
Grassmann Spaces
We exhibit the geometry and embeddings of the Grassmann space G(n,d) formed by the ddimensional subspaces of an ndimensional projective space over an arbitrary (skew) field F and related topics, like products of Grassmann spaces, Schubert spaces, or flag spaces.


Generalised Schubert Varieties
Italian Head of Project: Alessandro Bichara Dipartimento di Metodi e Modelli Matematici, Università di Roma "La Sapienza"
Austrian Head of Project: Hans Havlicek
Fourth Protocol of Scientific and Technological Cooperation between Italy and Austria (project no. 10).

Hermann Graßmann

Affine Spaces from Grassmann Spaces
If the ground field F is commutative then the Grassmann space G(3,1) admits a well known point model, namely the Klein quadric lying in a 5dimensional projective space. Via a stereographic projection a proper subset of the Klein quadric can be mapped bijectively onto a 4dimensional affine space. So all lines that are skew to a fixed line become the points of a 4dimensional affine space.
If the ground field F is a proper skew field then there is no point model like the Klein quadric. However, it is still possible to endow the set of all lines that are skew to a fixed line with the structure of an affine space. Thus the construction from above can be carried out irrespective of the existence of the Klein quadric. From a local point of view this affine space may serve as a substitute for the Klein quadric.
Similar constructions are possible for higher dimensions.


Publications

H. Havlicek: Zur Theorie linearer Abbildungen I, J. Geometry 16 (1981), 152167.

H. Havlicek: Zur Theorie linearer Abbildungen II, J. Geometry 16 (1981), 168180.

H. Havlicek: Liniengeometrische Modelle affiner 4Räume, In: N.K. Stephanidis (ed.): Proceedings of the Congress of Geometry  Thessaloniki 1987, 4753.

H. Havlicek: On sets of lines corresponding to affine spaces In: A. Barlotti et al. (eds.): Combinatorics '88, Proceedings of the International Conference on Incidence Geometries and Combinatorial Structures (Ravello, May 1988), vol. 1, Research and Lecture Notes in Mathematics, Mediterranean Press 1991, 449457.
Preprint (PDF)

H. Havlicek: Baer subspaces within Segre manifolds, Results Math. 23 (1993), 322329.
Preprint (PDF)

H. Havlicek: On isomorphisms of Grassmann spaces, Mitt. Math. Ges. Hamburg 14 (1995), 117120.
Preprint (PDF)

A. Blunck, H. Havlicek: Affine spaces within projective spaces, Resultate Math. 36 (1999), 237251.
Preprint (PDF)
 H. Havlicek, K. List, and C. Zanella: On automorphisms of flag spaces, Linear Multilinear Algebra 50 (2002), 241251.
Preprint (PDF)
 A. Bichara, H. Havlicek, and C. Zanella: On linear morphisms of product spaces, Discrete Math. 267 (2003), 3543.
Preprint (PDF)
 H. Havlicek, C. Zanella: On embedded products of Grassmannians, Discrete Math. 267 (2003), 153158.
Preprint (PDF)
 A. Blunck and H. Havlicek: On bijections that preserve complementarity of subspaces, Discrete Math. 301 (2005), 4656.
Preprint (PDF)
 H. Havlicek, V. Pambuccian: On the axiomatics of projective and affine geometry in terms of line intersection, Resultate Math. 45 (2004), 3544.
Preprint (PDF)
 H. Havlicek and M. Pankov: Transformations on the product of Grassmann spaces, Demonstratio Math. 38 (2005), 675688.
Preprint (PDF)


Quick Links
Andrea Blunck
Hans Havlicek
Klaus List
Spreads and Parallelisms
Line Geometry
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External Links
Corrado Zanella, University of Padua
