三次曲面的行列式表示的模空间与根系的不变量理论
The Moduli Space of Determinantal Representations of Cubic Surfaces and Invariant Theory of Root Systems
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中文总结 AI 辅助
该研究构建框架研究复射影三次曲面平展族的线性行列式表示模空间，针对E₆型有理二重点与Ẽ₆型简单椭圆奇点两种情形，分别建立相关同构结果，规避了局部单值群无限带来的框架失效问题。
中文摘要 AI 辅助
我们提出一个框架，用于研究复射影三次曲面的平展族在奇点边界上的线性行列式表示的模空间。首先，我们处理经典形变理论，其中曲面S₀具有E₆型有理二重点（RDP）。通过在全局参数切片的有限分歧伽罗瓦覆盖上建立全局同时解消，我们将相对模空间H_N实现为对角Weyl商空间(h×R)/W(E₆)。其次，我们将该分类扩展到临界边界构型，其中S₀包含唯一的孤立简单椭圆奇点，类型为Ẽ₆。由于局部单值群变为无限，经典同时解消框架完全失效。我们通过构造源自仿射Weyl群的Looijenga不变量代数滤过的几何替代物Z来规避这一障碍。我们还证明，向量丛Z在结构椭圆曲线E上分解为线丛的直和，其次数与E₆最高根的负Coxeter标记明确匹配。利用Riemann延拓定理和椭圆曲线上的向量丛刚性，我们确立同构结果，证明代表半通用族的模空间H̄在整体上同构于Z的全空间。
英文摘要
We present a framework for studying the moduli space of linear determinantal representations for flat families of complex projective cubic surfaces across singularity boundaries. First, we deal with the classical deformation theory where a surface S_0 possesses a rational double point (RDP) of type E_6. By establishing a global simultaneous resolution over a finite ramified Galois covering of the global parameter slice, we realize the relative moduli space H_N as a diagonal Weyl quotient (h x R)/W(E_6). Second, we extend this classification to the critical boundary configuration where S0 contains a unique isolated simple elliptic singularity of type E~_6. Because the local monodromy group becomes infinite, the classical simultaneous resolution framework completely breaks down. We bypass this obstruction by constructing a geometric substitute Z derived from the filtration of Looijenga's invariant algebra of affine Weyl groups. We also show that the vector bundle Z decomposes into a direct sum of line bundles over the structural elliptic curve E, with degrees explicitly matching the negative Coxeter marks of the highest root of E_6. Utilizing Riemann's extension theorem and vector bundle rigidity over elliptic curves, we settle our isomorphism result, proving that the moduli space H-bar representing the semiuniversal family is globally isomorphic to the total space of Z.
