Maximally symmetric stable curves

Michael A. van Opstall; Razvan Veliche

REVIEW

Maximally symmetric stable curves

Algebraic Geometry · Submitted 2 Mar 2006 · 3 references

Pith's verdict: The paper's sharp bound is false: the tetrahedral configuration of four elliptic tails on P1 yields 15552 automorphisms in genus 4, exceeding the claimed 10368.

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arXiv IDmath/0603061 (version 3)

Pith IDSO473R4A, Pith's permanent record of this paper and its review

CitationMichael A. van Opstall, Razvan Veliche. "Maximally symmetric stable curves." arXiv:math/0603061 (2006). https://pith.science/paper/SO473R4A

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Dear colleague,

The short version: the main theorem is false. A genus-4 counterexample is already visible from the paper's own assumptions. The paper asserts that n points on P1 can be permuted at most dihedrally (Section 1). That is wrong for n=4. The set {0,1,∞,t} with t^2−t+1=0 has a setwise stabilizer of order 12 in PGL2(C), isomorphic to A4, generated by z↦1/(1−z) and the three V4 involutions. Glue four copies of the j=0 elliptic curve at these points. The resulting stable curve has genus 4, dual graph a star with one rational vertex of valence (0,4), and automorphism group of order 6^4·12 = 15552. Theorem 4.9, for g=4, gives 6^4·2^4·(1/2) = 10368. So the claimed bound is exceeded by a factor of 3/2. This is the opposite failure mode of the reader's cross-ratio worry; the geometric symmetry is larger than the paper's dihedral count, not smaller.

What the paper does well: the graph-theoretic reduction program is serious. The lemmas about optimal trees, doubling, and strict optimal trees are interesting and may be reusable. The paper correctly solves the exercise in Harris–Morrison if one restricts to 'simple' curves where the only symmetries are cyclic or dihedral. The exposition is clear, and the proof is self-contained with no obvious circularity.

The soft spot is fundamental, not a technical edge case. The 'most efficient gluing' convention in Section 1 is an assumption that the maximum point-stabilizer on P1 is at most dihedral. That assumption is false; finite subgroups of PGL2(C) include tetrahedral, octahedral, and icosahedral groups. For n=4, the tetrahedral group has order 12, giving more automorphisms than any dihedral arrangement. The paper's own Lemma 3.2 lists (0,4) as an allowable valence, but Definition 3.3 limits elliptic valence to ≤3, so the 'simple' reduction is inconsistent with that lemma. In any case, the counterexample cannot be removed by tying point configurations to roots of unity.

Who is this for? Anyone working on automorphisms of stable curves or the local structure of moduli of curves. The paper deserves a careful referee because the flaw is non-obvious and the graph-theoretic scaffolding might be salvageable. But as it stands, the sharp bound and classification in Theorems 3.4, 4.9, and 5.7 are incorrect.

Recommendation: send to a knowledgeable referee; the counterexample should be reported to the authors. If they repair the geometric-symmetry part, the graph reduction might still yield a true bound once the finite subgroups of PGL2 are accounted for.

Best, [Your name]

Referee report

Major objections (5)

[Section 1; Theorem 4.9] The assertion that n points on P1 can be permuted at most dihedrally is false. The set {0,1,∞,t} with t^2−t+1=0 has setwise stabilizer in PGL2(C) of order 12: the order-3 map z↦1/(1−z) cycles 0,1,∞ and fixes t, and the three involutions swapping opposite pairs generate A4. Glue four copies of the j=0 elliptic curve to these four points. The result is a stable genus-four curve whose central vertex has valence (0,4), a case explicitly permitted by Lemma 3.2. Its automorphism group has order at least 6^4·12 = 15552, since each elliptic tail contributes six automorphisms fixing its node and the central P1 contributes the 12 automorphisms preserving the four nodes. Theorem 4.9 for g=4 (binary 100: k=0, l=1, N=4) gives 6^4·2^4·(1/2)=10368. Thus the claimed sharp bound is false, and the genus-four curve is not of the form described in Theorem 3.4.
[Section 2, Proposition 2.7] The proof of Proposition 2.7 contains the unsupported assertion that if exactly one attached tree Ti is non-degenerate, then edge-transitivity of H together with H not being a tree forces H to be 'tree-like (of diameter two), but with multiple edges'. No argument or reference is given for this structural claim, and this is exactly the case that closes the induction. Moreover, Lemma 1.6 had previously eliminated multiple edges from optimal graphs, so it is unclear how this configuration can arise at all. This step is load-bearing for Theorem 2.10, the reduction to trees.
[Section 2, Lemma 2.9] Lemma 2.9 is not proved in the necessary generality. The genus-two base case is dismissed as 'obviously true, inspecting case-by-case', and the inductive step asserts without proof that deleting the unique fixed rational vertex v and its k incident edges yields a connected graph G′ of genus g−k+1. No justification is supplied for connectivity, for the genus formula, or for the claim that the subgroup fixing the edges around v fixes their opposite ends in G′. Since Lemma 2.9 is used in Case 2 of Theorem 2.10 to argue that replacing components by optimal trees loses no symmetry, the reduction to trees is not fully established.
[Section 3, Definition 3.3 versus Lemma 3.2] There is an internal inconsistency in the definition of a simple curve. Lemma 3.2 lists (0,4) and (0,5) as permissible valences in an optimal graph, but Definition 3.3 requires that the elliptic valence be no greater than three. The text immediately afterward asserts that there exists a maximally symmetric stable curve that is simple. If the definition is intended to exclude (0,4) and (0,5), then the reduction lemma and the definition contradict each other; if it is not, then the subsequent graph enumeration is not based on a clearly defined class. The genus-four example with valence (0,4) is directly affected by this ambiguity.
[Proof of Theorem 3.4] The base cases of the proof of the Main Theorem are asserted rather than shown: the genus-three case is said to follow from a unique simple graph, the genus-four case from two simple dual graphs, and the genus-five case is described as 'easy'. No derivations are provided. In particular, a careful check of genus four would have encountered the curve with four j=0 tails at the tetrahedral points, whose automorphism group has order 15552, so the assertion that the genus-four maximally symmetric graphs are as stated is not a harmless omission.

