Result. Math. 32 (1997) 221-233 0378-6218/97/040221-13 $ 1.50+0.20/0 C> Birkhauser Verlag, Basel, 1997

Speech

I Results in Mathematics

IN HONOUR OF GIUSEPPE TALLINI pronounced at the opening cerimony of the Conference "Combinatorics '96", Assisi 8-14 September 1996

Pier Vittorio Ceccherini

This Conference "Combinatorics '96" is in honour of Giuseppe Tallini, who died in Rome on 4th April 1995. As a matter of fact he has been the inventor and the main organizer of the conference series "Combinatorics", which take place in Italy every other year. Of course we do not celebrate him because of that (or not only because of that), but because he has been an excellent mathematician. The list of his publications contains more than 160 items in algebra, differential geometry, algebraic geometry over finite fields, foundations of geometry, and in several branches of combinatorial theory. Together with Adriano Barlotti he has made the major contributions to the development of the Italian school of geometrical combinatorics founded by Beniamino Segre. Without any doubt, he was the inheritor of Segre's mantle at Rome. He was a member of the "Accademia di Scienze, Lettere ed Arti" of Naples, honorary member of the "Mathematische Gesellschaft in Hamburg", member of the "New York Academy of Sciences", member of the editorial board of "Journal of Geometry" as well of "Designs, Codes and Cryptography" and of several Italian journals. He was co-organizer of Oberwolfach conferences on "Grundlagen der Geometrie" (with Walter Benz), of Haifa conferences on "Geometry" (with Rafael Artzy and Joseph Zaks), of conferences on "Order in algebra and logic" (with Antonio Di Nola, Daniele Mundici and Angus Macintyre). He founded in Rome the famous "Seminario di Geometrie Combinatorie", and (with Gabor Korchmaros and Mario Marchi) the peripatetic "Scuola Estiva di Geometria Combinatoria". Both the "Seminario" and the "Scuola Estiva" are now named after him. He was very active in the Scientific Committees of the main Italian mathematical institutions: the "Unione Matematica Italiana", the "Istituto Nazionale di Alta Matematica" and the "Gruppo Nazionale Strutture Algebriche e Geometriche e loro Applicazioni", a branch of the "Consiglio Nazionale delle Ricerche"

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Giuseppe Tallini was born on 5th January 1930 in Fonnia, a seaside town, located between Rome and Naples. In Fonnia he attended primary and secondary school, and then in 1950 he moved to Rome to study mathematics at that University. There he studied with many distinguished mathematicians: especially Beniamino Segre and Enrico Bompiani, as well as Fabio Conforto, Mauro Picone, Giulio Krall, Enzo Martinelli and Lucio Lombardo-Radice. In Rome he also attended advanced seminars and courses held at the Istituto Nazionale di Alta Matematica founded in 1938 by Francesco Severi. He was specially attracted by the scientific personalities of Bompiani and Segre. From the former he learned the love of differential geometry; from the latter, algebra, projective and algebraic geometry, geometry over finite fields and combinatorics. He began his academic career at the University of Rome "La Sapienza" in 1954. He then moved, as full professor, to the University of Turin in 1966 and to the University of Naples in 1968. He returned to Rome as professor of geometry in 1974, a position he occupied till his death. In 1958 he had married Maria Scafati, a fellow student who has been a faithful companion both in his mathematical career and in all their life together. She is professor of algebra at the University of Rome "La Sapienza" and an internationally well-known mathematician herself. Giuseppe was blessed with a happy family life with Maria and their three sons, Giovanni (1960), a physician, Marco (1963), a geologist, and Luca (1967), a mathematician and computer scientist I will now sketch only some aspects of the scientific work of Giuseppe Tallini, following the lines of the commemorations I wrote for the Journal of Geometry and for Designs, Codes and

