Giuseppe Veronese and Ernst Witt -- Neighbours in PG(5,3)

Let $P$ be a point of the Veronese surface $\Vcal$ in \PG53. Then thereare four conics of $\Vcal$ through $P$. We show that the internal points of those conics form a 12-cap which is a point model for Witt's 5-$(12,6,1)$ design. In fact, this construction is "dual" to a similar construction that has been established by the author. We give an explicit parametrization of the cap $\Kcal$; the domain is a dual affine plane which arises from \PG23 by removing one point. Thus, as a by--product, we obtain an easy approach to the extended ternary Golay code $G_{12}$. Finally, we discuss some other procedures that yield 12-sets of points from the Veronese surface.