A partial geometry $S$ admitting an abelian Singer group $G$ is called of rigid type if all lines of $S$ have a trivial stabilizer in $G$. In this paper, we show that if a partial geometry of rigid type has fewer than $1000000$ points it must be the Van Lint-Schrijver geometry or be a hypothetical geometry with 1024 or 4096 or 194481 points, which provides evidence that partial geometries of rigid type are very rare. Along the way we also exclude an infinite set of parameters that originally seemed very promising for the construction of partial geometries of rigid type (as it contains the Van Lint-Schrijver parameters as its smallest case and one of the other cases we cannot exclude as the second member of this parameter family). We end the paper with a conjecture on this type of geometries.

中文翻译：

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具有刚性型阿贝尔辛格群的部分几何结果

如果$S$ 的所有线在$G$ 中都有一个平凡的稳定器，则承认一个阿贝尔辛格群$G$ 的部分几何$S$ 被称为刚性类型。在本文中，我们表明，如果刚性类型的部分几何具有少于 $1000000$ 点，则它必须是 Van Lint-Schrijver 几何或具有 1024 或 4096 或 194481 点的假设几何，这提供了刚性部分几何的证据类型非常少见。在此过程中，我们还排除了一组无限的参数，这些参数最初对于构建刚性类型的部分几何形状非常有希望（因为它包含 Van Lint-Schrijver 参数作为其最小情况，而我们不能排除的其他情况之一作为此参数系列的第二个成员）。我们以对这种几何形状的猜想结束论文。