The GIT Boundary of Quintic Threefolds

Abstract

We determine the strictly semistable boundary of the GIT compactification of quintic threefolds for the natural $\mathrm{SL}(5)$-action on $\mathbb{P}(\mathrm{Sym}^5\mathbb{C}^5)$. 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maximal supports with thirty-eight irreducible boundary components by overlooking S-equivalence; that claim is withdrawn and replaced by the twenty-one-component classification proved here.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 $\mathbb{P}^4$.