The Finite Field Kakeya Problem

Co-authored with Aart Blokhuis. Published in: Building Bridges Between Mathematics and Computer Science, Series: Bolyai Society Mathematical Studies , Vol. 19, 2008, Grötschel, Martin; Katona, Gyula O.H. (Eds.)

ABSTRACT: A Besicovitch set in the n-dimensional affine space over the Galois field with q element is a set of points containing a line in every direction. The Kakeya problem is to determine the minimal size of such a set. We completely solve this problem in the plane case.

Blocking Sets in PG(n,q^t)

Co-authored with Olga Polverino and Leo Storme. Published in:  Designs, Codes and Cryptography. vol. 44, 2007, pp. 97-113 ISSN: 0925-1022.

Blocking sets in PG(2,q^n) from Cones of PG(2n,q)

Co-authored with Olga Polverino. Published in: Journal of Algebraic Combinatorics. vol. 24, 2006, pp. 61-81 ISSN: 0925-9899.

Alcune linee di ricerca nelle geometrie di Galois

Published in:  Atti del XVI Congresso dell'Unione Matematica Italiana. Napoli, 13-18 Settembre 1999. Pubblicazione a cura dell' Unione Matematica Italiana

Gli argomenti trattati riguardano la geometria sui campi di Galois, che costituisce una delle aree più́ vaste ed importanti della combinatoria. Il suo interesse, al di là di quello intrinseco dovuto alla bellezza dei suoi risultati ed all’eleganza dei suoi metodi, è nato ed  è in continuo crescendo a causa delle innumerevoli applicazioni, sia in altri campi della matematica che in teorie e discipline maggiormente coinvolte in problemi concreti.
La vastità́ della letteratura esistente ci ha costretti ad una drastica selezione tra i possibili argomenti da presentare.  Abbiamo, così́ , limitato la nostra attenzione ad alcuni problemi classici, diversi dei quali non ancora completamente risolti, e tra questi abbiamo messo maggiormente in luce
quelli più́ vicini ai nostri gusti e ai nostri interessi di ricerca:
1. Caratteri di insiemi di punti in un piano proiettivo finito.
2. Le ovali di PG(2,q), q dispari.
3. Le iperovali di PG(2,q), q pari.
4. Insiemi di classe (0,n) in PG(2,q).
5. Ovoidi in PG(3,q).
6. (k, d)−calotte in P G(n,q).
7. Blocking sets.


     

On the structure of 3-nets embedded in a projective plane

Co-authored with Aart Blokhuis and Gabor Korchmaros. Submitted to Journal of Combinatorial Theory - Series A

We investigate finite 3-nets embedded in a projective plane over a (finite or infinite) field of any characteristic p. Such an embedding is regular when each of the three classes of the 3-net comprises concurrent lines, and irregular otherwise. It is completely irregular when no class of the 3-net consists of concurrent lines.
We are interested in embeddings of 3-nets  which are irregular but the lines of one class are concurrent. For an irregular embedding of a 3-net of order n> 4 we prove that,  if all lines from  two classes are tangent to the same irreducible conic,  then all lines from the third class are concurrent. We also prove the converse provided that the order n of the 3-net is smaller than p. In the complex plane, apart from a sporadic example of order n=5 due to Stipins, each known irregularly embedded 3-net has the property that all its lines are tangent to a plane cubic curve. Actually, the procedure of constructing irregular 3-nets with this property works over any field. In positive characteristic, we present some more examples for n>5 and  give a complete classification for n=4.

Blocking Sets, Linear Groups and Transversal Designs

Co-authored with Aiden A. Bruen and Olga Polverino. To appear on "Quaderni di Matematica", edited by Dipartimento di Matematica of the Seconda Università degli Studi di Napoli at Caserta

We exploit new methods involving affine groups to determine the complete geometric structure of perspective sets in PG(2,q). Using this we then, in a few pages, give a complete characterization of those blocking sets in PG(2,q) that contain at least two Rédei lines. Our characterization is analogous to, but slightly more detailed than, the characterization obtained by Sherman in the sequence [2, 3]. Finally, we use our results to sharpen known results (see [1]) by obtaining a detailed classification of transversal designs embedded in planes.

[1] A.A. Bruen and C.J.Colbourn, Transversal Designs in Classical Planes and Spaces, Journal of Combinatorial Theory, Series A, 92, (2000), 88-94.
[2] B.F. Sherman, Enclosed (k; n)-arcs, Journal of Geometry, 43, (1992), 166-177.
[3] B.F. Sherman, Minimal blocking sets in finite planes, Journal of Geometry, 43, (1992), 178-187.

Maximal arcs in Desarguesian planes of odd order do not exist

On (q+ t)-arcs of type (0, 2, i) in a Desarguesian plane of order q

On the non existence of blocking-setsin Pg/nq/and AG/nq/, for all large enough n.

Blocking sets with respect to special families of lines and nuclei of θn-sets in finite n-dimensional projective and affine spaces

On the classification of generalized quadrangles in a finite affine space AG (3, 2h), Boll

Caps and Veronese varieties in projective Galois spaces* 1

Bijections which preserve blocking sets

Blocking-sets in infinite projective and affine spaces

On the classification of generalized quadrangles in a finite affine space AG (3, 2h)

On a characterization of Dilworth truncations of combinatorial geometries

Nuclei of point sets of sizeq+ 1 contained in the union of two lines inPG (2, q)

Dual blocking sets in projective and affine planes

Extensions of combinatorial geometries by the addition of a unique line* 1

On a class of lattices associated with n-cubes* 1

3-nets embedded in a projective plane

Appunti del corso di Geometria Combinatoria

Appunti di Algebra Lineare