Minor objections (4)

[Throughout] The notation '6g', '2N(g)', '2k+1' and similar expressions is ambiguous because superscripts appear to have been lost; Theorem 4.9 should read 6^g · 2^{N(g)} · (3/8)^{k(g)} · (1/2)^{l(g)}. This makes the formulas difficult to verify.
[Section 1] The sentence 'n points on P1 can be permuted at most dihedrally' should be corrected to refer to the classical classification of finite subgroups of PGL2(C), which includes cyclic, dihedral, A4, S4, and A5 groups. The 'most efficient way' convention of placing branches at roots of unity is an assumption, not a consequence of the preceding discussion.
[Definition 3.3] If the elliptic-valence bound in Definition 3.3 is intentional, the list in Lemma 3.2 should be adjusted accordingly; if it is a typo, the definition should be corrected and the consequences for the later enumeration should be checked.
[Section 5, Remark 5.8] Remark 5.8 concedes that when four or more branches share a vertex, the automorphism group of P1 is only three-point transitive, so infinitely many attachment choices remain. This is in tension with the earlier assumption that all graph symmetries can be realized by curve automorphisms; the tension should be addressed explicitly.

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The paper

The paper claims

The automorphism group of a stable curve has a sharp, computable maximum.

The reading

The paper establishes the sharp upper bound on the order of the automorphism group of a stable curve of genus $g$, and describes every curve that reaches the bound. Hurwitz's classical bound $42(2g-2)$ for smooth curves fails once nodal singularities are allowed, so the paper replaces it with a formula read off from the binary expansion of $g$. The extremal curves are made entirely from copies of the $j=0$ elliptic curve $E$ and rational curves, arranged as a tree whose leaves are elliptic curves. The result matters because the moduli space of stable curves is locally a quotient by the automorphism group of a curve; the formula therefore bounds how singular the moduli space can be near its boundary.

What carries the argument

The load-bearing construction is the dual graph of the curve together with its group of geometric automorphisms $\mathrm{GAut}\, G$, meaning graph automorphisms that arise from genuine curve automorphisms. A sequence of replacement lemmas first reduces any candidate to a simple curve—components only $E$ or $\mathbb{P}^1$, dual graph a tree with elliptic leaves—where each replacement increases the number of automorphisms. The graph-theoretic core then uses the fact that every finite tree has an absolute root or virtual root, a Doubling Lemma that builds optimal trees with $2g$ leaves from optimal trees with $g$ leaves, and a valence analysis restricting the possible local configurations. The binary-digit pairing in Theorem 4.9 precisely tracks the doubling process: pairs '11' record branches that carry an extra 3-leaf component, pairs '10' record a particular loss of symmetry, and a remaining isolated 1 records a lone leaf.

Extended reading notes

Core claim

For genus $g$, the maximal order is $72$ when $g=2$; $6^g 2^{g-3} 6$ when $g=3\cdot 2^n$; $6^g 2^{g-5} 10$ when $g=5\cdot 2^n$; and in the general case $6^g 2^{N(g)} (3/8)^{k(g)} (1/2)^{l(g)}$, where scanning the binary expansion of $g$ from the left and grouping consecutive pairs that start with a 1 yields $k(g)$ pairs '11' and $l(g)$ pairs '10', with $N(g)=g-1$ when a lone 1 is left at the right end and $N(g)=g$ otherwise. The bound is attained by a dual graph that is a tree; all interior vertices correspond to $\mathbb{P}^1$'s, every leaf is a copy of $E$, and the tree is recursively built from binary trees as specified in Theorem 3.4. Theorem 5.7 completes the description: up to neutral moves, valence reduction, and stabilization, every maximizing curve has a strict optimal tree as its dual graph.

Load-bearing premise

The load-bearing premise is that the most efficient gluing recipe—placing isomorphic branches at roots of unity and pinning other attachments at zero and infinity—always realizes every graph symmetry the formulas count; if a vertex has four or more independent branch positions, the three-point transitivity of the projective-line automorphism group could make the true geometric automorphism group smaller than the graph count.

