The projective planes from which a point subset is deleted provided "classical" examples in the theory of linear spaces and many papers dealt with the finding of sufficient conditions for a linear space to be embeddable in a projective plane i.e. to construct an isomorphism with such a model. The theory further developed by considering planar linear spaces (cf. "L.M. Batten, A. Beutelspacher: "The theory of linear spaces. Cambridge University Press, 1993"), and classification theorems were proven as well as results on the embeddability in projective spaces. However, the following conjecture, due to W.M. Kantor, is still open: a planar space any two of whose planes either are disjoint or share a line is embeddable in a three-dimensional projective space. This conjecture was only proved in a weaker form under additional assumptions. The links among q-clans, generalised quadrangles of order (q,q2), flocks of the quadratic cone and translation planes, described in "J.A. Thas: Generalized quadrangles and flocks of cones. Europ. J. Comb 8 (1987)", really pushed the research in the field and lead to the introduction of BLT-sets when q is odd (cf. L. Bader, J.A. Thas, G. Lunardon: Derivation of flocks of quadratic cones. Forum Math. 2 (1990)") and herds of ovals when q is even ("W. Cherowitzo, T. Penttila, I. Pinneri, G.F. Royle: Flocks and ovals. Geom. Dedicata 60 (1996)"). Subsequently, research focused on semifield flocks, i.e. those flocks whose associated generalised quadrangle is a translation generalised quadrangle. Surprisingly, such a flock defines a translation ovoid of the non-singular quadric Q(4,q), and conversely (cf. "J.A. Thas: Symplectic spreads in PG(3,q), inversive planes and projective planes. Discrete Math. 174 (1997)" and "G. Lunardon: Flocks, ovoids of Q(4,q) and designs. Geom. Dedicata 66 (1997)"). Consequently, the Penttila-Williams ovoid defines a new semifield flock (cf. L. Bader, G. Lunardon, I. Pinneri: A new semifield flock, to appear in J. Comb. Theory (A)"). Therefore, the problem is still open of classifying such flocks and, more generally, all flocks of the quadratic cone. The linear blocking sets turned out to be especially useful to construct new examples of blocking sets. Moreover, the possibility of defining them in three equivalent manners allows their investigation by methods from projective geometry, linear algebra and finite field theory. Thus, new techniques are added to the traditional polynomial techniques which yields further information on their structure and properties. Presently, the main open problem concerns the existence of small minimal blocking sets which are not linear.
The researches which are planned by this unit mainly concern :
The ideas which will be developed are briefly described below.
The attention is focused on problems concerning some classes of finite planar spaces which can be summarized as follows:
Let (S, L, P) be a planar space and assume Lo is a subset of its lines such that (So, Lo) is a generalised quadrangle, So being the set theoretical union of the lines in Lo. Then (So, Lo) is embedded in the planar space. It is well known that, whenever (S, L, P) is a projective space, the generalised quadrangle is classical. The research aims at classifying (So, Lo) when the planar space is not a projective space and establishing whether the existence of a generalised quadrangle in a planar space forces it to be embeddable in a projective space.
The second topic concerns the characterisation of the Grassmann space of a planar space. Lines and pencils of lines in a planar space may be viewed as points and lines of a partial linear space (S,R). This space (S,R) was characterised by G. Tallini when the planar space is a projective space and, recently, by P.M. Lo Re and D. Olanda without such an assumption. The latter characterisation relies on the hypothesis (S,R) admits both a family of subspaces and a family of quasi-subspaces satisfying suitable incidence properties. The planned research aims at finding further characterisations of (S,R) by using only incidence properties of its lines or focusing on only one of the two subspace families considered by P.M. Lo Re and D. Olanda.
Several recent works in combinatorial geometries looked for sufficient conditions which guarantee the embeddability in PG(3,n) of a locally projective finite planar space of order n. To mention some already obtained results, the paper "P. Biondi: An embedding theorem for finite planar spaces, Rendiconti del Circolo Matematico di Palermo, Serie II, Tomo XLVII (1998)" yields a necessary and sufficient condition for such an embeddability in PG(3,n), provided the planar space contains an (n2+1)-cap. Next, finite planar spaces will be considered whose planes pairwise meet in a line and which can be endowed with an injective map between points and planes having the duality property.
The researches in section 2 will deal with the following topics:
The linear blocking sets, introduced by G. Lunardon, allowed the construction of blocking sets not of Rédei type in a desarguesian plane, disproving a nearly accepted conjecture. Blocking sets have been successfully studied by polynomial techniques; however, such techniques did not provide solutions to all open problems. In the case of linear blocking sets other methods can be applied and some problems become easier. Therefore, linear blocking sets will be investigated, focusing on those of minimal and maximal cardinality, in order to find new examples and characterisations..
The interest in these topics is widespread and the research will focus on investigating spreads of the hexagon H(q) associated with the group G2(q). A motivation is provided by a recent result, due to J.A. Thas et al., that a special spread of the considered hexagon defines the ovoid associated to Ganley flock.
Furthermore, since the ovoids of the quadric Q(6,q) are related to H(q), their investigation might lead to prove they do exist only when q is a power of 3 as many specialists in the field believe.
According to "L. Bader, G. Lunardon: On non-hyperelliptic flocks. Europ. J. Combinatorics 15 (1994)", any new example should be a sporadic one. In general, it is difficult to foresee the existence of sporadic objects, however some known facts show that the order of such a flock should be a power of 3. An attempt will be made to construct some examples of the above mentioned flocks relying on some recent results by J.A Thas, and with the help of some computer algebra package.
The most interesting topics seem to be:
The problem consists in finding combinatorial properties which are suitable to characterise some special (and well known) algebraic varieties. An example of such results is provided by "P. Biondi: A common characterization of ovoids, non singular ruled quadrics and non singular hermitian varieties in PG(d,n), to appear in Bull. Belg. Math. Soc.", where a common characterisation is provided of both quadrics and Hermitian varieties which relies on the property that all hyperplane sections with non-tangent hyperplanes have the same cardinality.
The classification of all geometries with a given diagram is a difficult problem and many mathematicians, all over the world, are engaged in its solution. A complete classification is only possible when the considered geometry satisfies additional assumptions. Theorems with this flavour are those in "G. Pica: On finite Cn-geometrieswith thick lines, Discr. Math. (1996)" e "A. Del Fra, G. Pica: Flag-transitive C2·Ln geometries, Discr. Math. (1997)". With this in mind, a classification is planned of geometries of type C3 admitting thin lines and whose residue at each point is a non-classical generalised quadrangle.
Algebraic curves will be studied which admit many rational points. In particular, the possibility is investigated of improving the bound, found by Stohr and Voloch, for the algebraic curves of order sqrt(q) - 1 over GF(q).