The Veronese Surface in PG(5, 3) and Witt's 5-(12, 6, 1) Design

✍ Scribed by Hans Havlicek

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 215 KB
• Volume: 84
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1998.2880

✦ Synopsis

A conic of the Veronese surface in PG(5, 3) is a quadrangle. If one such quadrangle is replaced with its diagonal triangle, then one obtains a point model K for Witt's 5-(12, 6, 1) design, the blocks being the hyperplane sections containing more than three (actually six) points of K. As such a point model is projectively unique, the present construction yields an easy coordinate-free approach to some results obtained independently by Coxeter and Pellegrino, including a projective representation of the Mathieu group M 12 in PG(5, 3).

1998 Academic Press

1. Introduction

Throughout this paper V is a 3-dimensional vector space over F := GF(3) and W denotes the symmetric tensor product V 6 V. Occasionally, it will be convenient to use coordinates. We fix an ordered basis (e 0 , e 1 , e 2 ) of V. It yields the ordered basis (e 0 6 e 0 , 2e 0 6 e 1 , 2e 0 6 e 2 , e 1 6 e 1 , 2e 1 6 e 2 , e 2 6 e 2 ) of W. All coordinate vectors are understood with respect to one of these bases. The projective plane on V is PG(2, 3)=(P(V), L(V), # ), where P(V) and L(V) denote the sets of points and lines, respectively. Likewise we have PG(5, 3)=(P(W), L(W), # ). The Veronese mapping is given by .

or, in terms of coordinates, by F(x 0 , x 1 , x 2 ) [ F(x 2 0 , x 0 x 1 , x 0 x 2 , x 2 1 , x 1 x 2 , x 2 2 ).