Models of hypersurfaces and Bruhat-Tits buildings

We propose a new approach to constructing semistable integral models of hypersurfaces over a discretely valued complete field K. For each stable hypersurface X over K we define a continuous stability function on the Bruhat-Tits building of PGL_{n+1}(K); its global minima control semistable hypersurface models after finite extensions of K. In particular, in residue characteristic zero the problem reduces to minimizing this function on the original building and then passing to a finite extension that turns a rational minimizer into a vertex. This extends work of Kollar and of Elsenhans-Stoll on minimal hypersurface models. We implement the resulting strategy for plane curves over p-adic number fields. In a follow-up article we use our results to compute the semistable reduction of smooth plane quartics.

💡 Research Summary

The paper introduces a novel method for constructing semistable integral models of hypersurfaces over a discretely valued complete field K. Let X⊂ℙⁿ_K be a hypersurface defined by a homogeneous polynomial F of degree d, assumed to be GIT‑stable (in particular smooth hypersurfaces are stable). Classical semistable reduction theorems guarantee the existence of a finite extension L/K after which X admits a semistable model, but no practical algorithm is known beyond the cases of binary forms (n=1) and plane cubics (n=2, d=3).

The authors observe that the set of isomorphism classes of hypersurface models of X is naturally identified with the set of vertices of the Bruhat–Tits building B_K of the group PGL_{n+1}(K). A vertex corresponds to a homothety class of full O_K‑lattices in V^* (the dual of the (n+1)‑dimensional K‑vector space V). Changing the coordinate system of X amounts to moving from one vertex to another.

Using this identification they define a “stability function”
 φ_X : B_K → ℝ,
which on a vertex measures the valuation of a suitable GIT invariant of F (equivalently the Hilbert–Mumford weight of the most destabilising one‑parameter subgroup). The function extends uniquely to the whole metric space B_K; on each apartment (an Euclidean affine space of dimension n) φ_X is piecewise affine, convex, and radially unbounded. Consequently φ_X attains a global minimum m_X, and the minimum locus M_X:= {b∈B_K | φ_X(b)=m_X} always contains at least one rational point (a point with rational barycentric coordinates).

The key results are:

1. Proposition 1.5. For a stable hypersurface X, φ_X exists with the properties above; a vertex b corresponds to a semistable model iff b∈M_X, and X is stable iff M_X consists of a single vertex after any finite extension.

2. Theorem 1.7 (residue characteristic 0). If the residue field k has char 0, any rational minimiser b∈B_K(ℚ) can be turned into a vertex after passing to a sufficiently ramified finite extension L/K. The corresponding lattice then yields a semistable model of X_L. Thus in char 0 the whole semistable reduction problem reduces to minimizing φ_X on the original building and a purely formal field extension step.

3. Theorem 1.10 (arbitrary characteristic). Considering all finite extensions L/K, the set of minima {m_{X,L}} has a smallest element. The proof avoids any GIT machinery, relying only on non‑Archimedean analysis and polyhedral geometry. Consequently a semistable model always exists after a suitable extension, even when the residue characteristic is positive.

The authors also introduce the notion of a stable minimiser: a rational point b∈B_K(ℚ) that remains a minimiser after any finite extension. If such a point becomes a vertex in some extension, the associated model is semistable.

From an algorithmic standpoint, the paper reduces the construction of semistable models to two concrete tasks:

• Problem 2.5. Compute φ_X explicitly on vertices (i.e. compute the Hilbert–Mumford weight for each lattice). This is essentially a p‑adic optimisation problem on the coefficients of F.
• Minimisation on B_K. For n=2 (plane curves) the authors solve Problem 2.5 using p‑adic linear algebra and convex geometry, yielding an effective procedure to locate a rational minimiser.

An implementation in SageMath is provided (reference