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Geometry of self-dual flats over a PID on a polarity
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Li-Ping Huang
Published/Copyright:
April 13, 2010
Abstract
Let R be any commutative principal ideal domain (PID), and let n be an integer ≥ 2. Denote by 𝕀n − 1 the set of all (n − 1)-dimensional self-dual flats of a polarity in the projective geometry ℙ(R2n). The geometric character of 𝕀n − 1 is discussed. Two self-dual flats A, B ∈ 𝕀n − 1 are said to be adjacent if dim(A ∩ B) = n − 2. We prove that ϕ : 𝕀n − 1 → 𝕀n − 1 is an adjacency preserving surjection if and only if ϕ is an adjacency preserving bijection in both directions. Chow's theorem on the self-dual flats is extended as follows: If the polarity is the symplectic polarity and ϕ : 𝕀n − 1 → 𝕀n − 1 is an adjacency preserving surjection, then ϕ is induced by a collineation on ℙ(R2n).
Key words.: Projective geometry; principal ideal domain; symmetric matrix; polarity; self-dual flat; adjacency; preserving
Received: 2008-05-29
Revised: 2008-11-28
Revised: 2009-01-06
Published Online: 2010-04-13
Published in Print: 2010-October
© de Gruyter 2010
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Keywords for this article
Projective geometry;
principal ideal domain;
symmetric matrix;
polarity;
self-dual flat;
adjacency;
preserving
Articles in the same Issue
- 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 Santaló 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
