International conference on geometry

Journal of Geometry Vol. 39 (1990) 0047-2468/90/020001-2751.50+0.20/0 (c) 1990 Birkh~user Verlag, Basel INTERNATIONAL CONFERENCE ON GEOMETRY From September 30 till October 6, 1990 an international conference on geometry was held at SchloB Rauischholzhausen (Justus-Liebig-Universitfit GieBen). The scientific or- ganizers were Adriano Barlott i (Firenze), Walter Benz (Hamburg) and Albrecht Beute lspacher (GieBen). There was a very fruitful interplay between various discipli- nes of geometry, such as foundations of geometry, topological geometries, finite geome- tries, applications and didactical aspects. Rafael Artzy (University of Haifa, Mount Carmel, Haifa 31 999, Israel) COLL INEAT ION GROUPS OF BENZ PLANES OVER F IN ITE PRME F IELDS Generators are found for the collineation groups F of Benz planes over finite fields of odd prime order and for the collineation groups r of their derived planes. In order to arrive at the group of a derived plane, one has first to study a stabil ity subgroup h of F. This A is then a subgroup of ~P. Relations between the generators of the groups F, A, and r are being determined in each of the three types of Benz planes. Walter Benz (Mathematisches Seminar der Universit~it Hamburg, BundesstraBe 55, W-2000 Hamburg) A GEOMETRIC VERSION OF THE BAXTER EQUATION Suppose that R is a ring which does not need to be commutative or associative and which does not need to contain an identity element 1. We nevertheless call subsets g(k) : = {(t, t + k) ] t ~ R} of R2 : = R• for k E R "lines of slope 1" of R2. The product of elements x = (Xl, x2), Y = (Yl, Y2) of R2 is defined by xy : = (XlYl, x2Y2). We then call S c R2 admissab le iff (i) S is closed under multiplication, 2 International Conference of Geometry (ii) every line of slope 1 has exactly one point in common with S. We show that every admissable set S c R2 leads to a solution f: R -~ R of the Baxter equation V x, y E R: f(xf(y) + f(x)y - xy) = f(x)f(y) by putting f(a-b) = a r (a, b) E S. and that furthermore every solution of the equation is induced by a suitable admissable set S of R2. This geometric version of Baxter's equation allows to construct 2~ dis- continuous solutions in case R : = R. Literature P. Volkmann, H. Weigel: Aequat. Math. 27 (1984). W. Benz, Abhandlungen Math. Sere. Hamburg (1987). Dieter Betten (Mathematisches Seminar der Universitat, Ludewig-Meyn-Str. 4, W-2300 Kiel) ORBITS IN 4 -D IMENSIONAL COMPACT PROJECT IVE PLANES Let (P,L) be a 4-dimensional compact projective plane and let E be the group of all its continous collineations. Since the fundamental papers of Salzmann MZ 1970, 71 who proved among other things that E ls a Lie group of dimension -< 16, much work has been done to classify these planes. Many translation planes and so-called shift planes have been found. As proved by Betten and L0wen (Geom. Ded. to appear) all planes with dim E _-> 7 are now determined. I f one is interested in flexible planes where E has an open orbit in the space offlag~, then only the case dim E = 6 is left. Assuming dim E = 6 then by L6wen (Geom. Ded. to appear) either E = R2>~ G12+(R) or E is solvable. Therefore, with the exception of one single group, we may suppose that E is solvable. We will prove that in this case the connected component E 1 fixes some flag (v,W), v E P, W E L, v E W. Using the action of E l on the pencil, Lv\{W} and the action of E 1 on W~{v}, we derive six orbit cases for the collineation group E 1. In two of these cases the possible planes are already known. Andrea B lunck (Fachbereich Mathematik der TH Darmstadt, SchloBgartenstr. 7, W-6100 Darmstadt) CONJUGACY IN LOCAL ALTERNATIVE R INGS An alternative ring with identity element 1 is called local, if the set of its non-units is an ideal. We introduce a class of local alternative rings, and show that the conjugacy re- International Conference of Geometry 3 lation "---" (a - b r 3 unit c: a = c-lbc) is transitive on the alternative field A and on A(e ) = A +Ae (with a2 _- 0, the "dual numbers" over A), but - - in the non-associative case - - on no other alternative ring of our class. Arrigo Bonisol i (Istituto di Matematica, Universit~ della Basilicata, Via N. Sauro, 85, L85100 Potenza) F IN ITE MINKOWSKI PLANES: SOME PROBLEMS The classification problems for finite Minkowski planes of odd order has been looked at from various points of view. In particular H. Wilbrink has characterized the known examples as those which admit a group of automorphisms acting transit ively on ordered pairs of non-parallel points. All of the known examples have an automorphism group acting transitively on point- circle flags as well as on point-line flags, and also acting primitively on points. Do these properties possibly characterize the known examples? I f not, what can be said in general on the derived affine planes? Benedito Cast rucc i (Avenida Higien6polis, 402, ap. 151, Sao Paulo, Brazil) AN AX IOMATIC SYSTEM FOR MOBIUS-SPACES I. Let be a triple, where P is a non-empty set of elements named points (indicated by A, B, C .... ), C is a set of subsets of P , named circles (c i rcumferences) indicated by a, b, c .... and S is a set of subsets of P, named spheres (spherical surfaces), indicated by a, 13, y .... The system is a MSbius-space, if the following axioms are valid. I. For every three points A, B, C with A ~ B ~ C ~ A there exists a unique circle a such that A, B, C E a. II. For every four points A, B, C, D not on a common circle, there exists a unique sphere u such that A, B, C, D ( a. HI. For every two points A and B, for every circle a and for every sphere a with A, B a, a C_ a, A ( a and B ~ a, there exists a unique circle b such that A, B ( b C a and a A b = {A}. IV. For every two points A, B and for every sphere a, if A ( a and B ~ a, then there exists a unique sphere [3 such that A, B ( ~ and a N ~ = {a}. V. For every three points A, B, C, for every circle a and for every sphere a , if A ~ B ~= C =~ A and A, B, C ( a A a, then a is contained in a. VI. There exist four points A, B, C, D such that A, B, C, D ~ a for any circle a. 4 International Conference of Geometry VII. There exist five points A, B, C, D, E such that A, B, C, D, E ~ a for all a. VIII. For every circle a there exist three points A, B, C with A =~ B =~ C =P A such that A,B,C E a. IX. For every sphere a and for every circle a c_ a there exist a point A E a-a. 2. Consistency of the given axiomatic system in the Eucl idean Space. Be A the Euclidean space, C the set of its circles and straight lines a, b, c . . . . . S the set of its spheres and planes a, [], u .... Now, we call A an M-space, whose points are those of A plus one point (improper point), whose M-circles are the circles and the extended straight lines of A, and whose M-spheres are the spheres and extended planes of