Cryptography (and I take this opportunity to thank James Hirschfeld who helped me with my poor English). You can find a more datailed analysis in the booklet (given to the participants at this Conference) "Giuseppe Tallini: his work and our reminiscences". An almost complete overview of his work will appear in a forthcoming commemoration in Bull. Un. Mat. Ital. Giuseppe Tallini had been awarded the degree in mathematics cum laude in 1954, with a thesis (under the supervision of Bompiani) on Kiihlerian varieties. This was the first of a series of papers in differential geometry, mostly concerning the theory of affine and projective connections. His range of publications included several books as well as volumes of lecture notes in geometry and topology, in algebra and in category theory. In algebra he studied archimedean rings, inversive semigroups, and non-standard Galois fields. Tallini made a special contribution to the foundations of geometry, developing several topics in this field: a generalized Desargues theorem for r-dimensional projective spaces, r>2; Lobachevski geometry in Galois spaces; topology associated with a projective space; connections between graph theory and geometric structures; incidence geometries, matroids and dimension theory; generalized

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quadrangles; ruled graphic systems; polar spaces; Lorentz transformations in a Galois plane; perspective sets; combinatorial problems in infinite Galois geometries and in inftnite linear spaces. These works are important enough to ensure him an eminent position among contemporary geometers. But Tallini's fame is specially due to his many papers in the theory of Galois geometries, a field where he became a major international figure. The leitmotiv of these papers is a special attention to the fascinating problem of characterizing algebraic varieties of the r-dimensional projective space over a Galois field, both from the 'embedded' and from the 'structural' point of view. The standard case to illustrate the embedded point of view is the following. Given a set of points in the space, one tries to rmding 'incidence properties' that are necessary and sufficient to ensure that it is an algebraic variety of a certain type. 'Incidence properties' of a set of points are those which describe the behaviour of the set with respect to subspaces of given dimensions, and in particular give some conditions for the cardinality of the set Such problems arose naturally after Beniamino Segre had shown in 1954 that the (q+ I)-arcs of the Galois plane of odd order q are precisely the irreducible conics of the plane. In the same class of ideas, in 1955, Adriano Barlotti and independently Gianfranco Panella had given a characterization of elliptic quadrics of the 3-dimensional Galois space of odd order q, as (q2+ I)-caps of the space. Actually the word 'cap' is derived from differential geometry and was originally introduced into finite geometry by Tallini in 1956. In 1955, during the 5th Conference of the Italian Mathematical Union, Tallini announced the characterization of quadrics of a Galois space of any dimension and any odd order q. He published in extenso this characterization, including also the case q even, in a series of papers dated 1956-57. In order to be a quadric, a given set of points must satisfy the condition of containing every line having more then two points in common with the set; a set with this property is now usually called a 'Tallini set'. Related to the case of elliptic quadrics, see also the article with his pupil Osvaldo Ferri, where a characterization is given for the set of lines which are secant to an elliptic quadric of the 3-dimensional Galois space of odd order q. Tallini obtained in 1957-59 a characterization of the Veronese surface of the 5-dimenional Galois space of odd order q, and of certain cubic surfaces of the 3-dimensional Galois space of odd order

q>3. In 1973, in a survey paper on characterizations of classical algebraic varieties of a Galois spaces of order q - including nonnal rational curves, and Hennitian varieties (which had been characterized in 1966 by Maria Scafati Tallini and, for the case q=4, by Adriano Barlotti) - he revisited his results on the Veronese surface in tenns of properties of the Grassmann variety of planes of an r-dimensional Galois space.

