The GIT Boundary of Quintic Threefolds
Abstract
We determine the strictly semistable boundary of the GIT compactification of quintic threefolds for the natural SL(5)-action on ℙ(Sym5ℂ5). Using the Hilbert--Mumford criterion and exact integer linear algebra, we enumerate thirty-eight maximal strictly semistable monomial supports up to coordinate permutation. After taking S-equivalence into account, their images in the GIT quotient reduce to twenty-one families, and we prove that these are precisely the irreducible components of the GIT boundary, with dimensions ranging from $1$ to $24$. Version 1 incorrectly identified the thirty-eight maximal supports with thirty-eight irreducible boundary components by overlooking S-equivalence; that claim is withdrawn and replaced by the twenty-one-component classification proved here. Four components recover Lakhani's first-level families and seventeen are additional components. For each component we construct a general polystable normal form and determine its generic connected projective stabilizer. Pairwise nonconjugacy of the resulting one-dimensional tori, together with Luna's etale slice theorem, excludes containments between distinct components. We also describe the singularities of general closed-orbit representatives. Their isolated singularities fall into exactly eleven weighted-homogeneous local families whose normalized weights occur in Yonemura's list, and every component has global minimal exponent $1$, the critical value (n+1)/d for quintic hypersurfaces in ℙ4.
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BibTeX
@misc{shibata2026boundary,
title = {{The GIT Boundary of Quintic Threefolds}},
author = {Yasutaka Shibata},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.00309},
eprint = {2605.00309},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.00309},
}
RIS
TY - UNPB
TI - The GIT Boundary of Quintic Threefolds
AU - Yasutaka Shibata
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2605.00309
UR - https://arxiv.org/abs/2605.00309
AB - We determine the strictly semistable boundary of the GIT compactification of quintic threefolds for the natural SL(5)-action on ℙ(Sym^5ℂ^5). Using the Hilbert--Mumford criterion and exact integer linear algebra, we enumerate thirty-eight maximal strictly semistable monomial supports up to coordinate permutation. After taking S-equivalence into account, their images in the GIT quotient reduce to twenty-one families, and we prove that these are precisely the irreducible components of the GIT boundary, with dimensions ranging from $1$ to $24$. Version 1 incorrectly identified the thirty-eight maximal supports with thirty-eight irreducible boundary components by overlooking S-equivalence; that claim is withdrawn and replaced by the twenty-one-component classification proved here. Four components recover Lakhani's first-level families and seventeen are additional components. For each component we construct a general polystable normal form and determine its generic connected projective stabilizer. Pairwise nonconjugacy of the resulting one-dimensional tori, together with Luna's etale slice theorem, excludes containments between distinct components. We also describe the singularities of general closed-orbit representatives. Their isolated singularities fall into exactly eleven weighted-homogeneous local families whose normalized weights occur in Yonemura's list, and every component has global minimal exponent $1$, the critical value (n+1)/d for quintic hypersurfaces in ℙ^4.
ER -