Affine Tallini Sets and Grassmannians
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Eva Ferrara Dentice
Abstract
In 1982/1983 A. Bichara and F. Mazzocca characterized the Grassmann space Gr(h, 𝔸) of index h of an affine space 𝔸 of dimension at least 3 over a skew-field K by means of the intersection properties of the three disjoint families of maximal singular subspaces of Gr(h, 𝔸) and, till now, their result represents the only known characterization of Gr(h, 𝔸). If K is a commutative field and 𝔸 has finite dimension m, then the image 
under the well known PlĂĽcker morphism 
is a proper subset 
of PG(M, K), 
, called the affine Grassmannian of the h-subspaces of 𝔸. The aim of this paper is to introduce the notion of Affine Tallini Set and provide a natural and intrinsic characterization of from the point-line geometry point of view. More precisely, we prove that if a projective space over a skew-field K contains an Affine Tallini Set Δ satisfying suitable axioms on “perp” of lines, then the skew-field K is forced to be a commutative field and Δ is an affine Grassmannian, up to projections. Furthermore, several results concerning Affine Tallini Sets are stated and proved.
© de Gruyter 2010
Artikel in diesem Heft
- A combinatorial approach to Alexander–Hirschowitz's Theorem based on toric degenerations
- Triply periodic minimal surfaces which converge to the Hoffman–Wohlgemuth example
- Total absolute horospherical curvature of submanifolds in hyperbolic space
- The case of equality for an inverse SantalĂł functional inequality
- Circle configurations in strictly convex normed planes
- Rank four vector bundles without theta divisor over a curve of genus two
- Affine Tallini Sets and Grassmannians
- Geometry of self-dual flats over a PID on a polarity
- Logarithmic comparison theorem versus Gauss–Manin system for isolated singularities
- A kinematic formula for the total absolute curvature of intersections
- Reducible Veronese surfaces
- Effective non-vanishing conjectures for projective threefolds
Artikel in diesem Heft
- A combinatorial approach to Alexander–Hirschowitz's Theorem based on toric degenerations
- Triply periodic minimal surfaces which converge to the Hoffman–Wohlgemuth example
- Total absolute horospherical curvature of submanifolds in hyperbolic space
- The case of equality for an inverse SantalĂł functional inequality
- Circle configurations in strictly convex normed planes
- Rank four vector bundles without theta divisor over a curve of genus two
- Affine Tallini Sets and Grassmannians
- Geometry of self-dual flats over a PID on a polarity
- Logarithmic comparison theorem versus Gauss–Manin system for isolated singularities
- A kinematic formula for the total absolute curvature of intersections
- Reducible Veronese surfaces
- Effective non-vanishing conjectures for projective threefolds
