对称性与可解性：等变枚举问题的伽罗瓦群
Symmetry and solvability: Galois groups of equivariant enumerative problems
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中文总结 AI 辅助
本文研究等变枚举问题的伽罗瓦群，刻画四种单值群关系，并证明在存在对称性时，求解三次曲面直线和平面四次曲线双切线问题可解。
中文摘要 AI 辅助
在19世纪末伽罗瓦理论诞生之初，枚举几何学家关注各种问题的可解性：例如，能否用定义三次曲面的系数的根式来表达其上的直线。根据埃尔米特的工作和哈里斯的著名定理，这等价于确定某个有限平展覆盖的单值群是否可解。该问题以及计算平面四次曲线的双切线的相关问题都是不可解的。当限制在$G$-对称三次曲面的轨迹上时，单值群会下降并可能变为可解。在此背景下，有四种单值群可供考虑：限制经典覆盖得到的单值群，模去射影变换后GIT商上的单值群，以及来自$G$-对称三次曲面上$G$-对称直线的层状覆盖的两种相关“单值群”概念——与非对称情形不同，这些概念各不相同。本文在商层的普遍背景下刻画了这四种单值群概念之间的关系，并利用这些联系计算了光滑三次曲面的所有11种可能自同构群以及光滑平面四次曲线的所有12种可能自同构群的这些单值群。我们证明，在存在任何对称性的情况下，求解三次曲面上的直线或平面四次曲线的双切线的问题都是可解的。
英文摘要
At the genesis of Galois theory at the end of the 19th century, enumerative geometers were concerned with the solvability of various problems: for instance whether one can express lines on a cubic surface in radicals in terms of the coefficients that define it. By work of Hermite and a celebrated theorem of Harris, this is equivalent to determining whether the monodromy group of a certain finite étale cover is solvable. This problem, as well as the related problem of computing bitangents to plane quartics, are both unsolvable. When one restricts to the locus of $G$-symmetric cubic surfaces, monodromy will drop and can become solvable. In this setting there are four flavors of monodromy one can consider: the one from restricting the classical cover, the one over the GIT quotient after modding out by projective transformations, and two related notions of "monodromy" coming from the stacky cover of $G$-symmetric lines on $G$-symmetric cubic surfaces - unlike in the non-symmetric setting these are all different. In this paper we characterize the relationships between all four notions of monodromy in the general setting of quotient stacks, and use these connections to compute all these monodromy groups for all 11 possible automorphism groups of smooth cubic surfaces, as well as for all 12 possible automorphism groups for smooth planar quartics. We demonstrate that, in the presence of any symmetry whatsoever, the problem of solving for lines on cubic surfaces or bitangents to plane quartics is solvable.