A. It is not difficult to verify the axioms I to IX. Judita Cofman (16, Cranford Lodge, 80, Victoria Drive, London, S.W.19 6HH, GB) THE THIRD DIMENSION - - PROBLEM SEMINARS FOR UNDERGRADUATES AND SIXTH-FORMERS In his lectures about great achievements in geometry Blaschke voiced the following opinion: "Ingenious ideas occur mostly to young research workers, however older col- leagues can be useful as midwives at their discoveries." Blaschke made this remark in connection with the 23-year old Cauchy's brilliant proof of the rigidity of convex polyhe- dra, encouraged by the ageing Lagrange. Cauchy's result, produced in 1812, belonged to three-dimensional euclidean geometry - - a subject, in those days extensively studied in schools and universities. Since then times changed, and so did syllabuses. One wonders: Would it be responsible from modern mathematical midwives to recommend their fledgelings the investigation of classical, three-dimensional geometrical problems? The aim of this talk is to outline ideas about Problem Seminars on the third dimension (instead of "old-fashioned" courses on stereometry) which could prove beneficial to un- dergraduates, and - - in reduced form - - to advanced sixth-formers. Such seminars could enable the students to improve their problem solving skills, to learn about famous topics, studied throughout the past centuries, to acquire better appreciation of objects in space, and to gain insight into a variety of interdisciplinary connection in mathematics. International Conference of Geometry 5 Paola De Vito (Dipartimento di Matematica, Via Mezzocannone, 8,1-80134 Napoli) ON SOME CLASSES OF FINITE LINEAR SPACES EMBEDDED IN A PAPPIAN PLANE (Joint work with P. M. Lo Re) In 1977 G. Tallini [7] posed the following question: Given a projective plane P, is every affine plane of order n > 3 embedded in P obtained from a projective subplane of P by deleting one of its lines? This question was motivated by the following observation. The affine plane of order 3 consisting of nine inflection points of a cubic curve is a counterexample. Korchm~ros [5] gave a positive answer to Tallini's question in the case that P is Pappian (finite or infinite). Using a result of Bruen [2] concerning finite linear spaces with n2 points, Bichara and Korchm~ros [1] extended Tallini's problem to more general configurations of points and lines. More precisely, ira finite linear space L = (S,L) with ISI = n2 and ILl = n2 + n is embedded in a Pappian plane P = PG(2,F) and S contains a non- degenerate quadrangle, then either n = 3, char P r 3 and L is the affine plane consisting of nine inflection points of a cubic curve or there exists in P a projective sub- plane nn of order n ~ 3 such that L is obtained from nn by deleting n + 1 points of which at least n are collinear. Starting from some results of ErdSs et al. [3], [4] and Metsch [6], we consider a similar embedding problem for finite linear spaces whose number v of points satisfies the inequality n2-n + 2 =< v ~ n2 + n + 1. References [1] A. Bichara, G. Korchm~ros, n2-sets in a projective plane which determine exactly n2+n lines, J. of Geometry 15 (1980), 175-181. [2] A. Bruen, The number of lines determined by n2 points, J. Combinat. Theory A 15 (1973), 225-241. [3] P. ErdSs, J.C. Fowler, V.T. SSs, R.M. Wilson, On 2-designs, J. Combinat. Theory A 38 (1985), 131-142. [4] P. ErdSs, R.C. Mullin, V.T. SSs, D.R. Stinson, Finite linear spaces and projective planes, Discrete Math. 47 (1983), 49-62. [5] G. Korchm~ros, On n2-sets of type (0,1,n) in projective planes, J. of Geometry, 15 (1980), 170-174. [6] K. Metsch, Linear spaces with few lines, Ph.D. thesis, University of Mainz 1989. [7] G. Tallini, Spazi di rette e Geometric Combinatorie, Sem. Geom. Comb. n. 3, Ist. Mat. Univ. Roma, (1977). 6 International Conference of Geometry Rolfdieter F rank (Mathematisches Seminar der Universitat, Bundesstr. 55, W-2000 Hamburg 13) ON SINGULAR PROJECT IVE TRANSFORMATIONS A pro ject ive t rans format ion is a map f between pappian projective spaces which is induced by a linear.map F between the corresponding K-vector spaces via f(Kx) := K(F(x)). It is called s ingular if Ker F =~ {0}; then f is not defined for the points of the s ingular subspace Ker f: = {Kx I x ~ Ker F-{0}}. Examples are the central projections; there the singular subspace is the center of projection. It is well known that any projectivity of an n-dimensional pappian projective space is uniquely determined by the images of n + 2 points (called reference points) in general position. We ask which sets of reference points determine a singular projective transformations. We have the following Theorem. Let f, g: P -~ P' be two singular projective transformations and dim Ker f = dim Ker g = : k-1. If C : = {A ( P I f(A) = g(A)} contains at least dim P + 2 + k points in general position, then C U Ker f U Ker g is a k-dimensional rational normal scroll. Remark. If K is algebraically closed, the theorem follows from Lemma 2.1 in [D. Eisenbud and J. Harris: On Varieties of Minimal Degree. Proceedings of Symposia in Pure Math. 46 (1987), 3-13]. Marcus Greferath (Fachbereich Mathematik der Universit~it, SaarstraBe 21, W-6500 Mainz) TWO STRUCTURE THEOREMS OF PROJECT IVE LATTICE GEOMETRIES We present an algebraic representation of hyperplanes in Desarguesian projective lattice geometries. Moreover we characterize semilinear maps between free unitary modules by means of the corresponding lattice maps. Harald Gropp (Mtihlingstr. 19, W-6900 Heidelberg) ON ORBITAL MATRICES Definit ion: An orbita l matr ix OM(a,b,c;s) is a square matrix integer entries aij (1 ~ i,j ---< a) with the following properties: 1. The sum of all entries in a row or column is equal to b. A with non-negative International Conference of Geometry 7 2. The sum of the squares of all entries in a row or column is equal to b + c. 3.The inner product of two different rows or columns is equal to s, where s -- (b2-b-c)/(a-1). The question is whether for given a, b, c there is an orbital matrix and if yes, how many non-isomorphic matrices there are. At first, some basic results about orbital matrices are obtained. Symmetric 2-designs turn out to be special orbital matrices, i.e. those with complexity 2. OM with higher complexity are related to many other combinatorial objects, like symmetric conference matrices, skew Hadamard matrices, weighing designs, semibiplanes and configura- tions. A few special eases are discussed and certain construction methods are shown using colourings of complete graphs. OM can be used to construct block designs with certain automorphisms. It seems to be better to have a general theory of OM instead of solving the problem "ad hoe" in every special ease. Moreover, OM are again combinatorial structures which generalize 2- designs (like configurations). The only