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1 - Classical Problems in Mechanics (R. Russo ed.)2 - Recent Developments in Partial Differential Equations (V. A. Solonnikov ed.)3 - Recent Progress in Function Spaces (G. Di Maio and Ľ. Holá eds.)4 - Advances in Fluid Dynamics (P. Maremonti ed.)5 - Methods of Discrete Mathematics (S. Löwe, F. Mazzocca, N. Melone and U. Ott eds.)6 - Connections between Model Theory and Algebraic and Analytic Geometry (A. Mac-

intyre ed.)7 - Homage to Gaetano Fichera (A. Cialdea ed.)8 - Topics in Infinite Groups (M. Curzio and F. de Giovanni eds.)9 - Selected Topics in Cauchy-Riemann Geometry (S. Dragomir ed.)

10 - Topics in Mathematical Fluids Mechanics (G.P. Galdi and R. Rannacher eds.)11 - Model Theory and Applications (L. Belair, Z. Chatzidakis et al. eds.)12 - Topics in Diagram Geometry (A. Pasini ed.)13 - Complexity of Computations and Proofs (J. Krajíček ed.)14 - Calculus of Variations: Topics from the Mathematical Heritage of E. De Giorgi (D.

Pallara ed.)15 - Dispersive Nonlinear Problems in Mathematical Physics (P. D’Ancona and V. Georgev

eds.)16 - Kinetic Methods for Nonconservative and Reacting Systems (G. Toscani ed.)17 - Set Theory: Recent Trends and Applications (A. Andretta ed.)18 - Selection Principles and Covering Properties in Topology (Lj.D.R. Kočinac ed.)19 - Trends in Incidence and Galois Geometries: a Tribute to Giuseppe Tallini (F. Maz-

zocca, N. Melone and D. Olanda eds.)20 - Mathematical Modelling of Bodies with Complicated Bulk and Boundary Behavior

(M. Šilhavý ed.)21 - Vector Bundles and Low Codimensional Subvarieties: State of the Art and Recent

Developments (G. Casnati, F. Catanese, and R. Notari eds.)22 - Theory and Applications of Proximity, Nearness and Uniformity (G. Di Maio and S.

Naimpally eds.)23 - On the Notions of Solution to Nonlinear Elliptic Problems: Results and Developments

(A. Alvino, A. Mercaldo, F. Murat, and I. Peral eds.)24 - Numerical Methods for Balance Laws (G. Puppo and G. Russo eds.)

Next issues

Trends in Incidenceand Galois Geometries:a Tribute to Giuseppe Tallini

edited by Francesco Mazzocca, Nicola Melone and Domenico Olanda

Received November, 2009

c 2006 by Dipartimento di Matematica della Seconda Università di Napoli

Photocomposed copy prepared from a LATEX file.

ISBN ?????ISBN 978-88-548-3571-9

Authors’ addresses:

Vito AbatangeloDipartimento di MatematicaPolitecnico di BariVia Orabona 4I-70125 Bari, Italyemail: abatvito@dm.uniba.it

Andrea BlunckFachbereich MathematikUniversität HamburgBundesstraße 55D-20146 Hamburg, Germanyemail: andrea.blunck@math.uni-hamburg.de

Aiden A. BruenDepartment of Electrical andComputer EngineeringSchulich School of EngineeringUniversity of Calgary2500 University Drive NWCalgary, Alberta T2N 1N4, Canadaemail: bruen@ucalgary.ca

Giorgio FainaDipartimento di Matematica e InformaticaUniversità degli Studi di PerugiaVia Vanvitelli 1I-06123 Perugia, Italyemail: faina@dipmat.unipg.it

Mario GionfriddoDipartimento di Matematica e InformaticaUniversità di Cataniaviale A. Doria 6I-95125 Catania, Italyemail: gionfriddo@dmi.unict.it

Luigia BerardiDepartment of Electricaland Information EngineeringUniversity of L’AquilaI-67040 Monteluco di RoioL’Aquila, Italyemail: berardi@ing.univaq.it

Bella BoseSchool of Electrical Engineering& Computer ScienceOregon State UniversityCorvallis, OR 97331 U.S.A.email: bose@eecs.orst.edu

Nicola DuranteDipartimento di Matematica e ApplicazioniUniversità di Napoli “Federico II”Complesso di Monte S. AngeloEdificio T, via CintiaI-80126 Napoli, Italyemail: ndurante@unina.it

