Elsevier

Discrete Mathematics

Volume 312, Issue 3, 6 February 2012, Pages 652-656
Discrete Mathematics

On finite Steiner surfaces

https://doi.org/10.1016/j.disc.2011.05.035Get rights and content
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Abstract

Unlike the real case, for each q power of a prime it is possible to injectively project the quadric Veronesean of PG(5,q) into a solid or even a plane. Here a finite analogue of the Roman surface of J. Steiner is described. Such an analogue arises from an embedding σ of PG(2,q) into PG(3,q) mapping any line onto a non-singular conic. Its image PG(2,q)σ has a nucleus, say Tσ, arising from three points of PG(2,q3) forming an orbit of the Frobenius collineation.

Highlights

► Any finite quadric Veronesean can be injectively projected into a plane. ► Non-singular Steiner embeddings in finite three-dimensional projective spaces. ► A finite analogue of the Roman surface of J. Steiner. ► An embedding of PG(2,q) into PG(3,q) mapping any line into a non-singular conic. ► A permutation of PG(2,q) mapping any line into a non-singular conic.

Keywords

Veronese surface
Steiner surface

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This work was funded by University of Padova, research project CPDA081192/08.