初等伪帽和良蛋的刻画
Characterisations of elementary pseudo-caps and good eggs
摘要：
在这篇文章中，我们使用了Desketesian传播理论来研究好鸡蛋。结果表明，一个含有$\mathrm{PG}(4n-1, q)$、$q$奇数、两个好元素的卵是基本的。通过一个简短的组合论证，我们证明了一个类似的陈述适用于大的伪上限，在奇数和偶数的特征。作为一个推论，这改进和扩展了Thas，Thas和Van Maldeghem（2006）的结果，其中一个鸡蛋至少需要4个好的元素才能得到相同的结论。我们改写这个推论，以获得山雀广义四边形$T_3(\mathcal{O})$的特征。Lavrauw（2005）将奇数特征的基本蛋描述为那些包含至少5个蛋元素的空间的好蛋，但不包含好元素。我们提供了一个适应这种特性弱鸡蛋奇，甚至特征。作为推论，我们得到了Lavrauw定理的一个直接的几何证明.
英文摘要：
In this note, we use the theory of Desarguesian spreads to investigate good eggs. Thas showed that an egg in $\mathrm{PG}(4n-1, q)$, $q$ odd, with two good elements is elementary. By a short combinatorial argument, we show that a similar statement holds for large pseudo-caps, in odd and even characteristic. As a corollary, this improves and extends the result of Thas, Thas and Van Maldeghem (2006) where one needs at least 4 good elements of an egg in even characteristic to obtain the same conclusion. We rephrase this corollary to obtain a characterisation of the generalised quadrangle $T_3(\mathcal{O})$ of Tits. Lavrauw (2005) characterises elementary eggs in odd characteristic as those good eggs containing a space that contains at least 5 elements of the egg, but not the good element. We provide an adaptation of this characterisation for weak eggs in odd and even characteristic. As a corollary, we obtain a direct geometric proof for the theorem of Lavrauw.
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