A remark on the uniqueness of embeddings of linear spaces into desarguesian projective planes

✍ Scribed by Klaus Metsch

• Publisher: John Wiley and Sons
• Year: 1995
• Tongue: English
• Weight: 294 KB
• Volume: 3
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.3180030406

✦ Synopsis

Suppose that q 2 2 is a prime power. We show that a linear space with a( q + 1)' + ( q + 1) points, where a 1 0.763, can be embedded in at most one way in a desarguesian projective plane of order q. 0 1995 John Wiley & Sons, he.

1. Introduction

A linear space consists of points and lines such that any two points are on a unique line, every line has at least two points, and there exist three noncollinear points. A rank two geometry G that can be embedded in a (desarguesian) projective plane of order q can be uniquely embedded if for any two embeddings cpi of G into projective planes ni of order q , i = 1,2, there exists an isomorphism cp form I l l onto n2 such that cp1 0 cp = cp2.

Let q 2 2 be an integer. Put f ( q ) := (1 + & ) q if q is a perfect square and, otherwise, f ( q ) = (& -i) ( q + 1) + 1. It follows from a result of [2] that a linear space with more than q2 + q + 1 -f ( q ) points can be embedded in at most dne way in a projective plane of order q (see Corollary 2.5). If q is a square of a prime power, then this bound is optimal. We show that a much better bound holds, if one considers only embeddings in desarguesian projective planes.

Theorem 1.1. Suppose that q is a prime power and that L is a linear space with v points that can be embedded in PG(2,q). Zfv 2 0.763(q + 1)2 + ( q + l), then L can be uniquely embedded in PG(2,q).