The Kodaira classification of the moduli space of pointed curves in genus $3$

Ruben de Preter

REVIEW 2 major objections 5 minor 1 cited by

The Kodaira classification of the moduli space of pointed curves in genus $3$

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For every n≥15, the moduli space $M_{3,n}$ of stable pointed genus-3 curves is of general type, completing the Kodaira classification in genus 3.

desk verdict Genuinely new result with a solid bigness computation, but the singularity analysis applies the Reid-Tai criterion to the wrong dual representation. read the letter →

arxiv 2506.23812 v1 pith:HSFYFDZT submitted 2025-06-30 math.AG

Ruben de Preter This is my paper · ORCID

classification math.AG MSC 14H1014E08

keywords moduliofcurvesKodairadimensiongeneraltypegenus3pointedReid-Taicriterioncanonicaldivisorsingularitiesspaces

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper completes the Kodaira classification of moduli spaces of stable pointed curves in genus 3. Its theorem is that $M_{3,n}$ is of general type whenever $n \geq 15$, and since $M_{3,n}$ was already known to be rational for $n \leq 14$, the Kodaira dimension of every genus-3 moduli space is now determined. The proof separates into two statements: the canonical class of $M_{3,n}$ is big for $n \geq 15$, and the singularities of $M_{3,n}$ impose no adjunction conditions for $n \geq 1$, so the big canonical class actually forces general type. The singularity analysis is the delicate part: it identifies the non-canonical locus as exactly the locus of curves with an unmarked elliptic tail of j-invariant 0.

What carries the argument

The argument is carried by two mechanisms. First, the Reid--Shepherd-Barron--Tai criterion and its refinements: a finite quotient $V/G$ has canonical singularities when $G$ contains no junior elements, where an element is junior when the sum of the fractional parts of its eigenvalues on $V$ lies strictly between 0 and 1. A variant from [CCM] gives an explicit inequality involving the order of vanishing of a pluri-canonical form along coordinate hyperplanes, allowing such forms to lift even at non-canonical quotient singularities. Second, a classification of all possible junior automorphisms of pointed genus-3 curves: Lemma 3.1 and Proposition 3.2 produce a finite list of cases, showing that outside the locus of unmarked elliptic tails of j-invariant 0, no junior automorphisms occur. For bigness, the machinery consists of the canonical divisor formula $K_{M_{3,n}} = 13\lambda + \psi_1 + \cdots + \psi_n - 2\delta - \delta_{1,\emptyset}$, the test-curve class obtained by gluing a fixed genus-2 pointed curve to a variable elliptic tail, the effective divisors coming from the hyperelliptic locus and from $D^3_{3,14}$, and the bigness of $a\lambda + b\psi$ for positive $a,b$.

What would settle it

Take a stable pointed genus-3 curve $[C,p_1,\ldots,p_n]$ with $n \geq 1$ and $[C,p_1,\ldots,p_n] \notin \Delta_{1,\emptyset}$. Compute the eigenvalues of a nontrivial automorphism $\varphi \in \operatorname{Aut}(C,p_1,\ldots,p_n)$ acting on $V = H^0(C, \Omega_C \otimes \omega_C(p_1+\cdots+p_n))$ and their age. If any such computation yields $0 < \operatorname{age}(\varphi) < 1$, Proposition 3.3 is contradicted; the entries of Table 1 are directly checkable by diagonalizing the listed elliptic, hyperelliptic, and rational cases.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: for every $n \geq 15$, the moduli space $M_{3,n}$ is of general type. This is deduced from Theorem 1.2, which shows that the canonical class $K_{M_{3,n}} = 13\lambda + \psi_1 + \cdots + \psi_n - 2\delta - \delta_{1,\emptyset}$ is big by writing it as a positive combination of big classes, effective divisor classes pulled back from the hyperelliptic locus and from the divisor $D^3_{3,14}$, and boundary classes; and from Theorem 1.3, which shows that all global $m$-canonical forms lift from the coarse moduli space to a resolution, so the singularities impose no adjunction conditions. The key singularity result is Corollary 3.5: for $n \geq 1$, the locus of non-canonical singularities is exactly $J_0$, the locus of pointed curves admitting an unmarked elliptic tail of j-invariant 0. Proposition 1.5 adds the needed rigidity: the pluri-canonical divisor $mK_{M_{3,n}}$ contains the boundary divisor $4m\Delta_{1,\emptyset}$ as a rigid component, so every global $m$-canonical form vanishes along $\Delta_{1,\emptyset}$ to order at least $4m$, which feeds into the lifting criterion at the non-canonical locus.

