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Quadragoni di Tits e sistemi rigati
No full textG. Tallini has studied the generalized quadrangles satisfying the two natural conditions (a) if P and Q are points that can be joined by a line there exists a point T such that neither P and T nor Q and T can be joined, and (b) there exists a point lying on three distinct lines. Such incidence structures arise in various ways in connection with polarities. On the other hand, the author of the present paper shows that if (a) or (b) are violated in a generalized quadrangle then it must belong to a list of four fairly degenerate types of examples
Sistemi rigati immersi in uno spazio proiettivo.
No full textThis paper give a complete classification of the generalized quadrangles whose lines are lines of a finite projective space
Seminversive planes
No full textAn H-inversive plane, where H is a set of positive integers, is a pair (Ω,C), where Ω is a set of points and C a family of subsets of Ω called circles such that (i) any three distinct points lie on a unique circle; (ii) given a circle B, a point x∈B and a point y∉B, the number of circles through x and y meeting B just at the point x belongs to H; (iii) there exist at least two circles and every circle contains at least three points. The integer n is defined to be the order of the plane if n+1={|B|:B∈C}.
The author investigates the {1,2}-inversive planes called seminversive planes under the assumption that Ω is finite. The main result of the paper under review is the following theorem: Suppose (Ω,C) is a finite seminversive plane of order n>5. Then (Ω,C) is either an inversive plane or a punctured inversive plane of order n. This extends results of M. Oehler [Geom. Dedicata 4 (1975), no. 2-3-4, 419--436; MR0405236 (53 #9030)]
Pseudoproduct spaces and C. Segre's varieties
No full textThe authors introduce the notion of pseudoproduct space as a particular type of projective partial
line space. Pseudoproduct spaces are studied in detail and a characterization is given for pseudoproduct spaces which satisfy a suitable regularity condition. The study leads to an interesting
graphic characterization of C. Segre’s varieties
A CHARACTERIZATION OF THE FAMILY OF SECANT OR EXTERNAL LINES OF AN OVOID OF PG)(3,q).
No full textA characterization of the family of external (secant) lines to an ovoid of PG(3,q) is given in terms of incidence with respect to points and planes
Ruled systems in combinatorial spaces
No full textn a generalized quadragon (S,R), if x and y are two distinct points of S, denote by tr(x,y) the subset of S consisting of points collinear to both x and y. The family A of such subsets defines in S a line space (S,A), because two distinct points of S belong to only one element of A. The degenerated blocks of (S,R), that is, the subsets of S consisting of points collinear to a fixed point x, are subspaces of (S,A).
In this work for the first time projective generalized quadragons are defined, namely, quadragons such that the elements of A and the degenerated blocks are closed sets of a combinatorial geometry G on S. It is proved that, like the classical examples, the dimension of G is always less than or equal to five. Moreover it is proved that if a projective generalized quadragon (S,R), with q+1=|r|, r∈R, is not embeddable in a projective space PG(n,q), necessarily each point is regular and the parameters of (S,R) are those of a Hermitian nonsingular form of PG(3,q) and then q is a square. It follows that, except for the previous case, a projective generalized quadragon is necessarily isomorphic with one of the classical examples
Sets of type (q, n) in PG(3, q)
No full textIn this paper a description for sets in PG(3,q) of type (q, n) with respect to planes is given
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