research in this field has been done in the construction of mary designs by E. Billington and colleagues. Orbital matrices are symmetrical examples of n-ary designs. Theo GrundhSfer (Fakulttit for Mathematik, Universit~it T~ibingen, Aufder Morgen- stelle 10, W-7400 Ttibingen) TOPOLOGICAL PROJECTIVE PLANES A projective plane with topologies on the set of points and on the set of lines is a topological plane, if the geometric operations of joining points and intersecting lines are continuous. Every topological skew field coordinatizes a topological (desarguesian) plane. There exist many non-desarguesian examples, e.g. the Moulton planes or the plane over the Cayley numbers O (octonions). The talk will deal mainly with compact connected planes (in fact, every locally compact connected plane is compact). The following fundamental result of L0wen (1983) deter- mines the topological dimensions of these planes: The point space of every compact con- nected plane has topological dimension 2, 4, 8, 16 or o~ (no example with infinite dimen- sion is known). In the finite-dimensional case, the lines are homotopy equivalent to spheres of dimension 1, 2, 4 or 8. There is precisely one Moufang plane for each of the fi- nite dimensions, viz. the plane over R, C, H (Hamilton's quaternions), 0 respectively. Let P be a compact connected plane. Many classification results have been obtained under the assumption that the group E of all continuous collineations of P is large in 8 International Conference of Geometry some sense. E.g. if E is transitive on points, then P is one of the Moufang planes men- tioned above (LSwen 1981, Salzmann 1975). Since E is always a (locally compact) topo- logical group with respect to the compact-open topology, the size of E can also be measured by its topological dimension dim E. For the four Moufang planes, one has dim E = 8, 16, 35, 78. The following results have been obtained by Salzmann and his school (1962-90): Let P have topological dimension 2. If dim Z > 4, then P is the plane over R. If dim E = 4, then P is a Moulton plane. If dim E = 3, then P is known (three families of planes). Let P have topological dimension 4. If dim Z > 8, then P is the plane over C. If dim E > 7, then P is known (one family and two "sporadic" planes for dim E = 8). Let P have dimension 8. If dime > 18, then P is the plane over H. If dim E >-- 17, then P is known (Hughes planes, certain translation planes and their duals). Let P have dimension 16. If dim E > 40, then P is the plane over O. If dim E = 40, then P is known (one family of division ring planes). Dirk Hachenberger (Mathematisches Institut, Justus-Liebig-Universitat, Arndtstr. 2, W-6300 GieBen) RESULTS AND PROBLEMS CONCERNING TRANSLATION NETS A translat ion net of order s and degree r ~ 3 with translation group G is a pair (N,G) where N is an (s,r)-net (an affine Sr(1,s,s2)-design) which admits G as an auto- morphism group acting regularly on the set of points of N and fixing each parallel class of N. The existence of an (s,r)-translation net (N,G) is equivalent to the existence of r mutual ly disjoint subgroups of order s of G. Such a set of subgroups of G is called an (s,r)-partial congruence part i t ion (short: (s,r)-PCP). Given an integer s > 1 and a group G of order s2, then the main problem is to determine the number T(G) := max{r International Conference of Geometry 9 Christoph Hering (Fakultht for Mathematik, Universit~it Ttibingen, Auf der Morgen- stelle 10, W-7400 Ttibingen) SOME DIOPHANTINE EQUATIONS RELATED TO GEOMETRIC PROBLEMS We present an algorithm to determine all integral solutions of Thue-Mahler equations of degree 2, that is Diophantine equations of the form ax2 + bx + c = c0"ClYl'...'CrYr for given integers a, b, c, co ..... Cr. This algorithm allows a uniform treatment of various Diophantine equations important for finite geometry. Examples are the Ramanuyan- Nagell equations x2 + 7 = 2 y, and the equations x2-x+6 = 2Y and x2-x+6 = 3.2 y, which are used to classify the perfect 2-error and 3-error correcting codes over GF(2). Also the two Diophantine equations (2)_1= qn-1 q-1 and 2)- 1 = qn-1 where q > 1 and n > 3 of Peter Cameron (if q is not too large). Finally, we can determine all pairs (n, x) such that n E N, x E Z, n ->- 3 and no prime divisor of On(X) is > 2n+ 1 under the additional assumption that n = 2a3~ and a :~ 1, or n = 6, 16 or 54. Erwin HeB (Siemens AG, ZFE IS KOM 4, Otto-Hahn-Ring 6, W-8000 MOnchen 83) THE USE OF ELLIPTIC CURVES IN CRYPTOGRAPHY Starting with the surprising solution to the 'key-exchange-problem' of classical crypto- graphy given by Diffie and Hellman, we present the concept of public-key-cryptogra- phy: - The algorithms for encryption and for decryption of data work with different keys KEnc and KDec. - It must be computational infeasable to deduce KDec from KEnc. (This allows pub l i sh ing Kenc.) Important examples of such public-key-systems are based on mathematical problems that are generally thought to be very difficult: The RSA-system. (Here the determina- tion of KDec from KEnc involves factorization of large integers), and the EIGamal- I0 International Conference of Geometry system. (In this case finding KDec from KEnt affords the ability to determine "discrete logarithms" in a large finite field GF(q).) The latter system is of special attraction, because it formally carries over to arbitrary finite groups. We discuss some necessary conditions that must hold for a group G, in order to be used in an E1Gamal-like public-key cryptosystem. The most important of these criteria are the following: (1) G must possess a "large" cyclic subgroup < g> of prime order p > 1040. (2) For an arbitrary element g' in it must be computational impossible to find the integer n modp with g' = gn. (The discrete logarithm problem in must be "hard".) An elliptic curve E over GF(q) is a nonsingular curve in the desarguesian plane PG(2,q) described by an equation of type Y2Z + aXYZ +bYZ2 = X3 + cX2Z + dXZ2 + eZ3. (a,..,e E GF(q)) The set of points on E can be made to an abelian group in a canonical way, this group is either cyclic or the direct product of two cyclic subgroups. For given parameters a ..... e it is a nontrivial problem to determine the exact order of E; in general we have only the following bounds, due to Hasse: q+l -2~/q < ]E I < q+l + 2x/q So for a given arbitrary elliptic curve we have the practical (but not unsolvable) pro- blem to check whether condition (1) holds or not. Comparing the ease of practical implementation of a certain group G with the diffi- culty of the discrete logarithm problem in G it seems that elliptic curves are a good compromise for an E1Gamal-analogue public-key cryptosystem. J.W.P. Hirschfeld (The University of Sussex, School of Mathematical and Physical Sciences, Brighton BN 1 9QH, England) A CHARCTERIZAT