146 research outputs found
The geometry of secants in embedded polar spaces
AbstractConsider a polar space S weakly embedded in a projective space P. A secant of S is the intersection of the point set of S with a line of P spanned by two non-collinear points of S. The geometry consisting of the points of S and as lines the secants is a so-called Delta space. In this paper we give a characterization of this and some related geometries
K-theory, LQEL manifolds and Severi varieties
We use topological K-theory to study non-singular varieties with quadratic
entry locus. We thus obtain a new proof of Russo's Divisibility Property for
locally quadratic entry locus manifolds. In particular we obtain a K-theoretic
proof of Zak's theorem that the dimension of a Severi variety must be 2, 4, 8
or 16 and so resolve a conjecture of Atiyah and Berndt. We also show how the
same methods applied to dual varieties recover the Landman parity theorem.Comment: 17 pages; added references; minor correction
Tight sets in finite classical polar spaces
We show that every i-tight set in the Hermitian variety H(2r + 1, q) is a union of pairwise disjoint (2r + 1)-dimensional Baer subgeometries PG(2r + 1, root q) and generators of H(2r + 1, q), if q >= 81 is an odd square and i < (q(2/3) - 1)/2. We also show that an i-tight set in the symplectic polar space W(2r + 1, q) is a union of pairwise disjoint generators of W(2r + 1, q), pairs of disjoint r-spaces {Delta,Delta(perpendicular to)}, and (2r + 1)-dimensional Baer subgeometries. For W(2r + 1, q) with r even, pairs of disjoint r-spaces {Delta,Delta(perpendicular to)} cannot occur. The (2r + 1)-dimensional Baer subgeometries in the i-tight set of W(2r + 1, q) are invariant under the symplectic polarity perpendicular to of W(2r + 1, q) or they arise in pairs of disjoint Baer subgeometries corresponding to each other under perpendicular to. This improves previous results where i < q(5/8)/root 2+ 1 was assumed. Generalizing known techniques and using recent results on blocking sets and minihypers, we present an alternative proof of this result and consequently improve the upper bound on i to (q(2/3) - 1)/2. We also apply our results on tight sets to improve a known result on maximal partial spreads in W(2r + 1, q)
On the smooth locus of aligned Hilbert schemes: the k-secant lemma and the general projection theorem
Let X be a smooth, connected, dimension n, quasi-projective variety imbedded
in \PP_N. Consider integers {k_1,...,k_r}, with k_i>0, and the Hilbert Scheme
H_{k_1,...,k_r}(X) of aligned, finite, degree \sum k_i, subschemes of X, with
multiplicities k_i at points x_i (possibly coinciding). The expected dimension
of H_{k_1,...,k_r}(X) is 2N-2+r-(\sum k_i)(N-n). We study the locus of points
where H_{k_1,...,k_r}(X) is not smooth of expected dimension and we prove that
the lines carrying this locus do not fill up \PP_NComment: 17 pages, revised versio
Field reduction and linear sets in finite geometry
Based on the simple and well understood concept of subfields in a finite
field, the technique called `field reduction' has proved to be a very useful
and powerful tool in finite geometry. In this paper we elaborate on this
technique. Field reduction for projective and polar spaces is formalized and
the links with Desarguesian spreads and linear sets are explained in detail.
Recent results and some fundamental ques- tions about linear sets and scattered
spaces are studied. The relevance of field reduction is illustrated by
discussing applications to blocking sets and semifields
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