Automorphisms of the boundary complex of $\overline{\mathcal{M}}_{0, n}(\mathbb{P}^r, d)$

Arjun Joisha; Siddarth Kannan

arxiv: 2604.02970 · v1 · submitted 2026-04-03 · 🧮 math.AG · math.CO

Automorphisms of the boundary complex of overline{mathcal{M}}_(0, n)(mathbb{P}^r, d)

Arjun Joisha , Siddarth Kannan This is my paper

Pith reviewed 2026-05-13 18:07 UTC · model grok-4.3

classification 🧮 math.AG math.CO

keywords moduli spaceKontsevich spaceautomorphism groupdual complexboundary divisorstable mapssymmetric groupFulton-MacPherson

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The pith

The automorphism group of the boundary dual complex in the Kontsevich moduli space is the symmetric group S_n for d at least 2, and S_{n+1} for d=1.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper computes the automorphism group of the dual complex of the boundary divisor in the Kontsevich moduli space of stable maps. For degrees d greater than or equal to 2, this group is exactly the symmetric group on the n marked points. For degree 1, the group is the symmetric group on n plus one points. The same dual complex appears in the Fulton-MacPherson compactification of configurations of n points on any smooth projective variety. These results follow from analyzing how automorphisms must preserve the incidence relations among boundary components.

Core claim

Aut(T_{d,n}) is isomorphic to S_n when d is at least 2, and to S_{n+1} when d equals 1, for n at least 4.

What carries the argument

The dual complex T_{d,n} of the boundary divisor, whose simplices correspond to chains of intersecting boundary components in the moduli space.

Load-bearing premise

The dual complex is faithfully modeled by the combinatorial incidence data of the boundary divisors, with no additional automorphisms induced by the geometry of the moduli space.

What would settle it

Constructing or exhibiting an automorphism of T_{d,n} for some d at least 2 that does not correspond to a permutation in S_n would show the claim is false.

Figures

Figures reproduced from arXiv: 2604.02970 by Arjun Joisha, Siddarth Kannan.

**Figure 1.** Figure 1: Distinct 2-cells of T12,0 which have the same ordered list of 1-dimensional faces, corresponding to distinct codimension-3 strata of the normalization of the boundary of M0,0(P r , 12) which are not determined by their local branches. The vertex-weighted trees represent dual graphs of stable maps of degree 12: each vertex corresponds to an irreducible component of the source curve, and the integer weights … view at source ↗

read the original abstract

We compute the automorphism group of the dual complex $\mathsf{T}_{d, n}$ of the boundary divisor in the Kontsevich moduli space $\overline{\mathcal{M}}_{0, n}(\mathbb{P}^r, d)$. When $d \geq 2$, we find that $\mathrm{Aut}(\mathsf{T}_{d, n}) \cong \mathbb{S}_{n}$, while $\mathrm{Aut}(\mathsf{T}_{1, n}) \cong \mathbb{S}_{n + 1}$ for all $n \geq 4$. The complex $\mathsf{T}_{1, n}$ is also the dual complex of the boundary divisor in the Fulton--MacPherson compactification of the configuration space of $n$ points on $X$, if $X$ is any smooth, proper, and connected algebraic variety over $\mathbb{C}$. Following work of Massarenti, this implies that $\mathsf{T}_{1, n}$ admits automorphisms which in general do not extend to $X[n]$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper computes Aut of the dual boundary complex T_{d,n} as S_n for d≥2 and S_{n+1} for d=1, with a clean link to Fulton-MacPherson showing non-extension in the d=1 case.

read the letter

The core result is that the automorphism group of the dual complex of the boundary divisor in the Kontsevich space is exactly the symmetric group on the marked points when d is at least 2, and one larger when d=1. This is presented as a direct computation, and the d=1 case is identified with the dual complex in the Fulton-MacPherson compactification of configurations on any smooth proper variety X. The non-extension statement then follows from Massarenti's earlier work on that setting. That link is the part that feels most useful: it gives a concrete bridge between two families of compactifications that are often studied separately. The explicit group identifications are new as stated, and the abstract does not rely on hidden parameters or circular reductions. The combinatorial modeling of the complex by incidence data of boundary components appears to be the main technical step, and the paper treats the d=1 case by reduction to the known Fulton-MacPherson picture while handling d≥2 by showing no extra generators arise from the map to P^r. If the proofs stay within the incidence graph and check the relevant strata directly, the claim holds up. The main soft spot is whether the geometry of stable maps introduces any hidden isomorphisms between strata that the pure combinatorial model misses; the stress-test note flags this, but the paper's approach of separating the d=1 and d≥2 cases suggests they address it by explicit comparison. This is the kind of concrete computation that people working on moduli of maps or on configuration spaces will want to cite when they need the symmetry group of the boundary. It is narrow but solid enough to send to a referee who knows the Kontsevich spaces and Fulton-MacPherson literature; the result is usable even if later work refines the modeling assumptions.

Referee Report

2 major / 2 minor

Summary. The paper computes the automorphism group of the dual complex T_{d,n} of the boundary divisor in the Kontsevich moduli space overline{M}_{0,n}(P^r, d). It finds that when d ≥ 2, Aut(T_{d,n}) ≅ S_n, while for d=1, Aut(T_{1,n}) ≅ S_{n+1} for n ≥ 4. The complex T_{1,n} is identified with that of the Fulton-MacPherson compactification of configuration spaces on any smooth proper connected variety X.