ION OF HERMIT IAN CURVES (Joint work with L. Storme, J.A. Thas, J.F. Voloch) The second largest, complete k-arc in PG(2,q2), q even and q > 2, has k = q2_q+ 1. An example KO of such a (q2-q+l)-arc is given by an orbit of , m = q2+q+l , where T is a projectivity of order q4 + q2 + 1 acting as a single cycle on the points of the plane. It is expected that any complete (q2_q+ 1)-arc K in this plane is projectively equivalent to Ko. The tangents to K belong to an algebraic envelope F of class q+ 1 and it is natural to show first that F is unique. More generally, it is shown that, if C is an algebraic curve of degree q+l defined over GF(q2), q > 2, with no l inear com- International Conference of Geometry II ponent, then C is Hermitian; that is, in a suitable coordinate system it has equation xq+l + yq+l + zq+l = 0. Herbert Hot je (Institut f~ir Mathematik, Welfengarten 1, W-3000 Hannover) CL IFFORD FLATS In any kinematic space (P,G,.) two parallelisms lie, lit on G • G are defined ([1]). For any G ,H E G, GfqH = {p} the set Gp-IH = {gp- lhEP ;gE G,h E H} i sea l leda Cl i f ford flat. It contains a 2-net, namely two different flocks of lines {gp-lH; g E G} and {Gp-lh; h E H} such that any line of one flock meets any line of the other flock in exactly one point. Properties of such clifford fiats will be investigated. For example: For G, H E G, G not parallel to H, the set GH is a Clifford fiat. For G, H E G, the set {GpH; p E P} is a partition of P. K inemat ic spaces can be defined by properties of such sets: a. Let P be a set of points and G a set of subsets of P such that (P, G) is a linear space. b. There are two parallelisms "lie, fir" on G such that any two different intersecting lines determine a 2-net with a flock of left and one of right parallel lines. c. Let GpH and GqK be the point sets of two 2-nets, G1 := Gph with h E H and G2 : = Gqk with k E K be different from G, g E G, and L : = {g lie H} n G1, {g tie K} n G2. Then the 2-net determined by G1 and L is independent from the choice of g E G for any two such 2-nets. d. Let GpK and HqK be the point sets of two 2-nets, K1 := gpK with g E G and K2 : = hqK with h E H be different from K, k E K, and L : = {k [[r G} fq K1, {k IIr H} fq K2. Then the 2-net determined by K1 and L is independent from the choice of k E K for any two such 2-nets. Refernces: [1] H. Karzel, H.-J. Kroll, K. S6rensen: Invariante Gruppenpartitionen und Doppel- r~iume. J. reine angew. Math. 262/263 (1973), 153-157. Heiner Kaiser (Mathematische Fakulttit der Friedrich-Schiller-Universit~it, Univer- sit~itshochhaus, 0-6900 Jena) SELF-S IMILAR DISSECTIONS OF POLYGONS An elementary-geometrical dissection of a polygon n in the euclidean plane into polygons nl, n2 ..... nk (k ~- 2) is called self-similar if ni is similar to n for i E {1, 2 ..... k}. A self-similar dissection is called perfect if its parts are mutually non-congruent. 12 International Conference of Geometry Theorems: 1. A triangle which is not right-angled admits a self-similar dissection into 5 parts if and only if its angles are of 120 ~ and 60 ~ 2. Every non-equilateral triangle can be perfectly dissected into 8 or less parts. Up to similarity there are precisely two triangles, which cannot be perfectly dissected into less than 8 parts. Open problems: 1. Can every equilateral triangle be perfectly dissected ? (This problem is due to W.T. Tutte.) 2. For which values of a natural number n does an arbitrary triangle admit a self- similar dissection into n mutually congruent parts? Vice versa: Which triangles admit self-similar dissections into n mutually isomteric parts for a given natural number n ? (In particular: Do there exist triangles which can be dissected in such a way into 6 or 7 parts?) Franz Kalhoff (Fachbereich Mathematik der Universit~it, Postfach 500500 W-4600 Dortmund 50) ON WITT RINGS OF PROJECTIVE PLANES After giving a brief account on the theory of the reduced Witt rings of orderable projec- tive planes and their planar ternary rings, we present a notion of quadrat ic semi- orderings in planar ternary rings that allows us to extend a good part of the classical relations between semiorderings and reduced forms of commutative fields to our setting. Reference F. Kalhoff, Witt Rings and Semiorderings of Planar Ternary Rings. In: Topics in Combinatorics and Group Theory (R. Bodendiek, R. Henn editors), Physica-Verlag Heidelberg 1990, 405-411 Helmut Karzel (Mathematisches Institut der TU Mtinchen, Arcisstr. 21, W-8000 Mtinchen 2) ARCHIMEDEISATION In order to study ordered kinematic spaces and to give an archimedeisation of these structures the following preliminary work has been done in collaboration with C.J. Maxson (the paper is submitted to "Results of Mathematics"): International Conference of Geometry 13 Connections between ordered groups, betweenness groups, cyclic ordered groups and separated groups; structure theorems on separated groups and archimedeisation of separated groups; complete discussion of three types of separated groups derived from unitary associative algebras (A, K) of degree 2 where K is a non-archimedean ordered field. Norbert Knarr (Institut ffir Analysis, TU Braunschweig, PockelstraBe 14, W-3300 Braunschweig) ON THE CONSTRUCTION OF GENERALIZED QUADRANGLES FROM POLAR SPACES OF RANK 3 It follows from recent work ofW.M. Kantor and J.A. Thas that with every flock of a qua- dratic cone in a finite projective space one can associate a generalized quadrangle. For the case of projective spaces of odd order, we give a direct geometric construction of this generalized quadrangle from a polar space of rank 3. Hans-Joachim Kroll (Mathematisches Institut der TU Mtinchen, Arcisstr. 21, W-8000 Munchen 2) SUBSPACES IN THE CHAIN GEOMETRY OF AN ALGEBRA A subspace of a chain geometry E(L,K) over an algebra (L,K) (cf. Benz [1]) will be de- fined as a subset S of points such that the chain joining any three pairwise non-parallel points of S is contained in S. In particular we study subspaces of M6bius geometries. The results enable us to construct examples of MSbius geometries having a quadric model of type A in the sense ofM. Werner [2]. References [1] Benz, W.: Vorlesungen t~ber Geometrie der Algebren. Berlin - Heidelberg - New York 1973. [2] Werner, M.: Kettengeometrien und Quadrikenmodelle. Dissertation Stuttgart 1981. Pia Maria Lo Re (Dipartimento di Matematica, Via Mezzocannone, 8,1-80134 Napoli) LINEAR SPACES' WITH p2 + p + 1 MUTUALLY INTERSECTING LONG LINES Let n be the maximum degree of a line in a finite linear space L and denote by bn the number of all lines of degree n (the long lines). 