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On Fano schemes of linear spaces of general complete intersections
Archiv der Mathematik ( IF 0.5 ) Pub Date : 2020-09-18 , DOI: 10.1007/s00013-020-01523-7
Francesco Bastianelli , Ciro Ciliberto , Flaminio Flamini , Paola Supino

We consider the Fano scheme $F_k(X)$ of $k$--dimensional linear subspaces contained in a complete intersection $X \subset \mathbb{P}^n$ of multi--degree $\underline{d} = (d_1, \ldots, d_s)$. Our main result is an extension of a result of Riedl and Yang concerning Fano schemes of lines on very general hypersurfaces: we consider the case when $X$ is a very general complete intersection and $\Pi_{i=1}^s d_i > 2$ and we find conditions on $n$, $\underline{d}$ and $k$ under which $F_k(X)$ does not contain either rational or elliptic curves. At the end of the paper, we study the case $\Pi_{i=1}^s d_i = 2$.

中文翻译:

一般完备交点线性空间的Fano格式

我们考虑包含在一个完整的交集 $X \subset \mathbb{P}^n$ 中的 $k$--维线性子空间的 Fano 方案 $F_k(X)$ 的多度 $\underline{d} = ( d_1, \ldots, d_s)$。我们的主要结果是 Riedl 和 Yang 关于非常一般的超曲面上线的 Fano 方案的结果的扩展:我们考虑 $X$ 是非常一般的完全交集和 $\Pi_{i=1}^s d_i > 2$ 并且我们在 $n$、$\underline{d}$ 和 $k$ 上找到条件,在这些条件下 $F_k(X)$ 不包含有理曲线或椭圆曲线。在论文的最后,我们研究了 $\Pi_{i=1}^s d_i = 2$ 的情况。
更新日期:2020-09-18
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