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This was the origin of a systematic investigation of the Grassmann variety representing the ddimensional subspaces of a projective space over a Galois field or more generally over any field, and, even more generally, of the 'Grassmann space' of the d-subspaces of any projective space, possibly reducible. Bruce Cooperstein had found (1977) a characterization of the Grasmmann space of the lines (d=l) in the finite case. Tallini was successeful in obtaining that characterization in the general case (1980-81), and then a characterization of the Grassmann space of the planes (d=2) (1981, with A. Bichara), and finally a characterization of the Grassmann space of the d-dimensional subspaces for all d;2:1 (1983, with A. Bichara). A further step are the characterizations of topological Grassmann spaces (1988, with J. Misfeld and C. Zanella) and of ordered Grassmann spaces (1991, with A. Bichara, J. Misfeld, and C. Zanella). These characterizations of the Grassmann spaces, in particular of the Grassmann varieties, are obtained from the beautiful 'structural point of view'. It can be summarized as follows. A Grassmann algebraic variety, regarded as the set of its points together with the set of its lines, is a sernilinear space satisfying certain combinatorial properties; more generally, any Grassmann space can be regarded as a sernilinear space where the points are the d-dimensional subspaces of the given space and the lines are the pencils of such d-subspaces. The characterization surprisingly says that any abstract semilinear space satisfying those properties is obtained from a Grassmann space. Further characterizations of Grassmann and of other classical varieties (Schubert, Veronese and Segre varieties) were obtained by his pupils A. Bichara, F. Mazzocca, C. Somma, P.M. Lo Re, D. Olanda, N. Melone, C. Zanella. In order to obtain a unified approach for the study of a wide class of algebraic varieties of a projective space over a field, Tallini developed in 1991 the general theory of what he called an (n)variety of the space, that is a set of points such that any line of the space is either external, tangent, or n-secant to the set, or lies on it. With this terminology a 'Tallini set' (as previously defined) is a (2)-variety and conversely. For example, quadrics, Grassmann varieties, Segre varieties and Schubert varieties are (2)-varieties, while any Hermitian variety of a Galois space of order q2 is a

(q+ I)-variety. Algebraic varieties are only a part of the work of Tallini in the geometry and the combinatorics of a fmite space. He obtained interesting results in several other directions, also related with the study of algebraic varieties of a Galois space. He investigated the theory of some key subsets of a Steiner system, in particular of a projective plane and of a Galois space, such as arcs, caps and subsets having prescribed characters. There were also papers with applications to statistics and to coding theory; on the theory of spreads; on blocking sets; on combinatorial designs; on linear and

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semilinear spaces; on planar spaces; on error-correcting codes; on authentication systems; on the algebra and geometry of hyperstructures. Galois geometry occurs in almost all Tallini's papers on these topics; projective Galois geometry is also used - in a joint paper with Albrecht Beutelspacher and Corrado Zanella - to construct two classes of essentially s-fold secure authentication systems with large s . Tallini studied all those topics using their close connections with Galois geometries, such as the one between caps and codes. A cap of kind s of a given Galois space is a set of points having s+2 as the minimum number of dependent points. For the applications to statistics and to coding theory, it is important to know - for given values of the parameters s (the kind of the cap), r (the dimension of the space spanned by its points) and q (the order of the Galois field) - the maximum value of the number k of points of such a cap. To any cap with those parameters is associated a code over the Galois field of order q, having length k, dimension k-r-l and distance s+2, and conversely. Take, as parity check matrix of the code, the matrix whose columns are

coordinate vectors of the points of the cap. The code is perfect if and only if the cap is complete, that is, if it is not contained in any larger cap with the same parameters. This connection makes possible the application of coding theory to the theory of caps, and conversely; for instance, this leads on the one side to finding an upper bound for the number of points of a cap and on the other to finding a geometrical proof that the only 2-error-correcting linear codes with minimum distance 5 are the binary repetition code and the ternary Golay code. The notion of cap is also used in a much more general context Let consider any incidence structure with v points, b blocks and incidence matrix M. Define d+ I to be the rank of Mover GF(2) and s+2 to be the minimum number of columns which are linearly dependent, and suppose that d+ I
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take as new blocks any set of blocks of the given structure of even (odd) type, i.e. such that in the

