Zur Giessener Elektronischen Bibliothek
Martin Bokler

Blockierende Mengen in endlichen projektiven Räumen

Abstract

In this work minimal $ m$-blocking sets of cardinality at most $ \theta_{m}+\theta_{m-1}\sqrt{q}$ in projective spaces $ PG(n,q)$ of square order $ q$, $ q\geq 9$, are characterized to be $ (t, 2(m-t-1))$-cones for some $ t$ with $ \max \{-1,2m-n-1\}\leq t \leq m-1$. In particular we will find the smallest $ m$-blocking sets that generate the whole space $ PG(n,q)$ for $ 2m\geq n \geq m$. Furthermore in projective spaces $ PG(n,q)$ of arbitrary order $ q \not= 2$ new lower bounds for the cardinality of minimal $ m$-blocking sets are determined. Let $ r_2(q)$ be the number such that $ q+r_2(q)+1$ is the cardinality of the smallest non-trivial line-blocking set in a plane of order $ q$. If $ B$ is a minimal $ m$-blocking set in $ PG(n,q)$ that contains at most $ q^m+q^{m-1}+\dots+q+1+r_2(q)\cdot(\sum^{m-1}_{i=2m-n'}q^i)$ points for an integer $ n'$ satisfying $ m\leq n'\leq 2m$, then the dimension of $ \langle B \rangle$ is at most $ n'$. If the dimension of $ \langle B \rangle$ is $ n'$, then the following holds. The cardinality of $ B$ equals $ q^m+q^{m-1}+\dots+q+1+r_2(q)\cdot(\sum^{m-1}_{i=2m-n'}q^i)$. For $ n'=m$ the set $ B$ is an $ m$-dimensional subspace and for $ n'=m+1$ the set $ B$ is a cone with an $ (m-2)$-dimensional vertex over a non-trivial line-blocking set of cardinality $ q+r_2(q)+1$ in a plane skew to the vertex. For $ n'>m+1$ and $ q$ not a prime the number $ q$ is a square and $ B$ is a Baer cone. If $ q$ is odd and $ \vert B\vert<q^m+q^{m-1}+\dots+q+1+r_2(q)\cdot(q^{m-1}+q^{m-2})$, it follows from this result that the subspace generated by $ B$ has dimension at most $ m+1$. Furthermore we prove that in this case, if $ \vert B\vert<\frac{3}{2}(q^{m}+1)$, then $ B$ is an $ m$-dimensional subspace or a cone with an$ (m-2)$-dimensional vertex over a non-trivial line-blocking set of cardinality $ q+r_2(q)+1$ in a plane skew to the vertex. For $ q=p^{3h}$, $ p\geq 7$ and $ q$ not a square we show this assertion for $ \vert B\vert\leq q^m+q^{m-1}+\dots+q+1+q^{2/3}\cdot (q^{m-1}+\dots+1)$.

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