Fulton-MacPherson Compactifications

Updated 13 January 2026

Fulton–MacPherson compactifications are a canonical method that compactifies configuration spaces via iterated blow-ups of all diagonals, yielding smooth and projective completions.
They feature a clear boundary stratification governed by the combinatorics of point collisions, often interpreted through nested set and tree structures.
Extensions to weighted, logarithmic, and operadic settings enhance their applicability in algebraic geometry, moduli theory, and topological field theories.

The Fulton–MacPherson compactification is a canonical and highly structured compactification of the configuration space of $n$ ordered points on a smooth algebraic variety or manifold. It provides a smooth, projective (or, in the topological and real-analytic setting, a manifold-with-corners) completion of the complement of all diagonals in $X^n$ , with boundary stratification governed by the combinatorics of point collisions. These spaces, their weighted and logarithmic analogues, and their operadic structures, have become central in algebraic geometry, topology, moduli theory, operad theory, and enumerative geometry.

1. Construction of the Fulton–MacPherson Compactification

Let $X$ be a smooth algebraic variety (or manifold), $n \ge 1$ . The open configuration space of $n$ ordered, distinct points is

$\mathrm{Conf}_n(X) = X^n \setminus \bigcup_{1 \leq i < j \leq n} \Delta_{ij},$

where each diagonal $\Delta_{ij} = \{(x_1,\ldots,x_n) \mid x_i = x_j\}$ parametrizes collisions of points.

Fulton–MacPherson compactification $X[n]$ is defined as a smooth, projective variety (or proper Deligne–Mumford stack in the unordered or relative setting) together with a birational morphism

$f_n: X[n] \to X^n$

which is an isomorphism over $\mathrm{Conf}_n(X)$ , with exceptional locus over the deeper diagonals. The fundamental construction proceeds by iterated blow-ups:

For each subset $S \subset \{1,\dots,n\}$ with $|S| \geq 2$ , the small diagonal $\Delta_S = \{x \mid x_i = x_j,\ \forall i, j \in S\}$ is blown up in $X^n$ or its strict transform, with the blow-up order refined by increasing $|S|$ (i.e., starting with pairwise diagonals).
After $n-1$ such stages, the resulting space $X[n]$ is smooth, contains $\mathrm{Conf}_n(X)$ as a dense open set, and has a boundary stratified by normal-crossing divisors $E_S$ lying above each $\Delta_S$ (Massarenti, 2016).

This construction is a particular case of the wonderful compactification in the sense of De Concini–Procesi–Li, for the building set of all diagonals in $X^n$ (Gallardo et al., 2016, Routis, 2014).

2. Boundary Stratification and Combinatorics

The boundary $\partial X[n] = X[n] \setminus \mathrm{Conf}_n(X)$ is a simple normal-crossing divisor, with irreducible components $E_S$ indexed by subsets $S$ with $|S| \geq 2$ . Each boundary stratum corresponds to a collection of nested or disjoint subsets, interpreted either as a stratification by forests of trees or by set partitions.

More precisely, the nonempty intersection

$\bigcap_{S_j} E_{S_j}$

is nonempty and transverse if and only if the $S_j$ form a nested or disjoint system—combinatorially, this asserts that point collisions are resolved hierarchically, encoding how multiple coincidences may happen in limit configurations. The moduli-theoretic interpretation is via trees where internal vertices parameterize the coinciding subsets and leaves parameterize the surviving points (Massarenti, 2016, Petersen, 2015).

3. Geometric, Functorial, and Moduli Properties

Smoothness and Universality

$X[n]$ is smooth and projective, and the blow-up process preserves these properties at every stage due to smooth and disjoint centers. The exceptional divisors $E_S$ meet transversely, yielding combinatorial control over the boundary.

Functoriality

Every morphism $f: X \to Y$ induces $f[n]: X[n] \to Y[n]$ commuting with the compactification structure. For unordered configurations or for moduli spaces, $X[n]$ admits natural symmetric group $\mathfrak{S}_n$ actions, and further admits forgetful maps $X[n+1]\to X[n]$ and universal families $X[n]^+ \to X[n]$ , encoding the addition of marked points or points on the universal curve/surface (Massarenti, 2016, Nesterov, 14 Jan 2025).