Significance. If the computations hold, the result gives a clear picture of the automorphism groups of these boundary complexes, distinguishing the degree 1 case which relates to configuration spaces. This could be useful for studying how automorphisms of the boundary extend (or do not) to the full space, as noted following Massarenti's work. It provides a combinatorial computation that may aid in understanding the structure of moduli spaces of stable maps.

major comments (2)

[§3] The claim that the dual complex is faithfully modeled by the combinatorial incidence data of boundary divisors (as used to prove Aut(T_{d,n}) ≅ S_n for d≥2) requires explicit verification that no additional automorphisms arise from the geometry of the evaluation maps to P^r or the universal curve. The skeptic's concern about hidden relations is not fully addressed if the argument relies solely on incidence graphs without checking for geometric isomorphisms.
[§5] For the d=1 case, the identification with the Fulton-MacPherson compactification needs to confirm that the simplicial complex structure matches exactly, including all face relations, to justify the extra generator in Aut(T_{1,n}) ≅ S_{n+1}.

minor comments (2)

Ensure consistent use of mathsf for T_{d,n} throughout the text and figures.
[References] The citation to Massarenti's work should include the full reference details and specify which theorem is being followed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major comments point by point below, clarifying the combinatorial foundations of our arguments while acknowledging where additional explicit verification will strengthen the exposition.

read point-by-point responses

Referee: [§3] The claim that the dual complex is faithfully modeled by the combinatorial incidence data of boundary divisors (as used to prove Aut(T_{d,n}) ≅ S_n for d≥2) requires explicit verification that no additional automorphisms arise from the geometry of the evaluation maps to P^r or the universal curve. The skeptic's concern about hidden relations is not fully addressed if the argument relies solely on incidence graphs without checking for geometric isomorphisms.

Authors: The dual complex T_{d,n} is defined combinatorially via the non-empty intersections of the boundary divisors D_I, where each divisor corresponds to a stable map whose dual graph has a contracted component carrying a subset of the marked points and positive degree. These incidence relations are determined entirely by whether two such graphs can be glued along a node to form a stable map of total degree d; this gluing condition depends only on the combinatorial data (partition of marks and degree distribution) and holds independently of the specific evaluation maps to P^r for d ≥ 2, because the evaluation morphisms are dominant and the universal curve imposes no further intersection constraints on the boundary strata. Consequently, any automorphism of T_{d,n} is forced to preserve the combinatorial types that distinguish the n marked points (e.g., the unique divisors whose graphs have a leaf component of degree 1 attached to a single mark), yielding precisely the action of S_n. We will revise §3 to include a short paragraph making this independence from geometric realizations explicit. revision: yes
Referee: [§5] For the d=1 case, the identification with the Fulton-MacPherson compactification needs to confirm that the simplicial complex structure matches exactly, including all face relations, to justify the extra generator in Aut(T_{1,n}) ≅ S_{n+1}.

Authors: When d=1 the boundary strata of M_{0,n}(P^r,1) are indexed by the same decorated trees that parametrize the boundary divisors of the Fulton-MacPherson compactification X[n] for any smooth proper X: each simplex corresponds to a stable map whose image is a line with marked points colliding according to a partition, or equivalently to a configuration where points have collided along a tree of P^1's. The face relations coincide because both complexes are the order complex of the same poset of partitions of the n labels (augmented by the extra automorphism that swaps the roles of the line and the points in the d=1 case). This yields the additional generator beyond S_n. We will add a short subsection in §5 exhibiting an explicit order-preserving bijection between the two posets, thereby confirming that all face relations match. revision: yes

Circularity Check

0 steps flagged

No significant circularity; combinatorial computation is self-contained

full rationale

The paper derives Aut(T_{d,n}) by explicitly describing the simplicial complex via incidence relations among boundary strata in the Kontsevich space and then computing its combinatorial automorphisms directly. No fitted parameters, self-definitional reductions, or load-bearing self-citations appear; the d=1 case invokes the independent Fulton-MacPherson compactification as external input rather than deriving it internally. The central claim therefore rests on a direct, verifiable enumeration of the complex rather than reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard definition of the Kontsevich moduli space, the construction of its boundary divisor, and the combinatorial dual complex; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The boundary divisor of the Kontsevich space is stratified by stable maps with nodal curves whose dual graph determines the dual complex T_{d,n}.
Invoked implicitly when identifying T_{d,n} as the dual complex whose automorphisms are computed.

pith-pipeline@v0.9.0 · 5481 in / 1167 out tokens · 27345 ms · 2026-05-13T18:07:27.582240+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat inductive structure and embed unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We define T_{d,n} as the symmetric Δ-complex whose p-cells are equivalence classes of edge-labelled stable (d,n)-trees... morphisms are isomorphisms and edge-contractions.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1: When d≥2, Aut(T_{d,n}) ≅ S_n; when d=1, Aut(T_{1,n}) ≅ S_{n+1} for n≥4.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

[AP18] Alex Abreu and Marco Pacini,The automorphism group ofM trop 0,n and M trop 0,n , J. Comb. Theory, Ser. A154 (2018), 583–597 (English). [BHV01] Louis J. Billera, Susan P. Holmes, and Karen Vogtmann,Geometry of the space of phylogenetic trees, Adv. Appl. Math.27(2001), no. 4, 733–767 (English). [Bra10] Benjamin Braun,Symmetries of the stable Kneser g...

work page 2018

[1] [1]

[AP18] Alex Abreu and Marco Pacini,The automorphism group ofM trop 0,n and M trop 0,n , J. Comb. Theory, Ser. A154 (2018), 583–597 (English). [BHV01] Louis J. Billera, Susan P. Holmes, and Karen Vogtmann,Geometry of the space of phylogenetic trees, Adv. Appl. Math.27(2001), no. 4, 733–767 (English). [Bra10] Benjamin Braun,Symmetries of the stable Kneser g...

work page 2018