Under the transformation T in which the set of all quadrics in PG(r,q) is mapped to the whole space \(PG(frac{1}{2}r(r+3),q),\) the repeated primes are mapped to the Veronese variety V. Among the properties that V satisfies are the following: (i) any two points of V belong to a conic on V; (ii) any two planes meeting V in conics either meet on V or are disjoint; (iii) if P is a point of V not lying on a conic C on V, then the tangents to C through P are coplanar.
The authors give an elegant characterization of V using these three properties. The case \(r=2\) is handled separately as at one stage in the general case an induction argument is required. The characterization does not depend on previous characterizations such as those of G. Tallini [Atti. Accad. Naz. Lincei, VIII, Ser., Rend., Cl. Sci. Fis. Mat. Natur. 24, 19-23 (1958; Zbl 0080.140)] or O. Ferri [ibid. 61(1976), 603-610 (1977; Zbl 0404.51004)] which only dealt with the case \(r=2\), but is closer in style to G. Tallini [Finite geometries and designs, Lond. Math. Soc. Lect. Note Ser. 49, 354-358 (1981; Zbl 0469.51006)].