arXiv math/04061952004-06-09math.AGmath.LO

一个非标准Bezout定理

A Non-Standard Bezout Theorem

Tristram de Piro

摘要：

本文给出Bezout定理的一个非标准类比。这是通过表明，在所有的特征，Zapriki重数的概念符合交重数，当我们考虑的投影度d和度e曲线的完整的家庭。结果是特别有趣的，因为它保持甚至当我们考虑交点在奇点的曲线或当曲线包含非约化组件。证明也提供了动机的事实，切线是一个可定义的关系，家庭的曲线内的非线性1维Zebriki结构X。这是一个至关重要的组成部分，在未发表的工作，由Peterzil和Zilber，任何这样的Zagliki结构解释一个纯代数闭域L与X作为一个可定义的有限覆盖。

英文摘要：

This paper provides a non-standard analogue of Bezout's theorem. This is acheived by showing that, in all characteristics, the notion of Zariski multiplicity coincides with intersection multiplicity when we consider the full families of projective degree d and degree e curves. The result is particularly interesting in that it holds even when we consider intersections at singular points of curves or when the curves contain non-reduced components. The proof also provides motivation for the fact that tangency is a definable relation for families of curves inside a non-linear 1-dimensional Zariski structure X. This is a crucial ingredient in unpublished work by Peterzil and Zilber that any such Zariski structure interprets a pure algebraically closed field L with X as a definable finite cover.

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