original structure through each point the number of blocks of the considered set is even (odd). He developed a general theory of the relationship between linear spaces (in particular Steiner systems) and combinatorial geometries (in the sense of Gian-Carlo. Rota); and made an important contribution to design theory with his studies on 'manifolds' of Steiner systems and on 'composition' of designs. Roughly speaking, composition of designs occurs when, in a given design, a design structure can also be defined in every block. This provides a method of contruction of several designs derived from a Galois geometry. A manifold of linear spaces is a linear space whose maximal subspaces are partitioned into certain classes such that the spaces of each class have the same size (depending on the class) and the number of members of each class through any point is constant (depending on the class, but being independent of the point). In a Galois space, nonsingular quadrics, non-singular Hermitian varieties, Grassmann varieties and Segre varieties are examples of manifolds of Steiner systems. Other examples are partial geometries and generalized quadrangles. Tallini states general properties of such structures, namely concerning non-collinear sets, intersection sets, blocking sets and spreads of lines. Tallini wrote many books and volumes of lecture notes in combinatorial geometry. These served not only as an exposition of the basic material, but also contain numerous research problems. In this way he sought to encourage young mathematicians to work in this field. With this aim in mind, he also created the series "Quaderni del Seminario di Geometrie Combinatorie". The 125 numbers so far published contain contributions from many of the major figures working in the area. We shall remember Giuseppe Tallini full of energy and enthusiasm, with an enormous vitality and a dominating personality. Even during the last period of his illness he continued to work very strongly. He had always the absolute necessity of communicating his ideas to other people. This he always did with a vigour that reflected his passion for geometry and mathematics. He devoted a great deal of time to his pupils, imbuing in them his love of geometry and combinatorics, and stimulating them with a continual flow of ideas. Many of them now occupy Chairs of Geometry, and continue to develop the Italian School of geometry and combinatorics, following his precepts. I think that his scientific work will continue to live in the work and in the love of his many pupils and friends. Pier Vittorio Ceccherini Dipartimento di Matematica "Guido Castelnuovo", Universita di Roma "La Sapienza" P.le AIdo Mom 2, 00185 Roma, Italy, E-mail: [email protected] Eingegangen am 9. August 1997

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PUBLICATIONS OF GIUSEPPE TALLINI [1J Sopra un teorema di A.Lichnerowicz suIla geometria kilhleriana, Rend. Acead. Naz. Lincei (8) 17 (1954). 204-209. [2J Sui sistemi a doppia composizione ordinati archimedei. Rend. Acead. Naz. Lincei (8) 18 (1955). 367-373. [3J Caratterizzazione deIle quadriche negli spazi lineari fmiti di dimensione qualunque e di online dispari.

Alii V Congresso UMl Pavia-Torino 1955. Cremonese. Roma (1956), 337-338.

[4} Su una estensione del teorema di Desargues. Boll. Un. Mat. llal. (3) 11 (1956). 46-48. [5} Sulle k-calotte degli spazi Iineari finiti. Note I. II, Rend. Aeead. Naz. Lincei (8) 20 (1956). 311-317. 442-446. [6] SuIle k-calotte di uno spazio Iineare fmito. Ann. Mat. Pura Appl. (4) 42 (1956). 119-164. [7] Caratterizzazione grafica deIle quadriche eIlittiche negli spazi finiti. Rend. Mat. Appl. (5) 16 (1957). 328-351. [8] Una proprietA grafica caratteristica deIla superftcie di Veronese negli spazi fmiti.

Convegno Internazionaie Relicoli e Geomelrie Proiellive. Palermo -Messina 1957. Cremonese. Roma (1958). 136-139. [9] Sui q-archi di un piano lineare fmito di caratteristica p=2. Rend. Aeead. Naz. Lincei (8) 13 (1957). 242-245. [10] Una proprieta grafica caratteristica deIla superficie di Veronese negli spazi finiti, Note I. II. Rend. Acead. Naz. Lincei (8) 24 (1958). 19-23. 135-138. [l1J Caratterizzazione grafica deIle quadriche ellittiche in uno spazio finito. La Rieerea Scientifiea (28) 4 (1958). 820-823. [12} Caratterizzazione grafica di certe superficie cubiche di S3.q. Note I, II. Rend. Aecad. Naz. Lincei (8) 26 (1959). 484-489. 644-648. [13] Le geometrie di Galois e Ie lora applicazioni alIa statistica e aIla teoria dell'informazione.

Rend. Mat. Appl. (5) 19 (1960). 379-400.

[14J On caps of kind s in a Galois r-dimensional space. Aela Arilh. 7 (1961), 19-28. [15J Le ipersuperficie irriducibili d'ordine minima che invadono uno spazio di Galois,

Rend. Acead. Naz. Lincei (8) 30 (1961). 706-712.

[16] Sulle ipersuperficie irriducibili d'ordine minimo che contengono tutti i punti di uno spazio di Galois Sr,q,

Rend. Mar. Appl. (5) 20 (1961). 431-479.

