The GIT Boundary of Quintic Threefolds (Announcement of Results)

TL;DR

This paper explicitly describes the boundary of the GIT moduli space of quintic threefolds, classifies boundary components, and analyzes their singularities and stability properties.

Contribution

It provides a detailed classification of the 38 boundary components, constructs normal forms, and analyzes singularities and stability criteria for quintic threefolds.

Findings

Identified 38 boundary components in the GIT moduli space.

Constructed normal forms for general polystable representatives.

Determined all boundary singularities are quasi-homogeneous with minimal exponent 1.

Abstract

We announce an explicit description of the strictly semistable boundary of the GIT moduli space of quintic threefolds. For the natural action of \(\mathrm{SL}(5)\) on \(\mathbb P(\mathrm{Sym}^5\mathbb C^5)\), we classify the 38 boundary components arising from maximal strictly semistable supports and construct closed-orbit normal forms for the general polystable representative in each component. We also determine the singular loci of these general representatives and compute their local and global minimal exponents. The isolated boundary singularities are quasi-homogeneous and fall into eleven analytic types, all with local minimal exponent equal to \(1\). Consequently, the global minimal exponent of a general closed-orbit representative in every boundary component is \(1=(4+1)/5\), the critical value in the stability criterion for quintic hypersurfaces in \(\mathbb P^4\). We further announce the pairwise non-inclusion of the 38 quotient-side boundary families and compute the codimension-one wall-adjacency graph, which has 38 vertices and 184 edges, is connected, and has diameter 4. Detailed proofs and complete case-by-case computations will appear in a forthcoming full-length paper.