Stefania FerriDepartment of Electricaland Information EngineeringUniversity of L’AquilaI-67040 Monteluco di RoioL’Aquila, Italyemail: stefania.ferri1@istruzione.it

Hans HavlicekInstitut für Diskrete Mathematikund GeometrieTechnische Universität WienWiedner Hauptstraße 8–10A-1040 Wien, Austriaemail: havlicek@geometrie.tuwien.ac.at

Olof HedenDepartment of MathematicsSchool of Engineering SciencesKTH - Royal Institute of TechnologySE-100 44 Stockholm, Swedenemail: olohed@math.kth.se

Helmut KarzelZentrum MathematikT.U. MuenchenD-80290 Muenchen - Germanyemail: h.karzel@freenet.de

Alexander KreuzerDepartment MathematikUniversität HamburgBundesstr. 55D-20146 Hamburg, Germanyemail: kreuzer@math.uni-hamburg.de

Mario MarchiDepartment of Mathematics and PhysicsCatholic UniversityVia Musei 41I-25121 Brescia, Italyemail: m.marchi@dmf.unicatt.it

Francesco MazzoccaDepartment of MathematicsSeconda Universitá degli Studi di Napolivia Vivaldi, 43I-81100 Caserta, Italyemail: francesco.mazzocca@unina2.it

Stefano InnamoratiDepartment of Electricaland Information EngineeringUniversity of L’AquilaI-67040 Monteluco di RoioL’Aquila, Italyemail: innamora@ing.univaq.it

Gábor KorchmárosDipartimento di MatematicaUniversità degli Studi della BasilicataViale dell’Ateneo Lucano 10Contrada Macchia RomanaI-85100 Potenza, Italyemail: korchmaros@unibas.it

Guglielmo LunardonDipartimento di Matematica e ApplicazioniUniversità di Napoli “Federico II”Complesso di Monte S. AngeloEdificio T, via CintiaI-80126 Napoli, Italyemail: lunardon@unina.it

Stefano MarcuginiDipartimento di Matematica e InformaticaUniversità degli Studi di PerugiaVia Vanvitelli 1I-06123 Perugia, Italyemail: gino@dipmat.unipg.it

Nicola MeloneDepartment of MathematicsSeconda Universitá degli Studi di Napolivia Vivaldi, 43I-81100 Caserta, Italyemail: nicola.melone@unina2.it

Klaus MetschMathematisches InstitutJustus Liebig UniversityArndtstrasse 2D-35392 Giessen, Germanyemail: klaus.metsch@math.uni-giessen.de

Vito NapolitanoDipartimento di MatematicaUniversità della BasilicataEdificio 3D, Viale dell’Ateneo Lucano 10I-85100, Contrada Macchia RomanaPotenza, Italyemail: vnapolitano@unibas.it

Udo OttInstitut für GeometrieTechnische Universität BraunschweigPockelsstraße 14D-38106 Braunschweig, Germanyemail: udo.ott@tu-braunschweig.de

Antonio PasiniDepartment of MathematicsUniversity of SienaPian dei Mantellini 44I-53100 Siena, Italyemail: pasini@unisi.it

Olga PolverinoDepartment of MathematicsSeconda Universitá degli Studi di Napolivia Vivaldi, 43I-81100 Caserta, Italyemail: olga.polverino@unina2.it

Salvatore MiliciDipartimento di Matematica e InformaticaUniversità di Cataniaviale A. Doria 6I-95125 Catania, Italyemail: milici@dmi.unict.it

Domenico OlandaDipartimento di Matematica e ApplicazioniUniversità di Napoli “Federico II”Complesso di Monte S. AngeloEdificio T, via CintiaI-80126 Napoli, Italyemail: olanda@unina.it

Fernanda PambiancoDipartimento di Matematica e InformaticaUniversità degli Studi di PerugiaVia Vanvitelli 1I-06123 Perugia, Italyemail: fernanda@dipmat.unipg.it

Silvia PiantaDepartment of Mathematics and PhysicsCatholic UniversityVia Musei 41I-25121 Brescia, Italyemail: s.pianta@dmf.unicatt.it

Luca G. TalliniDipartimento di Scienzedella ComunicazioneUniversità degli Studi di TeramoCoste Sant’AgostinoI-64100 Teramo, Italyemail: ltallini@unite.it

