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The line geometry of resonance varieties
"... Let R 1 (A, R) be the degreeone resonance variety over a field R of a hyperplane arrangement A. We give a geometric description of R 1 (A, R) in terms of projective line complexes. The projective image of R 1 (A, R) is a union of ruled varieties, parametrized by neighborly partitions of subarrangem ..."
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Let R 1 (A, R) be the degreeone resonance variety over a field R of a hyperplane arrangement A. We give a geometric description of R 1 (A, R) in terms of projective line complexes. The projective image of R 1 (A, R) is a union of ruled varieties, parametrized by neighborly partitions of subarrangements of A. The underlying line complexes are intersections of special Schubert varieties, easily described in terms of the corresponding partition. We generalize the definition and decomposition of R 1 (A, R) to arbitrary commutative rings, and point out the anomalies that arise. In general the decomposition is parametrized by neighborly graphs, which need not induce neighborly partitions of subarrangements of A. We use this approach to show that the resonance variety of the Hessian arrangement over a field of characteristic three has a nonlinear component, a cubic threefold with interesting line structure. This answers a question of A. Suciu. We show that Suciu’s deleted B3 arrangement has resonance
The Discovery Of My Completeness Proofs
 Bulletin of Symbolic Logic
, 1996
"... This paper deals with aspects of my doctoral dissertation 1 ..."
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This paper deals with aspects of my doctoral dissertation 1
Henkin’s Method and the Completeness Theorem
"... Let A be a firstorder alphabet and L be the firstorder logic in the alphabet A. For a sentence ϕ in the alphabet A, we will use the standard notation “ ⊢ ϕ ” for ϕ is provable in L (that is, ϕ is derivable from the axioms of L by the use of the inference rules of L); and “ = ϕ” for ϕ is valid (th ..."
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Let A be a firstorder alphabet and L be the firstorder logic in the alphabet A. For a sentence ϕ in the alphabet A, we will use the standard notation “ ⊢ ϕ ” for ϕ is provable in L (that is, ϕ is derivable from the axioms of L by the use of the inference rules of L); and “ = ϕ” for ϕ is valid (that is, ϕ is satisfied in every interpretation of L). The soundness theorem for
On codes generated from quadratic surfaces in PG(3, q)
, 2004
"... We construct two families of lowdensity paritycheck codes using pointline incidence structures in PG(3, q). The selection of lines for each structure relies on the geometry of the two classical quadratic surfaces in PG(3, q), the hyperbolic quadric and the elliptic quadric. ..."
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We construct two families of lowdensity paritycheck codes using pointline incidence structures in PG(3, q). The selection of lines for each structure relies on the geometry of the two classical quadratic surfaces in PG(3, q), the hyperbolic quadric and the elliptic quadric.
Transformations of Grassmann spaces
, 2004
"... This is a current version of a part of the book “Transformations of Grassmann spaces”. We study transformations of Grassmann spaces preserving certain geometrical constructions (related with buildings). The next part will be devoted to Grassmann spaces associated with polar spaces. ..."
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This is a current version of a part of the book “Transformations of Grassmann spaces”. We study transformations of Grassmann spaces preserving certain geometrical constructions (related with buildings). The next part will be devoted to Grassmann spaces associated with polar spaces.
I dedicate this paper to my wife Inna and my daugther SabinaStefany. ON TITS BUILDINGS OF TYPE An
, 2003
"... Abstract. Let P and P ′ be projective spaces having the same dimension, this dimension is denoted by n assumed to be finite. Denote by F and F ′ the sets of maximal flags of the spaces P and P ′ , respectively. A subset of F (F ′ ) is said to be an apartment if it is the intersection of F (F ′ ) and ..."
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Abstract. Let P and P ′ be projective spaces having the same dimension, this dimension is denoted by n assumed to be finite. Denote by F and F ′ the sets of maximal flags of the spaces P and P ′ , respectively. A subset of F (F ′ ) is said to be an apartment if it is the intersection of F (F ′ ) and an apartment of the Anbuilding associated with the projective space P (P ′). We show that any mapping f: F → F ′ sending apartments to apartments is induced by a strong embedding of P to P ′ or to the dual space P ′ ∗. Moreover this embedding is a collineation if f is surjective. 1.