Relationship with Other Moduli Spaces

For $X=\mathbb{P}^1$ (respectively $X$ a genus $g$ curve), $X[n]$ is isomorphic to $\overline{M}_{0,n+1}$ (respectively to the space of rational pointed stable curves or higher genus analogues).
$X[n]$ often arises as a boundary case of weighted or relative compactifications, or as a special fiber in logarithmic degeneration settings (Mok, 21 Mar 2025, Routis, 2014).

4. Weighted, Logarithmic, and Generalized Compactifications

Weighted Compactification

Given weights $w=(w_1,\dots,w_n)$ with $0 < w_i \leq 1$ , the weighted Fulton–MacPherson compactification $X[n,w]$ blows up only those diagonals $\Delta_I$ with $\sum_{i\in I} w_i > 1$ . This includes the classical $X[n]$ as the special case $w_i=1$ (Gallardo et al., 2016, Routis, 2014). Weighted compactifications connect directly with Hassett's weighted spaces of stable rational curves and provide a robust framework for wall-crossing and reduction morphisms.

Logarithmic and Degenerate Compactifications

Logarithmic analogues of Fulton–MacPherson spaces use logarithmic geometry to provide a log-smooth compactification of configuration spaces on pairs $(X,D)$ with normal crossings divisors $D$ , and admit well-behaved degenerations over smooth bases. The boundary structure is controlled by tropical data and the combinatorics of planted forests in tropicalizations, with central fibers admitting birational morphisms to products of log FM spaces on the components of degenerate fibers (Mok, 21 Mar 2025).

Generalized and GIT Quotient Compactifications

GIT quotients of Fulton–MacPherson or weighted FM compactifications, e.g., $X[n,w]/\!/G$ for a linear algebraic group $G$ acting on $X$ , are themselves wonderful compactifications under mild hypotheses on the stability conditions. This framework unifies classical, weighted, affine, and moduli-theoretic compactifications (Gallardo et al., 2016).

5. Operadic Structures and Topological Models

Fulton–MacPherson Operad

In the real-analytic/topological setting, the sequence $\{\mathrm{FM}_d(n)\}$ of Fulton–MacPherson compactifications of $n$ points in $\mathbb{R}^d$ embodies a topological $E_d$ -operad structure. The operad composition is defined by infinitesimal insertions ("blowing up" a configuration at a point), and the boundary stratification corresponds to rooted trees encoding how configurations degenerate (Tourtchine, 2012, Ching et al., 2020).

CW and simplicial decompositions of the 2-dimensional Fulton–MacPherson operad ( $\mathrm{FM}_2$ ) are known: the cells correspond to certain trees with decorated vertices, allowing for direct computation and modeling at the chain level. This is central for applications in homotopy theory, string topology, and symplectic cohomology, enabling direct modeling of Gerstenhaber and Batalin–Vilkovisky algebra actions (Salvatore, 2019, Bottman, 2021).

Koszul Duality and Homotopy Theory

The FM operad provides a small, cofibrant, $O(n)$ -equivariant $E_n$ -operad. Its Koszul dual is naturally $O(n)$ -equivariantly equivalent to its $n$ -fold desuspension, with explicit S-duality pairings constructed via bar/cobar constructions—a result foundational for topological field theory and the interplay between operads and (co)homology (Ching et al., 2020).

Polyhedral and Combinatorial Structures

For configurations on the circle, the FM compactification $S^1[n]$ is homeomorphic to $S^1 \times W_n$ , where $W_n$ is the cyclohedron, and the stratification is indexed by cyclic bracketings. Such structures underpin resolutions in application areas such as obstruction-theoretic approaches to problems of inscribed polygons in convex curves (Vrecica et al., 2008).