Joseph A. ThasDepartment of Pure Mathematicsand Computer AlgebraGhent UniversityKrijgslaan 281, S22B-9000 Gent, Belgiumemail: jat@cage.ugent.be

Corrado ZanellaDipartimento di Tecnica e Gestionedei Sistemi IndustrialiUniversità di PadovaStradella S. Nicola 3I-36100 Vicenza, Italyemail: corrado.zanella@unipd.it

Contents

Preface

Irreducible Hypersurfaces of Minimal Degree Containing AllPoints of a Finite Projective Space Over a Finite Field 1Vito Abatangelo and Gábor Korchmáros

Steiner Key Distribution Patterns 19Luigia Berardi, Stefania Ferri, and Stefano Innamorati

Blocking Sets in Chain Geometries 39Andrea Blunck, Hans Havlicek, and Corrado Zanella

Blocking Sets, Linear Groups and Transversal Designs 51Aiden A. Bruen, Francesco Mazzocca, and Olga Polverino

On Quadrics of P G(3, q) 67Nicola Durante, Vito Napolitano, and Domenico Olanda

The Maximal Size of a Maximal Partial Spread in P G(3, 9) 77Olof Heden, Giorgio Faina, Stefano Marcugini, and Fernanda Pambianco

Embedding of P4-Designs into 4-Cycle Systems 113Mario Gionfriddo and Salvatore Milici

Three-Reflection Theorems in the Hyperbolic Plane 127Helmut Karzel, Mario Marchi, and Silvia Pianta

A Remark on the Exchange Condition in Linear Spaces 141Alexander Kreuzer

Semifields and Linear Sets 153Guglielmo Lunardon

The Number of Lines Needed to Block All Planes in P G(5, q) 167Klaus Metsch

Construction and Classification of Some CyclotomicDifference Sets 187Udo Ott

The Smallest Underlying Field of a Lax Embedding 229Antonio Pasini

On Decoding Some Error Control Codes Using the ElementarySymmetric Functions 265Luca G. Tallini and Bella Bose

A Characterization of the Generalized Quadrangle W (q)and Applications 299Joseph A. Thas

Mathematics subject classification

Irreducible Hypersurfaces of Minimal Degree Containing . . . : 14G15, 11G20, 14H45.

Steiner Key Distribution Patterns: 94A62, 05B05.

Blocking Sets in Chain Geometries: 51A30, 51A45, 51B05, 51C05, 51E21.

Blocking Sets, Linear Groups and Transversal Designs: 51E21, 05B25, 51E20.

On Quadrics of P G(3, q): 51E210, 51E21.

The Maximal Size of a Maximal Partial Spread in P G(3, 9): 51E23.

Embedding of P4-Designs into 4-Cycle Systems: 05B05.

Three-Reflection Theorems in the Hyperbolic Plane: 51B20.

A Remark on the Exchange Condition in Linear Spaces: 51A05, 51E15.

Semifields and Linear Sets: 12K10, 51E14.

The Number of Lines Needed to Block All Planes in P G(5, q): 51D20, 52C17, 52C35.

Construction and Classification of Some Cyclotomic Difference Sets: 05B10, 20D60.

The Smallest Underlying Field of a Lax Embedding: 51A50, 51E24, 05B25.

On Decoding Some Error Control Codes Using . . . : 94B24, 94B27, 11R18.

A Characterization of the Generalized Quadrangle W (q) . . . : 51E30, 05B25, 51B25.

Preface

This volume of quaderni di matematica is in memory of Giuseppe Tallini, aleading figure in international Combinatorics, passionate and influential Edu-cator throughout his career and founder of the Italian research group in Combi-natorics. The volume consists of contributions from Mathematicians of variousnationalities, who met him and interacted with him in many ways. The se-lected papers range from various current research themes in Combinatorics:Galois Geometries, Algebraic geometry over finite fields, Coding Theory andIncidence Structures. The choice of non homogeneous subjects reflects the pe-culiarity of the Scientist and Teacher Giuseppe Tallini. For an idea regardinghis various research interests, namely his many scientific interests and results,his wish to be able to deeply understand problems, and his influential pas-sion for Mathematics, we refer to [1], where it clearly appears how broad andvarious his scientific production was. This expands from General Topology toDifferential Geometry, from Algebraic Geometry over finite fields to IncidenceGeometry of Special Algebraic Varieties, from Galois Geometry to Designs andError Correcting Codes theories. In conclusion, we refer to [2] for informationon the life and the scientific work of Giuseppe Tallini and we heartly thank allfriends and colleagues who have agreed to submit a paper for this volume.

