论文标题
与投影品种交集的Lefschetz定理
A Lefschetz theorem for intersections with projective varieties
论文作者
论文摘要
一个经典的Lefschetz超平面定理的一个版本指出，对于$ u \ subset \ mathbb p^n $，平滑的准标记尺寸至少$ 2 $，而$ h \ cap u $是一般超平面部分，是étalethy thy thy of thy thy thy thy of thy thy thy thy of thy thy thy thy thy thy thy thy thy thy thy thy thy thy thy thy thy thyétaleundamental Guttainal undamaltent $π_1（H \ iefter cap u）$ rirforivire \ rightarr。 We prove a generalization, replacing the hyperplane by a general $\operatorname{PGL}_{n+1}$-translate of an arbitrary projective variety: If $U \subset \mathbb P^n$ is a normal quasi-projective variety, $X$ is a geometrically irreducible projective variety of dimension at least $n + 1 - \dim U$, and $Y$ is a常规$ \ operatoRatorname {pgl} _ {n+1} $ - $ x $的翻译，然后地图$π_1（y \ cap u）\ rightarrowπ_1（u）$是冲销的。
One version of the classical Lefschetz hyperplane theorem states that for $U \subset \mathbb P^n$ a smooth quasi-projective variety of dimension at least $2$, and $H \cap U$ a general hyperplane section, the resulting map on étale fundamental groups $π_1(H \cap U) \rightarrow π_1(U)$ is surjective. We prove a generalization, replacing the hyperplane by a general $\operatorname{PGL}_{n+1}$-translate of an arbitrary projective variety: If $U \subset \mathbb P^n$ is a normal quasi-projective variety, $X$ is a geometrically irreducible projective variety of dimension at least $n + 1 - \dim U$, and $Y$ is a general $\operatorname{PGL}_{n+1}$-translate of $X$, then the map $π_1(Y \cap U) \rightarrow π_1(U)$ is surjective.
