散在群的广义通用性、特殊点与预解式次数
Generalized Versality, Special Points, and Resolvent Degree for the Sporadic Groups
AI总结：
本文针对散在群，利用其射影表示的不变量理论建立预解式次数上界，并提出弱 $\operatorname{RD}_k^{\leq d}$-通用性以联系簇上特殊点的存在性。
摘要：
预解式次数是衡量代数与几何现象复杂性的不变量，其中也包括有限群的复杂性。迄今为止，有限单群 $G$ 的预解式次数仅在以下情形中得到研究：$G$ 为循环群、交错群、$E_6$、$E_7$ 或 $E_8$ 型 Weyl 群的单因子，或 $\operatorname{PSL}\left(2, \mathbb{F}_7\right)$。本文利用散在群射影表示的不变量理论，建立了其预解式次数的上界。为此，我们引入（弱）$\operatorname{RD}_k^{\leq d}$-通用性概念，并将其与簇上“特殊点”的存在性联系起来。
英文摘要：
Resolvent degree is an invariant measuring the complexity of algebraic and geometric phenomena, including the complexity of finite groups. To date, the resolvent degree of a finite simple group $G$ has only been investigated when $G$ is a cylic group; an alternating group; a simple factor of a Weyl group of type $E_6$, $E_7$, or $E_8$; or $\operatorname{PSL}\left(2, \mathbb{F}_7\right)$. In this paper, we establish upper bounds on the resolvent degrees of the sporadic groups by using the invariant theory of their projective representations. To do so, we introduce the notion of (weak) $\operatorname{RD}_k^{\leq d}$-versality, which we connect to the existence of "special points" on varieties.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