Francesco Mazzocca, Nicola Melone and Domenico Olanda

References

[1] Ceccherini, P.V.: Commemorazione di Giuseppe Tallini, Bollettinodella Unione Matematica Italiana (8) 1-B (1998), 451–474.

[2] Tallini, G.: Opere scelte, edited by Unione Matematica Italiana, EdizioniCremonese (2004).

Irreducible Hypersurfacesof Minimal Degree ContainingAll Points of a Finite ProjectiveSpace Over a Finite Field

Vito Abatangelo and Gábor Korchmáros

Contents

1. Introduction (3).2. G. Tallini’s paper entitled “Irreducible hypersurfaces of minimal order invading

a Galois space” (4).3. Irreducible curves of minimal degree containing all points of a desarguesian plane

over a finite field (13).

Irreducible Hypersurfaces of Minimal Degree Containing All Points of . . . 3

1. Introduction

The fundamental result on algebraic curves over finite fields is the Hasse–Weil bound

(1.1) 𝑞 + 1 − 2𝑔√𝑞 ≤ 𝑁𝑞 ≤ 𝑞 + 1 + 2𝑔

√𝑞.

on the number 𝑁𝑞 of 𝐺𝐹 (𝑞)-rational points of an irreducible non-singular alge-braic curve 𝒳 of genus 𝑔 defined over the finite field 𝐺𝐹 (𝑞). The bound (1.1)was originally proved by Hasse for 𝑔 = 1 in 1933, and then by Weil for 𝑔 > 1

in 1948. From (1.1) via the zeta function, the Riemann hypothesis for functionfields over finite fields follows, see Bombieri [5]. An extension of (1.1) to pri-mals (hypersurfaces) of 𝑃𝐺(𝑟, 𝑞) of degree 𝑚 is the Hasse-Dwork-Deligne-Weilbound, see for instance [8],

(1.2) 𝜃(𝑟 − 1, 𝑞) − 𝑑𝑞𝑟−1𝑞 ≤ 𝑁𝑞 ≤ 𝜃(𝑟 − 1, 𝑞) + 𝑑𝑞𝑟−1𝑞,

where 𝑑 = [(𝑚 − 1)/𝑚][(𝑚 − 1)𝑟 − (−1)𝑟], and 𝜃(𝑠, 𝑞) = 1 + 𝑞 + . . . + 𝑞𝑠, thenumber of points of 𝑃𝐺(𝑠, 𝑞).

It is easy to show that every hypersurface containing all points of 𝑃𝐺(𝑟, 𝑞)

has degree at least 𝑞 + 1. So, the interesting problem arises to classify theirreducible (possibly singular) hypersurfaces 𝑓 of minimum degree containingall points of 𝑃𝐺(𝑟, 𝑞). This problem was completely solved by G. Tallini in1961. His paper [13], reproduced in English in Section 2, contains a numberof elegant results and characterizations of such hypersurfaces. An expandedversion of [13] is [14].

For 𝑟 ≥ 3, G. Tallini showed the existence of irreducible hypersurfaces ofminimum degree 𝑞 + 1, defined over 𝐺𝐹 (𝑞), containing all points of 𝑃𝐺(𝑟, 𝑞)

and gave a classification for such hypersurfaces, see 𝑛.3 in Section 2.G. Tallini also pointed out that his result does not hold for 𝑟 = 2, as a

plane curve 𝑓 of degree 𝑞 + 1, defined over 𝐺𝐹 (𝑞) and containing all points of𝑃𝐺(2, 𝑞) is reducible and splits into 𝑞+1 lines of a pencil. Then he classified allirreducible plane curves of degree 𝑞+2 with the desired property. Let 𝑓 be such

∗The research was supported by Italian MIUR PRIN Strutture Geometriche, Combinato-ria e loro applicazioni.

