МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2008 MATHEMATICS AND EDUCATION IN MATHEMATICS, 2008 Proceedings of the Thirty Seventh Spring Conference of the Union of Bulgarian Mathematicians Borovetz, April 2–6, 2008 CLASSIFICATION OF THE 2-SPREADS OF PG(5, 2)
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چکیده
1.1 Projective spaces and spreads. A projective space is a geometry consisting of a set of points and a set of lines, where each line is a subset of the point set, such that the following axioms hold: • Any two points are on exactly one line. • Let A, B, C, D be four distinct points no three of which are collinear. If the lines AB and CD intersect each other, then the lines AD and BC also intersect each other. • Any line has at least 3 points. Let V be a vector space of dimension d + 1 over the division ring F . The geometry P (V ) that has as its points the 1-dimensional subspaces of V and as its lines the 2dimensional subspaces of V , is a projective space. Any projective space that is not a projective plane is isomorphic to some P (V ), which is also denoted by PG(d, F ). If F is a finite field with q elements, then the notation PG(d, q) is used, where d is called dimension, and q order of the projective space, and any line has q + 1 points. An automorphism of PG(d, q) is a bijective map of the point set that preserves collinearity, i.e. maps the lines into lines. A linear subspace of a projective space is a set U of points such that if A, B ∈ U , then any point on the line AB is contained in U . Any subspace together with the lines contained in it, is a projective space. For two lines A and B of PG(d, q), denote by 〈A, B〉 the subspace of smallest dimension containing them. A t-spread in PG(d, q) is a set S of t-dimensional subspaces such that any point of the geometry is on exactly one element of S.
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تاریخ انتشار 2008