The reflexivity, the (semi-)ordinariness, the dimension of dual varieties and the structure of Gauss maps are discussed for Segre varieties, where a Segre variety is the image of the product of two or more projective spaces under Segre embedding. A generalization is given to a theorem of A. Hefez and A. Thorup on Segre varieties of two projective spaces. In particular, a new proof is given to a theorem of F. Knop, G. Menzel, I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky that states a necessary and sufficient condition for Segre varieties to have codimension one duals. On the other hand, a negative answer is given to a problem raised by S. Kleiman and R. Piene as follows: For a projective variety of dimension at least two, do the Gauss map and the natural projection from the conormal variety to the dual variety have the same inseparable degree?
ASJC Scopus subject areas
- Geometry and Topology