Load-bearing premise

The load-bearing premise is that the classification of junior automorphisms collected in Lemma 3.1 and Proposition 3.2 is exhaustive: if even one nontrivial automorphism of a pointed genus-3 curve outside the exceptional loci has age below 1 and is missing from the tables, the identification of the non-canonical locus with $J_0$ and the lifting argument both break down.

Editorial extensions

If this is right

For every $n \geq 15$, $M_{3,n}$ has maximal Kodaira dimension equal to its dimension $6+n$, so it is not uniruled and not rational.
Combined with the known rationality of $M_{3,n}$ for $n \leq 14$, the theorem gives the complete Kodaira classification in genus 3: Kodaira dimension $-\infty$ for $n \leq 14$ and maximal for $n \geq 15$.
For $n \geq 1$, the non-canonical singular locus of $M_{3,n}$ is exactly $J_0$, the locus of pointed curves with an unmarked elliptic tail of j-invariant 0; all other quotient singularities are canonical.
Every global $m$-canonical form on $M_{3,n}$ vanishes along $\Delta_{1,\emptyset}$ to order at least $4m$, a rigid-boundary phenomenon that makes the adjunction condition automatic.
The theorem confirms the expectation that in each fixed genus only finitely many pairs $(g,n)$ fail to be of general type, with genus 3 now settled at the threshold $n=15$.

Reading between the lines

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

Beyond the paper, the refined lifting criterion should transfer to higher genera: failure of the Reid--Tai hypothesis at a non-canonical locus need not be fatal if one can exhibit a rigid boundary component of high multiplicity, so the known thresholds for higher genera could be attacked by the same combination.
Extending the test-curve computation of Proposition 1.5 to $M_{g,n}$ for $g \geq 4$ would show whether the factor $4m$ along $\Delta_{1,\emptyset}$ is a general low-genus phenomenon; if it is, the adjunction step in higher genera becomes purely a statement about bigness of the canonical class.
The sharp jump from rational at $n=14$ to general type at $n=15$ in genus 3 suggests that for other fixed $g$, the Kodaira dimension may also jump directly from $-\infty$ to maximal as $n$ grows, with no intermediate values; checking this pattern at the current thresholds in higher genera would test the numerical mechanism behind the bigness proof.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

The main result is genuinely new and, on the face of it, right: M_{3,n} is of general type for n ≥ 15, completing a case that had been open. The bigness argument in Section 5 is clean and I checked the arithmetic; the chosen constants t and s do cancel λ and ψ, and the boundary coefficients become positive for n ≥ 15. The rigid component argument for 4m·Δ_{1,∅} is also standard and the test-curve computation in Lemma 4.2 checks out.

The soft spot is in the singularity analysis, and it is not merely a matter of unexpanded “one checks” steps. The paper computes ages on V = H^0(C, Ω⊗ω(p_1+...+p_n)), but the local quotient model is V^*/G where V is that space of differentials. The age on the dual is generally different: for eigenvalues {−1, i} on V the age is 3/4, but on V^* it is 5/4. Proposition 3.2 only bounds the age on V, yet Proposition 3.3 and Corollary 3.5 use this to conclude that outside Δ_{1,∅} there are no junior elements for the quotient V^*/G. That inference does not follow as written. The proof needs to rule out age_{V^*}(φ) < 1 for automorphisms outside Δ_{1,∅}, and the paper does not compute that.