4 V. Abatangelo and G. Korchmáros

an irreducible plane curve defined over 𝐺𝐹 (𝑞), and write 𝜑𝑖𝑗 = 𝑥𝑞𝑖𝑥𝑗 − 𝑥𝑞

𝑗𝑥𝑖

for 0 ≤ 𝑖, 𝑗 ≤ 2. Then the equation of 𝑓 in a suitable homogeneous coordinatesystem (𝑥0, 𝑥1, 𝑥2) is one of the following type:

(i) for every 𝑞, 𝐾(𝑥0 + 𝑥1)𝜑01 + 𝑥0𝜑20 + 𝑥2𝜑12 = 0, where 𝑋3 −𝐾𝑋 −𝐾 isan irreducible polynomial over 𝐺𝐹 (𝑞);

(ii) for every 𝑞 ≡ 1 (mod 3), 𝐾𝑥1𝜑01 + 𝑥0𝜑20 + 𝑥2𝜑12 = 0 where 𝑋3 −𝐾 isan irreducible polynomial over 𝐺𝐹 (𝑞);

(iii) for 𝑝 = 3, (𝑥1 +𝐾𝑥2)𝜑01 − 𝑥0𝜑20 − 𝑥2𝜑12 = 0, where 𝑋3 +𝐾𝑋 + 1 is anirreducible polynomial over 𝐺𝐹 (𝑞);

If we allow plane curves defined over an algebraic extension of 𝐺𝐹 (𝑞), thereexist irreducible plane curves of degree 𝑞 + 1 containing all points of 𝑃𝐺(2, 𝑞).This will be shown in Section 3.

The above results of G. Tallini rise the problem of determining the planecurves of minimum degree containing all points of a given subset Δ in 𝑃𝐺(2, 𝑞).This problem has been recently investigated in the case where Δ consists of allinternal points to an irreducible conic in 𝑃𝐺(2, 𝑞), see [2] and the survey paper[9]. For this case, the minimum degree is equal to 𝑞−1, and those curves whichattain the minimum form a linear system of dimension 𝑞 − 1. Interestingly,this result is an essential technical tool for the study of combinatorial charac-terizations of point-sets blocking every external lines to a conic 𝒞. Analogousresults hold when Δ consists of all external points (or all internal points plusthe points of 𝒞), see [3, 1, 4, 7].

2. G. Tallini’s paper entitled “Irreducible hypersurfaces of minimal orderinvading a Galois space”

Let 𝑃𝐺(𝑟, 𝑞) be a Galois space, that is, an 𝑟-dimensional projective spaceover a finite field 𝛾𝑞 of order 𝑞 (where 𝑞 = 𝑝ℎ and 𝑝 is a prime number which isthe characteristic of the field and ℎ is a positive integer). 𝑃𝐺(𝑟, 𝑞) is thought tobe embedded in the 𝑟-dimensional projective space defined over the algebraicclosure Γ of 𝛾𝑞 (see [10]). However, the term of points (or geometric objects) of

Irreducible Hypersurfaces of Minimal Degree Containing All Points of . . . 5

𝑃𝐺(𝑟, 𝑞) without any specification indicates those with coordinates (or coeffi-cients) in 𝛾𝑞, whereas we will explicitly say when we refer to points (or objects)with coordinates (or coefficients) not in 𝛾𝑞 but in Γ.

By definition, an algebraic hypersurface (that is, an algebraic primal) in𝑃𝐺(𝑟, 𝑞) is the set consisting of points whose coordinates are in Γ, that is, in𝛾𝑞 or in any of its algebraic extensions, satisfy a form in 𝑟 + 1 variables withcoefficients in 𝛾𝑞 . There are several interesting problems related to algebraichypersurfaces, for instance the calculation of the number of their points in 𝛾𝑞.Another question (related with the classification of distributive quasifields, see[6]) concerns hypersurfaces without points in 𝛾𝑞, and the characterization ofirreducible hypersurfaces of minimum degree.

A problem, which is in some sense complementary to the previous one,consists in investigating hypersurfaces of 𝑃𝐺(𝑟, 𝑞) with the maximum numberof points; especially those containing all points of 𝑃𝐺(𝑟, 𝑞), with particularregards to the irreducible ones. The hypothesis that a hypersurface contains allpoints of 𝑃𝐺(𝑟, 𝑞) implies interesting, specific, geometrical properties that areworth investigating. We are concerned with this for irreducible hypersurfaces ofminimum degree. Here we summarize our results together with some outlinesof proofs. The reader is referred to the forthcoming paper [14] for all details.