14 International Conference of Geometry Melone [4] proved that, if L is not a projective plane and the long lines intersect each other, then bn =< v-l; he characterized the finite linear spaces for which equality holds. This result suggests to study finite linear spaces satisfying the following condition: (*) The long lines are mutually intersecting. Many results have been obtained in this direction, see for example [1], [2], [3]. We study finite linear spaces satisfying (*) in which the number of long lines is p2 + p + 1 (p a fixed integer), without any hypothesis on the number of points of L. References [1] P. De Vito, Su una classe di spazi lineari con le rette lunghe a due a due incidenti, Pubbl. Dip. Mat. e Appl. "R. Caccioppoli" Univ. Napoli, 19, (1990), (submitted). [2] P. De Vito e P. M. Lo Re, Spazi lineari su v punti con v-3 rette lunghe a due a due incidenti (to appear in Ricerche di Matem.) [3] P. De Vito, P.M. Lo Re and N. Melone, Linear spaces on v points with v-2 mutually intersecting long lines, J. of Geometry 37 (1990), 87-94. [5] N. Melone, Sugli spazi lineari finiti in cui le rette di massima lunghezza sono a due a due incidenti. Pubbl. Dip. Mat. e Appl. Univ. Napoli, 21, (1988). Rainer LSwen (Abteilung ffir Topologie, TU Braunschweig, Pockelstral3e 14, W-3300 Braunschweig) TOPOLOGICAL PROJECTIVE SPACES Presenting joint work with R. Kuhne, we describe a greatly simplified proof of Misfeld's results about topological projective spaces, namely that every finite dimensional pro- jective space over a topological field can be made into a topological projective space in a natural manner, and that all finite dimensional arguesian topological projective spaces arise in this way. The simplifications are achieved by systematic use of the general linear group as a topological transformation group and of quotient topologies. Topological properties of the GraBmann set of all k-dimensional subspaces of a given projective space are also discussed. International Conference of Geometry 15 Helmut Mi iurer (Fachbereich Mathematik, TH Darmstadt, Schlol3gartenstr. 7, W-6100 Darmstadt) AUTOMORPHISMS OF THE GEOMETRY (R2, R[x]) OF THE REAL POLYNOMIALS (R2, R[x]) denotes the incidence structure consisting of the point set R2 and the set {{(x,f(x)) ] x E R} I f ~ R[x]} of blocks. It will be shown that every automorphism of this geometry has the form (x,y) -* (ax+b, cy+d(x)) where a, b, c are real numbers with a. c ~ 0 and where d is a real polynomial. Vasili C. Mavron (Department of Math., University College of Wales, Aberystwyth, Dyfed SY 23 3 BZ) ARCS IN SOME SYMMETRIC DESIGNS In a symmetric 2-(v,k,h) design, an a-arc is a non-empty set which is met by every block in either 0 or a points. By considering the dual design, if necessary, one may assume a =< x/(k-k). We shall look at the situation when H is the symmetric 2-(v,k,h) correspon- ding to a (v,k,~)-graph F and investigate the subsets of F giving rise to a-arcs in H for the case a = V'(k-2~). The graphs P may, for example, be constructed from certain Steiner systems or sets of mutual ly orthogonal latin squares. In the case of the symmetric 2-(36,15,6) constructed using a single latin square L of order 6, we show that the 3-arcs correspond precisely to the transversals of L. We give examples where L has and does not have transversals. The corresponding designs are therefore non-isomorphic. More generally one can prove for nets of degree 3 and order n ~> 3 that a non-empty subset of points which is such that any point of the net is joined to either 0 or 3 points of the subset is necessarily a transversal of n points. Francesco Mazzocca (Universit~ di Napoli, Dipartimento di Matematica, Via Mezzo- cannone, 8,1-80125 Napoli) SOME SPEClAL POINT SETS IN PG(n,q) (Joint work with A. Blokhuis) 16 International Conference of Geometry For every point set B in PG(n,q) we denote by L(B) the set of all hyperplanes ~ such that I B - n I =- 0 rood p, where p is the characteristic of GF(p). We are interested in the set R(B) of all points a such that every hyperlane through a belongs to L(B). Points in R(B) are called B-regular and one can show that a point is B-regular if and only if every hyperplane through it meets B in a constant number of points mod p. Moreover, the set B is said to be regu lar if every hyperplane belongs to L(B), that is every point in PG(n,q) is B-regular. Main result. A polnt-set B is regular if and only if the polynomial FB(X0' Xl' '" ' Xn) = I 0~ + blXl + "" + bnXn )q-1 [b = (b0, bl,,.. ,bn)] b~B is identically zero. Moreover, if B is not regular, a point a = (al, a2 ..... an) is B-regular if and only if the polynomial a0x0 + alxl + ... + anXn divides FB(X0, xl ..... Xn). It follows that IR(B)[ < q-1 for every non-regular set B. We show some consequences of the previous result and point out some related pi'oblems. References [1] A. Blokhuis and F. Mazzoeea, On maximal sets of nuclei in PG(2,q) and quasi-odd sets in AG(2,q). Proceedings of the Isle of Thorns conference, 1990, to appear. [2] A. Blokhuis and H. Wilbrink, Characterization of exterior lines of certain sets of points in PG(2,q). Geom. Dedieata 23, 253-254. [3] A. Bruen and F. Mazzoeea, Nuclei of sets in finite projective and affine spaces. Submitted to Combinatoriea. Thomas Meixner (Mathematisches Institut, Justus Liebig-Universitat, Arndtstr. 2, W-6300 GieBen) Geometr ic character izat ions of some sporadic groups We continue Buekenhout's program of characterizing the sporadic simple groups by their flag-transitive action on diagram-geometries. There have been considered "circle- extensions" of generalized quadrangles, which correspond to locally polar spaces. Now we consider extensions of other classical geometries. International Conference of Geometry 17 Nicola Melone (Universit~ di Napoli, Dipartimento di Matematica, Via Mezzocannone, 8,1-80125 Napoli) ON DOWLING-WILSON CONFIGURATIONS IN FINITE LINEAR SPACES In 1989 K. Metsch has given a positive answer to the so-called Dowling-Wilson con- jecture: If a finite linear space with v points and b lines has t lines through a point admitting a common missing line, then b ~ v+t. A Dowling-Wilson configration is, by definition, any family C of lines of a finite linear space with a missing line such that the inequality b > v + IC I holds. Examples of Dowling-Wilson configurations, other than those proposed by Dowling and Wilson are (i) all the lines ofa subspace admitting a missing llne, (ii) all the lines missing a fixed line of maximum degree. We deal with the following natural question: (*) If a finite linear space has t+ 1 mutually disjoint lines, is it true that b > v + t? In other words, does a partial spread of lines give rise to a Dowling-Wilson configuration? The h-punctured projective planes easily show that the answer to (*) is negative. U. Ott and myself have proved the following Theorem. Let t+ 1 be the maximum number of mutually disjoint lines in a finite linear space L with v points and b lines: I f there exists in L a partial spread F of t+ 1 lines of the same degree k, then either L is a punctured projective plane or b ~ v + t. Moreover, i f all the points on the lines of F have degree k + 1, equality holds if and only if (i) t = 0, and L is a projective plane or a near-pencil, or (ii) t > 0, and L is an affine plane of order t+ 1 with one point at infinity. This result supports the following problem: Determine "general" Dowling-Wilson configurations C and characterize the finite linear spaces for which the equality b = v + [C] holds. Klaus Metsch (Mathematisches Institut, Justus-Liebig-Universit~t, Arndtstr. 