2014/01/18 by Yasutaka Shibata, Shibata, Yasutaka
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1401.4525
openalex publication_date 2014/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the GIT compactification ℙ(Sym3ℂ7)//SL(7) of the moduli space of cubic fivefolds X⊂ℙ6 and give an explicit description of its strictly semistable boundary. We construct closed-orbit normal forms and show that the strictly semistable locus has exactly 21 irreducible components. For a general polystable member in each component we determine Sing(X): besides finitely many isolated points, the singular locus may contain a one-dimensional component which is a line, a smooth conic, a (2,2) complete-intersection curve, or an elliptic quartic. The isolated boundary singularities are quasi-homogeneous and fall into precisely six analytic types; we single them out as extremal cubic fivefold singularities. Using Park's framework relating minimal exponents to hypersurface GIT stability, we prove that each boundary component is characterized by the critical value α=(n+1)/d=7/3 for (n,d)=(6,3), both locally for the isolated extremal types and globally for a general member of the component. Finally, via Kirwan's stratification we compute the codimension-one wall-adjacency relation among the 21 components, obtaining an explicit graph with 21 vertices and 56 edges (in particular, with no isolated vertices).