The moduli space of marked cubic surfaces is one of the most classical moduli spaces in algebraic geometry, dating back to the nineteenth‐century work of Cayley and Salmon. Modern interest in was restored in the 1980s by Naruki's explicit construction of a ‐equivariant smooth projective compactification of , and in the 2000s by Hacking, Keel, and Tevelev's construction of the Kollár–Shepherd‐Barron–Alexeev (KSBA) stable pair compactification of as a natural sequence of blowups of . We describe generators for the cones of ‐invariant effective divisors and curves of both and . For Naruki's compactification , we further obtain a complete stable base locus decomposition of the ‐invariant effective cone, and as a consequence find several new ‐equivariant birational models of . Furthermore, we fully describe the log minimal model program for the KSBA compactification , with respect to the divisor , where is the boundary and is the sum of the divisors parameterizing marked cubic surfaces with Eckardt points.

中文翻译：

---

标记立方曲面模空间的 W(E6)$W(E_6)$-不变双有理几何

标记立方曲面的模空间是代数几何中最经典的模空间之一，可以追溯到 19 世纪 Cayley 和 Salmon 的工作。 2000 年代，Hacking、Keel 和 Tevelev 对 的 Kollár-Shepherd-Barron-Alexeev (KSBA) 稳定对紧化的构造使得现代人对 的兴趣在 2000 年代得到了恢复。的自然爆炸序列。我们描述了 不变有效除数锥体以及 和 曲线的生成器。对于 Naruki 的紧致化，我们进一步获得了不变有效锥的完整稳定基轨迹分解，并因此发现了几个新的等变双有理模型。此外，我们充分描述了 KSBA 紧致化的对数最小模型程序，关于除数 ，其中 是边界， 是用 Eckardt 点参数化标记立方表面的除数之和。