1994, J. Combin. Theory Ser. A67, 140?160) using our model and classify all eggs of PG(7, 2) which is equivalent to the classification of all translation generalised quadrangles of order (4, 16).