The Moduli Space of Determinantal Representations of Cubic Surfaces and Invariant Theory of Root Systems

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.