[17] Una proprietA in grande delle varietA a connessione affine compatte con applicazioni aile varietA a connessione proiettiva, Rend. Acead. Nm. Lincei (8) 32 (1962), 644-648. [18] SuIle connessioni proiettivamente equivalenti di una Vn compatta, Rend. Aeead. Naz. Lincei (8) 33 (1962). 244-252. [19] Intomo aIle forme di uno spazio di Galois ed agli spazi subordinati giacenti su esse.

Rend. Acead. Naz. Lincei (8) 33 (1962). 421-428.

[20] Un'applicazione delle geometrie di Galois a questioni di statistica, Rend. Acead. Naz. Lincei (8) 3S (1963). 479-485. [21] Lezioni di geomelria dijJerenziale: Variela riemonniane eompatle, 1st. Mat. "G. Castelnuovo". Univ. Roma, a.a. 196263.

Ill] Sulle connessioni di Weyllocalmente metriche. Rend. Acead. Naz. Lincei (8) 36 (1964). 1-7. [23J Calotte complete di S4,q contenenti due quadriche ellittiche quali sezioni iperpiane, Rend. Mar. Appl. (5) 23 (1964), 108-123. [24] Ujabb eredmenyek a Galois-Geometri~ban. A Magyar Tudomanyos Akademia III (14) 2 (1964). 183-192. [lSJ Connessioni dotate di metriche locali su una varieta differenziabile.

Rend. Sem. Mar. Univ. Polil. Torino lS (1965-66). 1-10.

[26] Sulla struttura algebrica delle trasformazioni tra parti di un insieme. Ann. Mar. Pura Appl. (4) 71 (1966). 295-322.

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[27] Local metrics with a global connection, Annales Univ. Sci. Budapestinensis, Sectio Math. 9 (1966), 23-26. [28J Lezioni di geometria differenziale, 1st. Mat. "G. Castelnuovo", Univ. Roma, 1966. [29] Appunti di algebra, 1st. Mat., Univ. L'Aquila, 1966-67. [30] Metriche locali dotate di una connessione globale su una variedl differenziabile, Periodico di Matemalica (4) 46 (1968), 340-358. [31] Lezioni di istituzioni di geometria superiore, 1st. Geometria, Univ. Torino, 1968. [32] Una solucian geometrica a ciertos problemas de estadfstica, Atti SimpPanamericano de MatemOtica Aplicada, 1968. [33] Lecciones de geometria superios: cohomologia de las formas sobre una variedad diferenciable, Parle I, II. Inst. Mat. Univ. Rosario, 1968. [34] Introduzionealla coomologia a coefficienti in un fascio, Con[. Sem. Mal. Univ. Bari 117 (1969), 1-24. [35] Geometria de Lobachewsky en un espacio finito Pn[GF(q)], (n=I,2; q impar), Revista Mat. Fis. Teor. Univ. Tucuman (A) 20 (1970), 203-232. (With S. Bruno). [36] Calegorie e funtori, 1st. Mat. Univ. Napoli, 1969. [37] Geometria iperbolica in un piano di Galois S2,q' con q dispari, Ricerche di Mal. 19 (1970), 48-78. (With S. Bruno). [38J Appunti sui fondamenli di geomelria affine e proiettiva, 1st. Mat. Univ. Napoli, 1970. [39] Ruled graphic systems, Alii del Convegno di Geomelria Combinaloria e sue Applicazioni (perugia, 11-17 settembre 1970), Oderisi, Gubbio (1971), 385-393. [40] Geometrie di Galois e loro applicazioni alia fisica, Sem. Lab. Naz. CNEN, Frascati (1970), LNF-70/63, 1-27. (With E. G. Beltrametti). [41] Slrullure geometriche: spazi lopologici e varieta differenziabili, Liguori. Napoli, 1970. [42] Cohomologia de Cech y cohomologia a coeficientes en un haz, Malhemalicae Nolae 22 (1970-71), 27-47. [43J Lezioni di geometria superiore: spazi proieltivi, 1st. Mat. Univ. Napoli, 1970-71. [44J Topologia associata ad uno spazio grafico, Ricerche di Malematica 20 (1971), 253-259. [45] Sistemi grafici rigati, lSI. Mat. Univ. Napoli, Relaz. 8 (1971), 1-47. [46J Strutture d'incidenza dotate di polaritA, Rend. Sem. Mat. Fis. Milano 41 (1971), 1-42. [47] Lezioni di geometria superiore: coomologia a coefficienti in un fascio, 1st. Mat. Univ. Napoli, 1971-72. [48] Slrutture grafiche proieltive, Liguori, Napoli, 1973. [49] Variela differenziabili e coomo/ogia di De Rham, Cremonese, Roma, 1973. [50] Probemi e risultati sulle geometrie di Galois, lSI. Mal. Univ. Napoli, Relaz. 30 (1971), 1-30 [51] Una dimostrazione del teorema di De Rham, Con[. Sem. Mal. Univ. Bari 139 (1975), 1-16. [52] I k-insiemi di classe [O,I,n,q+l] regolari di Sr,q' Quad. Gruppi Ric. Mat. del CNR, Alii Convegno del GNSAGA, (Modena, 10-11 gennaio 1975), Firenze, 1976, 101-110. [53J Lezioni sulla teoria dei gruppi di Lie, Sem. Lab. Naz. CNEN, Frascati (1975), LNF-75/16(L), 1-109. (With C. Mencuccini and A. Reale). [54] Graphic characterization of algebraic varieties in a Galois space, Teorie Combinatorie, Vol. II, Atti dei Convegni Lincei 17 (Rom a, 3-15 settembre 1973), 1976, 153-165. [55] Questioni di geometria combinatoria, Sem. 1st. Mat. Univ. L'Aquila , 1977, 1-16.

