The flexibility of $m$-suspensions constructed via a local slice

Abstract

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 $\operatorname{Susp}(X, f, 1, k_2, \dots, k_m)$.