Editorial extensions

If this is right

For every genus the maximum is attained, so the answer to the open exercise posed for stable curves is a finite, explicitly constructible list of extremal curves.
The Hurwitz bound is replaced by a formula that can be evaluated directly from the binary expansion of $g$; for instance, $g=215$ gives $k=2$, $l=1$, and $N=214$.
The automorphism groups of boundary curves of the moduli space $\overline{\mathcal{M}}_g$ are uniformly bounded by the formula, giving a quantitative handle on the singularities of the moduli space.
In some genera, such as $g=1+4+16+64$, the maximizing curves form a family of positive dimension equal to $k(g)+l(g)+g-N(g)-3$, rather than a single curve.
Curves with $5\cdot 2^n$, $3\cdot 2^n$, or $2^n$ leaves have a unique maximally symmetric model; other genera can have several, connected by neutral moves.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The extremal problem is ultimately a weighted-tree optimization: replace each elliptic leaf by a factor of 6 and each graph symmetry by its geometric realization, and the result is a purely combinatorial maximum. The same binary-pairing formula would apply to any species of rooted trees with the same local symmetry rules.
The tacit assumption that all counted graph symmetries are geometrically realizable could be tested independently by computing cross-ratios of branch points on each rational vertex; a vertex with four or more independent branches is the natural place for a discrepancy to appear.
Over fields other than the complex numbers, where the automorphism group of the projective line is smaller or has different transitivity, the same reduction scheme would produce a different bound, so the binary formula is specific to working over $\mathbb{C}$.
Because the maximal curves are trees, the moduli-space strata they occupy are contractible in a natural sense; the positive-dimensional families from Remark 5.8 are cross-ratio deformations of attachment points, not deformations of the components.

Original abstract

We give a sharp bound on the number of automorphisms of a stable curve of a given genus and describe all curves attaining this bound.

The compiled record

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.94 · compares with prior results 0.08 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

We give a sharp bound on the number of automorphisms of a stable curve of a given genus and describe all curves attaining this bound.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We give a sharp bound on the number of automorphisms of a stable curve of a given genus and describe all curves attaining this bound.	contribution 0.71	0.95	0.90	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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arXiv listing abstract v3

not examined

full text
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levels run

P abstract

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T body sentences
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Figures

**Figure 1.** Illustrating Lemmas 1.4-1.6 Lemma 1.5. There exists a maximally symmetric stable curve whose components are all P 1 or E. Proof. By Lemma 1.4, all components may be taken smooth, and it is clear that replacing an elliptic curve less symmetric than E with E only helps. If C1 is a genus h > 1 component of C, replace it with a rational curve, attach a rational tail to this, and then glue h copies of E to this second rational curve. Ignoring graph symmetry,C1 contributed at most 42(2h−2) automorphisms 2

**Figure 2.** Illustrating Lemmas 1.6-2.2 2 Breaking cycles This section contains the main part of the reduction: we may assume that the dual graph of a maximally symmetric stable curve is a tree. To achieve this, we need to break cycles in the graph, producing a new graph with more automorphisms. In one case, this is easy. In this entire section we assume that all of the reductions from the previous section have been carried out: we are now studying simple graphs whose interior vertices all correspond to smooth rational curves and whose leaves are copies of E. Definition 2.1. A cycle in a graph…

**Figure 3.** Illustrating three of the cases of the Main Theorem

**Figure 4.** Attachment rules for attaching appendages of type

**Figure 5.** Splitting a 3-3-3 configuration in the Doubling Lem

**Figure 6.** Nonuniqueness in genus seven As the genus increases, even worse nonuniqueness can occur. However, we can describe all maximally symmetric curves of a given genus. Proposition 5.1. A maximally symmetric curve of genus g has 1. all components P 1 or E, 2. dual graph a tree with all leaves elliptic and other vertices rational. Proof. In all of the reduction steps, we actually gain automorphisms with one exception: reducing valence at a vertex where neighboring vertices are non-isomorphic does not necessarily add any automorphisms. Therefore, conditions of valence are dropped from the co

Figures reproduced from arXiv:math/0603061 by the authors.

References

Works cited (3)

[1]

J. Harris and I. Morrison. Moduli of curves. Springer-Verlag, New York, 1998

View on Pith 1998
[2]

A. Hurwitz. “ ¨Uber algebraische Gebilde mit eindentigen Transformation en in sich.” Math. Ann. 43, 403-442 (1893)

View on Pith
[3]

J.P. Serre. Trees. Springer-Verlag, Heidelberg, 1980. 16

View on Pith 1980

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

J. Harris and I. Morrison. Moduli of curves. Springer-Verlag, New York, 1998

View on Pith 1998

[2] [2]

A. Hurwitz. “ ¨Uber algebraische Gebilde mit eindentigen Transformation en in sich.” Math. Ann. 43, 403-442 (1893)

View on Pith

[3] [3]

J.P. Serre. Trees. Springer-Verlag, Heidelberg, 1980. 16

View on Pith 1980