2, W-6300 GieBen) AN IMPROVEMENT OF BRUCK'S COMPLETION THEOREM In 1963 R.H. Bruck proved his famous completion theorem for mutually orthogonal latin squares. It states that a net of order k and deficiency 5 can be completed if 2k > (5-1)(53-82+5+2). In this talk an improvement of this bound is presented: It is 18 International Conference of Geometry sufficient to demand 3k > 853-1852+85+8. This result has applications in various fields of finite geometry. In particular, we can prove the following assertions. �9 For every maximal partial spread with deficiency 5 > 0 in PG(3,q), q not a square, we have 853-1852 + 85 ~ 3q 2. �9 There exists a constant c such that for all integers n and a with n > c.a3 every l inear space with constant point degree n + 1 and minimum line degree n + 1-a can be embedded in a projective plane of order n. Wolfgang Nolte (Fachbereich Mathematik, TH Darmstadt, Schloftgartenstr. 7, W-6100 Darmstadt) COORDINAT IZAT ION OF PAPP IAN PROJECT IVE KL INGENBERG PLANES Let F be a projective Klingenberg plane with at least four nonneighbour points on each line. I f the dual pappus theorem holds in F then F can be coordinatized by a commuta- t i re local ring. Antonio Pas in i (Dipartimento di Matematica, Via Doz Capitano 15,1-53100 Siena) D IAGRAMS FOR BENZ PLANES The following result is well known: The MSbius planes are precisely the geometries belonging to the following Buekenhout diagram: c Af (1) : : : points edges circles where c denotes the class of all complete graphs and Af denotes the class of affine planes. The objects called "edges" are the pairs of points in the same circle. We now consider the following diagrams International Conference of Geometry 19 c N* (2) : : : s s+ l points edges circles and c N* (3) : : S points edges s+l circles where N* N denote the class of dual nets and the class of nets, respectively, and s and s+ 1 are finite parameters. Given a geometry G belonging to (2) or (3), the point-graph P(G) of G is the sub- geometry of G consisting of the elements called "points" and "edges". We prove the following assertions. Theorem 1. A geometry G belonging to the diagram (2) is a Laguerre plane iff the following conditions hold in F(G): (LL) The graph F(G) has no multiple edges, (T) every 3-clique of F(G) belongs to some circle. Theorem 2. A geometry G belonging to the diagram (3) with s > 1 is a Minkowski plane iffboth (LL) and (T) hold in F(G). Jochen Pfalzgraf (Research Institute for Symbolic Computation, Johannes Kepler University A-4040 Linz) A NET THEORETIC ASPECT OF GEOMETRIC REASONING In this contribution we give a short survey of the emerging interdisciplinary field of geometric reasoning (GR), in particular constructive algebraic methods of GR, as they are represented by the local RISC-Linz project group in the ESPRIT project MEDLAR. Methods of GR are applied there e.g. to particular robotics problems and automated geometric theorem proving. 20 International Conference of Geometry Here we want to extend principles and methods of GR to systems which can be modeled by net theory in a general sense, such as communication (processor) nets and connectionist systems/artificial neural networks. Our interest concentrates in particular on nets which carry the structure of a homogeneous space (group space) given by a group action. The corresponding local and global symmetries are of particular interest. As an example we consider a Hopfield net. The question concerning GR is now: Is there a method to find out whether a given net is generated by a group action? We discuss this problem from the point of view of (noncommutative) geometric spaces; these can natural ly be interpreted in terms of net theory Ccoloured directed graphs") and vice versa. With the help of simplex configurations we can treat the above problem: a net is a homogeneous space w.r.t, a group action if it has a suitable simplicial structure (in the geometric sense). Concerning applications, methods of symbolic computation are interesting w.r.t, the im- plementation of a procedure which can solve such a problem as mentioned before. Generalizing the previous considerations we present some arguments that it is of prac- tical interest to extend the concept of a geometric net to "net fiberings" consisting of a base space (a net) and over each base point a state space (fiber). The whole system forms a fiber space. We conclude with a remark concerning a certain category of nets occuring in computer science which is a universally topological category and which in particular is a certain topos. Categories, fiberings/sheaves and topoi seem to be of increasing interest as formal mathematical concepts in the foundation of computer science. Gfinter P icker t (Mathematisches Institut, Justus Liebig-Universit~t, Arndtstr. 2, W-6300 GieBen) DOUBLE PLANES AND LOOPS A doub le p lane is a pair (n,n') of projective planes with the same point set P, such that every line of n' is an oval of n and every line of n is an oval of n'. For p, q E P with p ~ q the notations pq, (pq)' mean the n- and n'-line resp.joining p, q. Then in P relative to an "origin" o (6 P) an operation + is defined by the following equations (for all p, q 6 P): o +p=p =p +o, and (*) (op)'A oq = {o,r} (rq)' A rp = {r, p+q}, if p,q#= o. International Conference of Geometry 21 I f r = q or r = p, then (rq)', rp mean the tangents in r to the n'-oval oq and the n- oval (op)' resp. In [DO] it is shown that (P,+) is a loop with neutral element o and equality of left and right inverses (i.e. p + (-p) = o = (-p) + p ); D being any n-line through o, the lines of n, n' are the p + D, -D + p (p ~ P) resp. These concepts and con- structions are motivated by the special case: Existence of an abelian group of n-colli- neations, transitive on P; equivalently: Existence of an operation + such that (P, +) is an abelian group with a difference set D. Then the p + D (p ~ P) are the lines of n and the -D + p (p ~ P) are ovals, lines of another plane n', which together with n forms a double plane, and (*) gives a reconstruction of + by n, n'. In [DO] configuration theorems are given, which together are equivalent to commutativity and associativity of + and thus describe those double planes constructed by an abelian difference set. Now, generalizing the concept of abelian difference set, the problem arises: Given a loop (P,+) with equality of left and right inverses and a subset D of P with ID[ ~ 3, conditions for (P, +), D, more general than those for an abelian difference set, are sought for, such that (i) p+D, -D+p (p E P) are the lines ofplanes n,n ' with point set P; (ii) (n, n') is a double plane; (iii) (*) is valid. A partial answer is given in the Theorem. I f (P, +) is a communtative loop with neutral element o, D a subset of P with at least 3 elements and Vp ~ P-{o} 3! (d,d') ~ D2:p = d -d ' Yp E P ,a ,b E D: (p -a ) + b = p + (b -a ) = (p + b) -a , then (i), (ii), (iii) hold. Literature [DO] G. Pickert: Differenzmengen und Ovale. Discrete Math. 73 (1988/89), 165-179. Stefan E. Schmidt (FB Mathematik der Universittit Mainz, Saarstr. 