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

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 Wittʹs 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 groupM12in PG(5, 3).