Boundary of the moduli space of stable cubic fivefolds
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arXiv
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| Format: | Preprint |
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2014
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| _version_ | 1866915825974247424 |
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| author | Shibata, Yasutaka |
| author_facet | Shibata, Yasutaka |
| contents | We study the GIT compactification $\mathbb{P}(\mathrm{Sym}^3\mathbb{C}^7)//\mathrm{SL}(7)$ of the moduli space of cubic fivefolds $X\subset\mathbb{P}^6$ 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 $\mathrm{Sing}(X)$: 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1401_4525 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Boundary of the moduli space of stable cubic fivefolds Shibata, Yasutaka Algebraic Geometry We study the GIT compactification $\mathbb{P}(\mathrm{Sym}^3\mathbb{C}^7)//\mathrm{SL}(7)$ of the moduli space of cubic fivefolds $X\subset\mathbb{P}^6$ 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 $\mathrm{Sing}(X)$: 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). |
| title | Boundary of the moduli space of stable cubic fivefolds |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/1401.4525 |