21, W-6500 Mainz) A UNIF IED APPROACH TO PROJECT IVE AND AFF INE LATT ICE GEOMETRIES We shall develop a set of general geometric axioms which unify the so far existing concepts in the literature. This framework also yields new results in which geometries of unitary modules are synthetically characterized. 22 International Conference of Geometry Eberhard SchrSder (Mathematisches Seminar der Universit/it, BundesstraBe 55, W-2000 Hamburg 13) METRIC GEOMETRY Metric geometry may be considered as "incidence geometry together with metric". More precisley, one can think of an incidence substructure of a pappian projective space, where the metric is induced by a quadratic form. For example, if A(V, F) is the affine space corresponding to a vector space V over a field F and if q: V ~ F is a quadratic form, then the corresponding congruence relation =q is defined by (A, B) ~-q (C, D) :r q(A-B) = q(C-D) for A ..... D ~ V. Metric geometry is concerned with sets (points, lines, subspaces, circles, quadrics), with relations (congruence, orthogonality, conjugacy, conformity), with mappings (reflec- tions, motions, similarities, metric automorphisms), and with measures (distances, measures of angles). From the foundational point of view it is natural to ask for geo- metric characterizations of metric structures, of metric automorphisms, and of quadrics. There have been a lot of contributions to the characterization of metric structures and of quadrics in the past, but it seems that in the future the characterization of metric automorphisms will play an outstanding role. In my lecture I give a small review on results concerning characterizations of metric geometry. In any case, several mathematicians have contributed ideas on the way to the solution. In detail, I report on a characterization - of the euclidean R3 by means of seven axioms, - of regular euclidean spaces within the frame of affine geometry by means of an axiom of synunetry, - ofaffine metric spaces of characteristic =~ 2 by means of three axioms, - of the class of all projective metric spaces by means of a generalized 3-reflection condition, and - of a class of subgeometries of projective metric spaces which plays an important role in reflection geometry. International Conference of Geometry 23 Andreas Schroth (Institut for Analysis der Techn. Universittit, Pockelstr.14, W-3300 Braunschweig) THREE-D IMENSIONAL QUADRANGLES WITH LARGE AUTOMORPHISM- GROUPS As in other topological geometries the question is investigated how large the automorphismgroup of a nonclassical three-dimensional quadrangle can become. It will be shown that if the group is of dimension at least six, the quadrangle is classical. Also examples for three-dimensional quadrangles with a 5-dimensional group will be given. The proof relies heavily on the fact that every three-dimensional quadrangle is the associated Lie geometry of a flat Laguerre plane. Ralph-Hardo Schulz (II. Mathematisches Institut der FU Berlin, Arnimallee 3, W-1000 Berlin) ON CHECK CHARACTER SYSTEMS 1. A check character system is a way to define an error detecting code over an alphabet Q by adding a check digit an to a word al...an-1 of information digits. This can be done either in explicit form: an = f(al ..... an-l) or implicitly by a check equation g(al ..... an-l, an) = c. The functions f and g may be defined by using a given operation on Q, for instance a group operation or that ofa quasigroup (i.e. Latin square). Although the two coding schemes (the direct and the implicit one) are regarded as equivalent one should observe that they may lead to different requirements on the quasigroup used. 2. Now let Q be the affine group G1 = A(1,q) defined over K = GF(q). Its elements are the mappings K ~ K with x ~ ax+b attached to (a,b) ~ K* x K. P ropos i t ion 1: Let q = 2m > 2 and choose t with gcd (t,q-1) = 1 = gcd(t--1,q-1); furthermore, put da = i for a ~ K*-{1} and choose dl ~ K-{0, 1}. Then the mapping T: (a,b) --~ (at, da'b) is a permutation of G1 = A (1,q), and the code with alphabet GI and check equation IITi(ai) = c detects all single errors and all neighbour transpositions (that are errors of the form ... ai ... ~. . . ai' ... or... aiai + 1 . . . - -~ . . . ai + 1 ai... respectively). As in the dihedral 'groups, other mappings T with that property cannot be group automorphisms. 3. Let Q = G2 be one of the groups found by ITO. We recall: Let q be a prime power greater 2, let t be a prime divisor of q- l ; here we choose t > 2; furthermore, let dl, d2 K = GF(q) with dl ~ d2 and dl t = 1 = d2 t. To abbreviate, we associate (i, k) with 24 International Conference of Geometry (10) (d 0) k l 0d i2 These elements form a subgroup G2 of GL(2,q). Proposition 2: Choose e E {2,3 ..... t - l} and f0 ~ GF(q)-{0,1}; furthermore, put fi = (d1-ld2) i for i E {1 ..... t-l}. Then the mapping T: (i,k) --~ (e.i, fi'k) (with e-i reduced mod t) is a bijection of G2, and the code with alphabet G2 and check equation HTi(ai) = c detects all single errors and all neighbour transpositions. Twin errors and jump twin errors are errors of the form ... aa ... --~ ...bb... and ... aca ... -~... bcb.. respectively. The parameters of the smallest cases of Proposition 2 where alle single, twin, jump twin errors and neighbour transpositions can be detected are: q=23, t= 11, e=2, f0=2~GF(23) and q=29, t=7, e=2, f0=4EGF(29) . The alphabet then contains 253 and 203 elements, respectively. Helmut S iemon (P/idagogische Hochschule Ludwigsburg, Abteilung Mathematik, W-7140 Ludwigsburg) CYCLIC STE INER QUADRUPLE SYSTEMS SQS(2p) WITH p = 1,5 mod 12 The existence problem for cyclic Steiner Quadruple Systems SQS(2p), p -= 1, 5 mod 12 is not yet solved completely. A first major approach was made by E. KShler in 1978 who associated an orbit graph G(p) with a cyclic SQS(2p) in order to reduce the existence problem to finding a one-factor in this graph. In our talk we classify all G(p) according to their vertices of degree smaller than 3 and show that the graphs in each class have a one-factor if a certain number-theoretic