论文标题
在最小的集合中非常宽敞
Very ampleness in strongly minimal sets
论文作者
论文摘要
受Zariski几何形状的极大吸引力的启发，我们介绍并研究了任何非常最小的集合中非常宽敞的平面曲线家族的概念，以及相应的非常最小的集合的概念（以这种家族的确定性为特征）。我们显示各种基本属性；例如，任何强烈的最小设置内置的代数封闭场的内部设置都非常足够，并且任何非常极端的非常小的固定设置非正交设置，而不是强烈的最小套件$ y $ y $内部为$ y $。然后，我们将这些结果与Zilber的限制性三分法一起应用来表征这些结构$ \ MATHCAL M =（m，\ dots）$在代数封闭的字段中解释的$ \ Mathcal M =（m，\ dots）$，从而恢复了所有可构造的$ m $ powers集。接下来，我们表明非常最小的集合可以承认所有维度的平面曲线的家庭家庭，并用它来表征可定义的假旋转器。最后，我们表明可分配的最小群体非常足够，并且推断出 - 回答了G. Martin的一个旧问题 - 在纯代数封闭的字段中，$ k $ $ k $在$（k，+，\ cdot）$和$（k，\ cdot）$之间没有减少。
Inspired by very ampleness of Zariski Geometries, we introduce and study the notion of a very ample family of plane curves in any strongly minimal set, and the corresponding notion of a very ample strongly minimal set (characterized by the definability of such a family). We show various basic properties; for example, any strongly minimal set internal to an expansion of an algebraically closed field is very ample, and any very ample strongly minimal set non-orthogonal to a strongly minimal set $Y$ is internal to $Y$. We then apply these results with Zilber's restricted trichotomy to characterize using very ampleness those structures $\mathcal M=(M,\dots)$ interpreted in an algebraically closed field which recover all constructible subsets of powers of $M$. Next we show that very ample strongly minimal sets admit very ample families of plane curves of all dimensions, and use this to characterize very ampleness in terms of definable pseudoplanes. Finally, we show that divisible strongly minimal groups are very ample, and deduce -- answering an old question of G. Martin -- that in a pure algebraically closed field, $K$ there are no reducts between $(K,+,\cdot)$ and $(K, \cdot)$.
