The Grassmann Space of a Planar Space

AbstractIn this paper we give a characterization of the Grassmann space of a planar space

Sistemi rigati immersi in uno spazio proiettivo.

This paper give a complete classification of the generalized quadrangles whose lines are lines of a finite projective space

Quadragoni di Tits e sistemi rigati

G. 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

seminversive planes

An 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 ymeeting 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

Seminversive planes

An 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)]

Sets of type (q, n) in PG(3, q)

In this paper a description for sets in PG(3,q) of type (q, n) with respect to planes is given

A CHARACTERIZATION OF THE FAMILY OF SECANT OR EXTERNAL LINES OF AN OVOID OF PG)(3,q).

A 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

PROJECTIVE SPACES AND INVERSIVE PLANES

In this paper, a common characterization of the finite projective space of dimension four and order n and of a finite inversive plane of order n + 1 in terms of regular (k, n) finite planar spaces is given