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

Hans Havlicek

TL;DR

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 groupM12 in PG(5,3).

Abstract

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 H.S.M. Coxeter and G. Pellegrino, including a projective representation of the Mathieu group $M_{12}$ in PG(5,3).

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

TL;DR

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 groupM12 in PG(5,3).

Abstract

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 for Witt's 5- design, the blocks being the hyperplane sections containing more than three (actually six) points of . As such a point model is projectively unique, the present construction yields an easy coordinate-free approach to some results obtained independently by H.S.M. Coxeter and G. Pellegrino, including a projective representation of the Mathieu group in PG(5,3).

Paper Structure

This paper contains 2 sections, 4 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Variations on $13-4+3=12$

Key Result

Theorem 1

Write ${\cal K}$ for that set of points in $\hbox{\rm PG}(5,3)$ which is obtained from the Veronese surface $\hbox{\rm im,}\varphi$ by replacing the planar quadrangle $\Gamma_\infty$, i.e. the $\varphi$--image of the line at infinity, with its diagonal triangle $\Delta_\infty$. Then the following ho

Theorems & Definitions (11)

  • Theorem 1
  • Remark 1
  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • ...and 1 more