1. Let 𝐹𝑛 be a hypersurface of degree 𝑛, defined over Γ and containingall points of 𝑃𝐺(𝑟, 𝑞). Then it must be 𝑛 ≥ 𝑞 + 1; in fact this property isclearly true in the case 𝑟 = 1. Then we can go on inductively with respect to𝑟. Two cases are possible indeed, according as 𝐹𝑛 contains or does not, as itscomponents, each of the 𝑞𝑟 + 𝑞𝑟−1 + . . . + 𝑞 + 1 hyperplanes of 𝑃𝐺(𝑟, 𝑞). Inthe former case, clearly 𝑛 ≥ 𝑞 + 1. In the latter case, there exists a hyperplane𝑃𝐺(𝑟−1, 𝑞) which is not a component of 𝐹𝑛. Then 𝐹 ′𝑛 = 𝑃𝐺(𝑟−1, 𝑞)∩𝐹𝑛 isa hypersurface of 𝑃𝐺(𝑟 − 1, 𝑞) containing every point of 𝑃𝐺(𝑟 − 1, 𝑞), whence𝑛 ≥ 𝑞 + 1 follows, again by the inductive hypothesis.

By a suitable application of a result due to B. Segre (see [11, Lemma 1,p.272]), we are able to prove that the most general 𝐹𝑛 containing every pointof 𝑃𝐺(𝑟, 𝑞) has equation in homogeneous coordinates (𝑥0, 𝑥1, . . . , 𝑥𝑟) of thefollowing type

6 V. Abatangelo and G. Korchmáros

(2.1) 𝐹𝑛 ≡0,𝑟𝑖<𝑗

𝐴𝑖𝑗 𝜑𝑖𝑗 = 0 ,

where 𝜑𝑖𝑗 = 𝑥𝑞𝑖𝑥𝑗 −𝑥𝑖𝑥

𝑞𝑗 , and 𝐴𝑖𝑗 are forms of degree 𝑛− 𝑞−1 depending only

on 𝑥𝑖, . . . , 𝑥𝑟.Now we address the question weather among the hypersurfaces of equation

(2.1) there exist anyone which is irreducible (over Γ). We want to determineand investigate those of minimum degree.

The case of the plane is different from the case of higher dimensional spaces.In (2.1), if 𝑟 = 2 and 𝑛 = 𝑞+1 (and therefore the 𝐴𝑖𝑗 ’s are constant), then it iseasily checked that (2.1) splits into the 𝑞+1 lines of the pencil with centre in thepoint (𝐴12,−𝐴02, 𝐴01). Therefore each curve 𝒞𝑞+1 defined over 𝛾𝑞 containingevery point of 𝑃𝐺(2, 𝑞) consists of the 𝑞 + 1 lines of a pencil. Now, let (2.1) ifwe put

𝑟 ≥ 3, 𝑛 = 𝑞 + 1, 𝐴2𝑖 2𝑖+1 = 1

for 𝑖 = 0, 1, . . . , [ 𝑟−12 ], and assume that the remaining 𝐴𝑖𝑗 vanish. We may

immediately observe that (2.1) represents a hypersurface with no singularity inΓ for 𝑟 odd, but a cone projecting from a point a hypersurface of the previouskind lying in a space 𝑃𝐺(𝑟 − 1, 𝑞) for 𝑟 even. In any case, when 𝑟 ≥ 3 thereexist 𝐹 𝑞+1 which are irreducible (over Γ) and contains every point of 𝑃𝐺(𝑟, 𝑞).

In the next subsection 2 we show that there exist curves 𝒞𝑞+2 which areirreducible (in Γ) and contain every point of 𝑃𝐺(𝑟, 𝑞). We provide a completeclassification from the projective point of view, based on some their specificproperties. Finally, in subsection 3, we outline certain remarkable properties ofthe hypersurfaces 𝐹 𝑞+1 which are irreducible (over Γ) and contain every pointof 𝑃𝐺(𝑟, 𝑞). Moreover, we determine the projectively inequivalent classes andcharacterize them from a geometrical point of view.

2. Suppose that there exists a curve 𝒞𝑞+2 defined over 𝛾𝑞 which is irre-ducible (in Γ) and contains each of the 𝑞2+𝑞+1 points of 𝑃𝐺(2, 𝑞). Obviously,each line of the plane must have intersection multiplicity two with the curveat a unique point (in 𝛾𝑞) of the curve. Hence, a priori, either the line is tan-gent to the curve at that point or the point is double for the curve. However,