Abstract
In Dedicata 16 (1984), pp. 291–313, the representation of Desarguesian spreads of the projective space PG(2t − 1, q) into the Grassmannian of the subspaces of rank t of PG(2t − 1, q) has been studied. Using a similar idea, we prove here that a normal spread of PG(rt − 1,q) is represented on the Grassmannian of the subspaces of rank t of PG(rt − 1, q) by a cap V r, t of PG(r t − 1, q), which is the GF(q)-scroll of a Segre variety product of t projective spaces of dimension (r − 1), and that the collineation group of PG(r t − 1, q) stabilizing V r, t acts 2-transitively on V r, t . In particular, we prove that V 3, 2 is the union of q 2 − q + 1 disjoint Veronese surfaces, and that a Hermitan curve of PG(2, q 2) is represented by a hyperplane section U of V 3, 2. For q ≡ 0,2 (mod 3) the algebraic variety U is the unitary ovoid of the hyperbolic quadric Q + (7, q) constructed by W. M. Kantor in Canad. J. Math., 5 (1982), 1195–1207. Finally we study a class of blocking sets, called linear, proving that many of the known examples of blocking sets are of this type, and we construct an example in PG(3, q 2). Moreover, a new example of minimal blocking set of the Desarguesian projective plane, which is linear, has been constructed by P. Polito and O. Polverino.
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Lunardon, G. Normal Spreads. Geometriae Dedicata 75, 245–261 (1999). https://doi.org/10.1023/A:1005052007006
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DOI: https://doi.org/10.1023/A:1005052007006