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[56] Spazi di rette e geometrie combinatorie, Quad. Sem. Geom. Comb. lSI. Mat. "G. Caslelnuovo" Univ. Romo 3 (1977). [57] Beniamino Segre (1903-1917), Archimede (1977), 143-145. [58] Lezioni di Geomelria III: Spazi dei cerchi e delle stereo Lo spazio delle coniche e la superficie di Veronese. Teorie delle coniche in un piano di Galois, 1st. Mat. "G. Castelnuovo", Univ. Roma, 1917-78.

[59] Lezioni di Geomelria III: Archi, ovali, insiemi quadralici, calolle negli spazi finili, 1st. Mat. "G. Castelnuovo", Univ. Roma, 1977-78. [60] Grafi e spazi geometrici, Sem. lSI. Mat. Univ. L'Aquila, 1978, 1-18. [61] lpergruppoidi di Steiner e geometrie combinatorie, Alii Convegno "Sislemi binari e lora applicazioni" (Taormina 1978), 119-125. [62] Spazi parziali di rette, spazi polari, geometrie subimmerse in spazi proiettivi, Appunti di A. Venezia. Quad. Sem. Geom. Comb. 1st. Mat. "G. Caslelnuovo" Univ. Romo 14 (1979), 1-58. [63] Introduzione alia leoria dei codici, Appunti redatti da O. Ferri, Sem. lSI. Mat. Univ. L'Aquila , 1979. 1-11. [64] Spazi combinatori e sistemi di Steiner, Riv. Mal. Univ. Parma (4) 5 (1979), 221-248. [65] La categoria degli spazi di rette, Sem. lSI. Mal. Univ. L'Aquila, 1980, 1-25. [66] Codici e geometrie combinatorie, Appunti redatti da A. Di Concilio e P.M. Lo Re, Quad. Sem. Geom. Comb. lSI. Mat. "G. Caslelnuovo" Univ. Romo 23 (1980), 1-16. [67] I k-insiemi di rette di PG(d,q) stodiati rispetto ai fasci di rette, Parte I: Appunti redatti da P.M. Lo Re; Parte il: Appunti redatti da A. Venezia, Quad. Sem. Geom. Comb. lSI. Mat. "G. Castelnuovo" Univ. Roma 28 (1980), 1-17, 1-16. [68] La geometria delle grassmanniane di uno spazio di Galois, Convegno "Slrulture combinatorie e lora applicazioni", CIRM, Trento 20-25 ottobre 1980, 1-33. [69] On a characterization of the Grassmann manifold representing the lines in a projective space, in PJ. Cameron et aI. (eds), Finite geometries and designs, Proc. of the 2nd Isle of Thorns Conference, Cambridge University Press, 1981, 354358. (London Math. Soc., LNS 49). [70] Codes, caps and linear spaces, in PJ. Cameron et aI. (eds.) Finile geomelries and designs, Proc. of the 2nd Isle of Thorns Conference, Cambridge University Press, 1981, 72-80. (London Math. Soc., LNS 49). (With P.V. Ceccherini). [711 Commemorazione di Beniamino Segre, Rend. Mal. Appl. (7) 1 (1981). 