claim is true. Giuseppe Tal l ini (Dipartimento di Matematica, Istituto "Guido Castelnuovo", Univer- sit~ di Roma, Piazzale Aldo Moro 2,1-00185 Roma) THE GEOMETRY OF THE COUNTABLE D IMENSIONAL GALO1S SPACE PG(N,q) Let PG(N,q) be the Galois space of countable dimension. As in the classical case, the characters , the c lass and the type of a subset K with respect to the d-dimensional subspaces (d finite, d => 0) are defined. The study of these subsets can then proceed by increasing the number of non-zero characters. Various characterization theorems are proved. In particular, the sets of class [0, 1, n, q+l ] with respect to lines are cha- International Conference of Geometry 25 racterized; it turns out that these are quadrics and hermitian varieties. For instance, the following assertion is proved: Every regular non-singular subset K of PG(N,q) of class [0, 1, n, q+ 1] with respect to lines is a quadric or a hermitian variety. Here, K is called non-singular if there is an n-secant through any point of K; the set K is said to be regular if for every point P of K, the tangents to K through P and the lines through P lying on K form a hyperplane or the whole space. One can pose many problems, for instance on the characterization of classical varieties (quadrics, hermitian varieties, Greassmannian manifiolds, C. Segre product manifolds, and so on) with respect to their behaviour to finite dimensional subspaees of PG(N,q). Helga Teeklenburg (Institut f~ir Mathematik der Universit~it, Welfengarten 1, W-3000 Hannover) QUASI-ORDERED AFFINE SPACES Generalizing the concept of order in Euclidean spaces as far as possible, we postulate only that the betweenness function is compatible with the congruence relation. Starting from the incidence geometric properties of such betweenness functions there will be introduced quasi-orderings in affine spaces. Quasi-ordered affine spaces will be described algebraically by quasi-ordered ternary rings. Suitable examples of quasi-ordered ternary rings show on the one hand that every non-empty Lenz-Barlotti class contains quasi-orderable affine planes and on the other hand that there are non quasi-orderable affine planes as well as quasi-orderable affine planes without proper quasi-orderings. Zsolt Tuza (Computer and Automation Institute, Hungarian Academy of Sciences, H-1111 Budapest, Kende u. 13 - 17, Hungary) 1. UNSOLVED PROBLEMS RELATED TO GEOMETRY We present open problems motivated by studies in various fields including biology, formal language theory, Helly-type properties of finite set systems, extremal graph theory, computational complexity, and discrepancy theory. The common feature of those problems is that we expect their solution by geometric methods. 2. EQUILATERAL TRIANGLES HAVE NO INCONGRUENT DISSECTION Solving an old problem of Tutte, we prove that equilateral triangles cannot be de- composed into a finite number of equilateral triangles with mutually distinct size. 26 International Conference of Geometry This theorem completes the results of H. Kaiser who proved that any other triangle is decomposable into at most eight similar, incongruent parts. Johannes Ueberberg (Mathematisches Institut, Justus Liebig-Universittit, Arndt- str. 2, W-6300 GieBen) COMPUTER GEOMETRY IN FINITE PROJECTIVE SPACES In this talk we report on the project "Computer Geometry in finite projective spaces". This project is a common initiative of the universities of Giel~en, Bayreuth and Kiel. The project has the following two tasks: 1. The Computer Geometry package will allow geometers to use the computer for their research in projective spaces. There will be possibilities to investigate geometric structures such as linear subspaces, projective subspaces, quadrics, arcs or spreads, for example - to generate these structures systematically or at random, - to calculate parameters, - to manipulate these structures, for instance by deleting or adding a point. On the other hand, the user of such a system will also be able to define new incidence structures or geometries coming from a projective space, for instance the geometry defined by the points, lines and conics in a projective plane. Investigations of these "new" geometries are also possible, for example the determination of its parameters or checking if it is a linear space. 2. The second aspect of the Computer Geometry in projective spaces is to be an exemplary investigation for a more general Computer Geometry for finite geometries. In this talk we will make precise the nature of the problems that may be handled by the Computer Geometry package. Furthermore we will present the fundamental data structures for the description of geometries. Corrado Zanella (Dipartimento di Matematica, Istituto "Guido Castelnuovo", Universith di Roma, Piazzale Aldo Moro 2,1-00185 Roma) ON THE ORDER STRUCTURE IN THE LINE GEOMETRY OF A PROJECTIVE SPACE (Joint work with A. Bichara, J. Misfeld, G. Tallini) The order structure of a projective space P, according to E. Sperner's [2,3] definition, can be described by an order function. The domain of an order function r 1 is the set of all quadruples (H,K,a,b), where H and K are hyperplanes, and a and b are points of P International Conference of Geometry 27 not on H U K. An order function may take only the values • 1. We say that the pairs (H, K) and (a, b) separate each other if, and only if, rl(H,K,a,b) = -1. Let Pn,F be given, where F is an ordered commutative field. To define an order structure in the Grassmann space FI(Pn,F) (whose "points" are the lines of Pn,F), one may use the existent isomorphism between Pl(Pn, F) and the Grassmann variety GI,n,F embedded in PN,F, where N§ = (n§ (Observe that PN,F is an ordered projective space.) The above order structure can also be defined in a purely geometric way. Thus, for any ordered projective space P one can speak of the ordered Grassmann space FI(p). We have shown that the ordered Grassmann spaces are characterized by axioms which are quite similar to those of Sperner. It follows that every partial linear space of finite rank which satisfies Tallini's [4] and Sperner's axioms and the continuity axiom is geometrically isomorphic to a real Grassmann manifold, with an isomorphism which preserves the order structure. References [1] Misfeld, J., Tallini, G. and Zanella, C.: Topological Grassmann spaces. Rend. Mat. Appl. (7) 8 (1988), 223-240. [2] Sperner, E.: Die Ordnungsfunktionen einer Geometrie. Math. Ann. 121 (1949), 107- 130. [3] Sperner, E.: Beziehungen zwischen geometrischer und algebraischer Anordnung. S.- B. Heidelberger Akad. Wiss. Math.-Natur. K1. 1949, 10. Abh., 38 S. [4] Tallini. G.: On a characterization of the Grassmann manifold representing the lines in a projective space. Finite geometries and designs (Chelwood Gate 1980), London Math. Soc. Lecture Notes Ser. 49, Cambridge 1981, 354-358.