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SEMINARS

Generalized ovals and generalized ovoids

时间 Datetime

2017-05-11 14:00 — 15:00

地点 Venue

Large Conference Room

报告人 Speaker

Professor J. A. Thas

单位 Affiliation

Ghent University

邀请人 Host

图论与组合组

A non-singular conic of the projective plane PG(2, q) over the finite field GF(q) consists of q+1 points no three of which are collinear. For q odd, this non-collinearity condition for q+1 points is sufficient for them to be a conic; see Segre (1954). Generalizing, Segre considers sets of k points in PG(2, q), k ≥ 3, no three of which are collinear. The concept of a k-arc in PG(2, q) was generalized to that of a k-cap in PG(n, q); a k-cap of PG(n, q), n ≥ 3, is a set of k points no three of which are collinear. An elliptic quadric of PG(3, q) is a cap of size q2+1. In 1955, Barlotti and Panella independently showed that, for q odd, the converse is true. Also, q2+1 is the maximum size of a k-cap in PG(3, q) for q ≠ 2. This leads to the definition of an ovoid of PG(3, q) as a cap of size q2+1 for q ≠ 2 and, for q = 2, a cap of size 5 with no 4 points in a plane. Ovoids of particular interest were discovered by Tits (1962).

Arcs and caps can be generalized by replacing their points with m-dimensional subspaces to obtain generalized k-arcs and generalized k-caps.

In the talk we will consider generalized ovals and generalized ovoids. Also applications and open problems will be mentioned.

A non-singular conic of the projective plane PG(2, q) over the finite field GF(q) consists of q+1 points no three of which are collinear. For q odd, this non-collinearity condition for q+1 points is sufficient for them to be a conic; see Segre (1954). Generalizing, Segre considers sets of k points in PG(2, q), k ≥ 3, no three of which are collinear. The concept of a k-arc in PG(2, q) was generalized to that of a k-cap in PG(n, q); a k-cap of PG(n, q), n ≥ 3, is a set of k points no three of which are collinear. An elliptic quadric of PG(3, q) is a cap of size q2+1. In 1955, Barlotti and Panella independently showed that, for q odd, the converse is true. Also, q2+1 is the maximum size of a k-cap in PG(3, q) for q ≠ 2. This<