1-29. [72] Su di una caratterizzazione della varietA di Grassmann rappresentativa dei piani di uno spazio proiettivo, Quad. Sem. Geom. Comb. 1st. Mal. "G. Castelnuovo" Univ. Romo 34 (1981). 1-24. (With A. Bichara). [73] On a theorem by W. Benz characterizing plane Lorentz transformations in laemefelt's world, 1. Geometry 17 (1981), 171-173. [74] Su una caratterizzazione della grassmanniana delle rette di uno spazio proiettivo, Rend. Sem. Mal. Brescia 6 (1981), 82-86. [75] Geometrie d'incidenza e matroidi, Quad. lAC del CNR (3) 127 (1981), 1-34. [76] Fibrazioni in rette di PG(r ,q), Quad. Sem. Geom. Comb. lSI. Mat. "G. Caslelnuovo" Univ. Romo 37 (1981), 1-17. [77] Fibrazioni in rette di PG(r,q), Appunti curati da O. Ferri, Sem. lSI. Mat. Univ. L'Aquila , 1981,1-16. [78] La superficie di Veronese: aspetti geometrici e combinatorici, Appunti redatti da A. Bichara. lSI. Mal. Appl. Fac. Ing. Univ. L'Aquila , 1982, 1-20. [79] The geometry on Grassmann manifolds representing subspaces in a Galois space, Ann. Discrele Math. 14 (1982), 9-38. [SO] On a characterization of the Grassmann manifolds representing the planes in a projective space, Ann. Discrele Math. 14 (1982), 129-150. (With A. Bichara).

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[81] Caps related to incidence structures and to linear codes.

Ann. Discrete Math. 14 (1982). 175-182. (With P.V. Ceccherini). [82] On line k-sets of type (0. n) with respect to lines in PG(d,q). Ann. Discrete Math. 14 (1982), 283-292. [83] Fibrazioni mediante rette in PG(r,q). Le Matemotiche (I) 37 (1982), 8-27. [84] Campi di Galois non standard. Redazione di P.M. 10 Re

1st. Mat. e Mecc. Raz. Fac.lng. Univ. Napoli, Serie rapporti interni 44 (1982), 1-22. [85] Problemi e risultati in geometria combinatoria. A cura di L. Berardi, F. Eugeni. O. Ferri.

1st. Mat. Appl. Fac. Ing. Univ. L'Aquila • 1982. 1-19

[86] Spazi di rette finiti e k-insiemi di PG(r.q). Quad. Sem. Geom. Comb. 1st. Mat. "G. Castelnuovo" Univ. Romo 42 (1982). 1-24. [87] k-insiemi e blocking sets in PG(r.q) e in AG(r,q). Quad. Sem. Geom. Comb. 1st. Mat. Appl. Fac.lng. Univ. L'Aquila 1 (1982). 1-36. [88] On the non existence of blocking-sets in PG(n,q) and AG(n,q) for all large enough n. Pubbl.lst. Mat. Univ. Napoli (3) 31 (1982-83), 1-8. (With F. Mazzocca). [89] Beniamino Segre. Ann. Discrete Math. 18 (1983), 5-12. [90] On a characterization of Grassmann space representing the h-dimensional subspaces in a projective space. Ann.

Discrete Math. 18 (1983). 113-131. (With A. Bichara).

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