AI 中文总结
研究通过局部切片构造的\(m\)-悬架，确定其一般灵活和灵活的充分条件，对于灵活的\(X\)，给出构造局部切片\(f\)的方法，以保证悬架\(\operatorname{Susp}(X, f, 1, k_2, \dots, k_m)\)的灵活性。
AI 中文摘要
我们称\(\operatorname{Susp}(X, f, k_1, \dots, k_m) = \mathbb{V}(y_1^{k_1} \dots y_m^{k_m} - f(x)) \subset X \times \mathbb{A}^m\)为仿射簇\(X\)上通过局部切片\(f(x) \in \mathbb{K}[X]\)构造的\(m\)-悬架，若\(X\)上存在局部幂零导子\(\delta\)使得\(\delta (f) \neq 0\)，\(\delta^2 (f) = 0\)。本文确定了此类簇一般灵活和灵活的充分条件。此外，对于灵活的\(X\)，我们提出了一种构造局部切片\(f\)的方法，以保证悬架\(\operatorname{Susp}(X, f, 1, k_2, \dots, k_m)\)的灵活性。
英文摘要
We refer to the variety $\operatorname{Susp}(X, f, k_1, \dots, k_m) = \mathbb{V}(y_1^{k_1} \dots y_m^{k_m} - f(x)) \subset X \times \mathbb{A}^m$ as an $m$-suspension over affine variety $X$, constructed via a local slice $f(x) \in \mathbb{K}[X]$, if there exists a locally nilpotent derivation $δ$ on $X$ such that $δ(f) \neq 0, δ^2 (f) = 0$. In this paper, we determine the sufficient conditions under which such a variety is generically flexible and those under which it is flexible. Furthermore, for a flexible $X$ we propose a construction of a local slice $f$ that guarantees the flexibility of the suspension $\operatorname{Susp}(X, f, 1, k_2, \dots, k_m)$.