6. Intersection Theory and Cohomological Properties

The Fulton–MacPherson compactification admits an explicit presentation of its Chow ring $A^*(X[n])$ as an algebra generated by:

Pullbacks from $A^*(X^n)$ ,
Formal boundary divisor classes $D_I$ for $|I|\ge 2$ , with relations reflecting:
Disjointness of overlaps,
Support (i.e., vanishing on the strict transform of the diagonal),
Chern class compatibility for the normal bundles of the diagonals.

This explicit presentation, established via iterated applications of Keel's blow-up formula, provides a foundational tool for recursive computation of intersection numbers, especially in cases where $X[n]$ corresponds to well-studied moduli spaces (e.g., $\overline{M}_{0,n+1}$ for $X = \mathbb{P}^1$ ) (Petersen, 2015, Routis, 2014).

Weighted analogues follow the same structure, with boundary divisors and relations reflecting only those diagonals whose total weight exceeds $1$ (Routis, 2014).

7. Applications and Structural Results

Moduli and Automorphism Groups

The automorphism group of $X[n]$ is closely controlled by that of $X$ :

For $n \neq 2$ or $\dim(X)\ge 2$ , $\mathrm{Aut}^0(X[n]) \cong \mathrm{Aut}^0(X)$ via the diagonal action.
For $X$ a genus $g$ curve ( $g\neq 1$ ), $\mathrm{Aut}(C[n]) \cong S_n \times \mathrm{Aut}(C)$ for $n\neq 2$ (Massarenti, 2016).

Morphisms between Fulton–MacPherson spaces (e.g., $C[n] \to C[r]$ for a curve $C$ ) are classified: any dominant morphism factors through "forgetful" maps corresponding to projection on subsets (Massarenti, 2016).

Enumerative Geometry and Hilbert Schemes

The Fulton–MacPherson compactification is critical in relating Gromov–Witten invariants of Hilbert schemes of points and orbifold symmetric products, via wall-crossing formulas and one-parameter families interpolating between Hilbert schemes and FM spaces. The wall-crossing structure identifies a universal I-function and expresses tautological integrals for Hilbert schemes in terms of FM integrals and combinatorial data (Nesterov, 14 Jan 2025, Nesterov, 6 Jan 2026).

Logarithmic and Degeneration Formulas

Log FM spaces appear naturally as special fibers in degenerations, with their structure controlling the gluing and cut-and-paste properties necessary for degeneration formulas in algebraic and logarithmic Gromov–Witten theory. Each central-fiber component of the log degeneration is a birational modification of a product of boundary log FM spaces (Mok, 21 Mar 2025).

Birational Geometry

For $n > d+8$ ( $d = \dim X$ ), the FM compactification $X[n]$ fails to be a Mori dream space: its Cox ring is not finitely generated, and its effective cone is not polyhedral. This negative result places strict constraints on the birational models and the birational rigidity of these configuration spaces when $n$ is sufficiently large in relation to the dimension (Gallardo et al., 2021).

References:

(Massarenti, 2016): "On the biregular geometry of the Fulton-MacPherson compactification"
(Gallardo et al., 2016): "Wonderful compactifications of the moduli space of points in affine and projective space"
(Petersen, 2015): "The Chow ring of a Fulton-MacPherson compactification"
(Routis, 2014): "Weighted compactifications of configuration spaces and relative stable degenerations"
(Tourtchine, 2012): "Context-free manifold calculus and the Fulton-MacPherson Operad"
(Salvatore, 2019): "A cell decomposition of the Fulton MacPherson operad"
(Bottman, 2021): "A simplicial version of the 2-dimensional Fulton-MacPherson operad"
(Ching et al., 2020): "Koszul duality for topological E_n-operads"
(Nesterov, 14 Jan 2025): "Hilbert schemes of points and Fulton-MacPherson compactifications"
(Nesterov, 6 Jan 2026): "On the Hilbert-Chow crepant resolution conjecture"
(Mok, 21 Mar 2025): "Logarithmic Fulton--MacPherson configuration spaces"
(Gallardo et al., 2021): "The Fulton-MacPherson compactification is not a Mori dream space"
(Vrecica et al., 2008): "Fulton-MacPherson compactification, cyclohedra, and the polygonal pegs problem"