The likely fix is straightforward: age_{V^*}(φ) < 1 is the same as age_V(φ^{-1}) < 1, so the same geometric case analysis should go through after inverting the automorphism, with the eigenvalue lists in Table 1 dualized. But the manuscript does not contain that step, so the no-adjunction theorem has a genuine gap in its current form.

There are also two minor textual issues: the reference [BMS] appears to duplicate [AB], and the general definition of the Farkas divisor says n = (2r+1)(g+1) while the proof uses D^3_{3,14}, consistent with (2r+1)(g−1). These are easy corrections.

Bottom line: the paper deserves a serious referee. The bigness part is solid and the final theorem is important, but the singularity analysis needs to be reworked or clarified on the dual-age issue before the proof is complete.

Referee Report

2 major / 5 minor

Summary. The paper aims to complete the Kodaira classification of the moduli space of pointed genus-3 curves by proving that \overline{M}_{3,n} is of general type for n \ge 15. The strategy is standard: first show that the canonical class K_{\overline{M}_{3,n}} is big, and then show that the singularities of \overline{M}_{3,n} impose no adjunction conditions. The bigness argument is an explicit divisor-class computation using the pullback of the hyperelliptic locus in \overline{M}_3 and a Farkas-type divisor on \overline{M}_{3,14}; the no-adjunction argument is a Reid–Shepherd–Barron–Tai analysis of the local actions of automorphism groups of pointed stable curves, together with a test-curve argument identifying a rigid component of the canonical divisor. The paper concludes that Theorem 1.1 follows from Theorem 1.2 (bigness) and Theorem 1.3 (no adjunction).

Significance. If Theorem 1.1 is correct, it is a clean and valuable completion of the Kodaira classification in genus 3, closing the last open genus in which only finitely many cases were known. The paper also proves a structural statement, Theorem 1.3, about adjunction conditions for all n \ge 1, which is of independent interest. The bigness computation in Section 5 is explicit, arithmetically checkable, and, modulo the cited divisor classes, correct; the choice of the constants t and s is not fitted to the conclusion but is forced by matching the \lambda- and \psi-coefficients. The singularity analysis is extensive and is the part of the paper that carries the main technical risk; as explained in the major comments, that part is not yet established as written.

major comments (2)

[Section 3, Propositions 3.2–3.4] The age used for the Reid–Tai criterion is computed on the wrong representation. Proposition 3.2 and Proposition 3.3 compute ages on V = H^0(C, \Omega_C \otimes \omega_C(p_1+\cdots+p_n)). Proposition 3.4, however, recalls that a neighborhood of a point of \overline{M}_{3,n} is isomorphic to a neighborhood of the origin in V^*/G, where G = Aut(C,p_1,\ldots,p_n), and its Case 1 invokes Proposition 3.3 to conclude that G has no junior elements for the quotient singularity V^*/G. The age on the dual representation is not equal to the age on V: replacing an eigenvalue e^{2\pi i r} by e^{-2\pi i r} replaces the fractional part r by 1-r when r>0. For example, the eigenvalue pair (\zeta_4,-1) in Table 1 row (6) has age 3/4 on V but age 5/4 on V^*. Conversely, an automorphism with age_V(\varphi) \ge 1 can have age_{V^*}(\varphi) < 1, so the bound in Proposition 3.2 does not, by itself, rule out junior elements of V^*/G outside \Delta_{1,\emptyset}. Since Proposition 3.4, Corollary 3.5, and hence Proposition 1.4 and Theorem 1.3 depend on this exclusion, the no-adjunction theorem is not proved by the present argument. The eigenvalue lists and inequalities in Section 3 need to be recomputed for the dual representation, or a proof must be given that the two ages agree in all cases that occur.
[Section 3, Corollary 3.5 and Example 2.6] The use of Example 2.6 to conclude that the general point of J_0 is a non-canonical singularity is also affected by the same dual-representation issue. The text says that a generator for the elliptic-tail case acts on H^0(\Omega_C \otimes \omega_C(\sum p_i)) with eigenvalues (\zeta_6, \zeta_6^2, 1, \ldots, 1). On the deformation space V^*, the corresponding eigenvalues are (\zeta_6^{-1}, \zeta_6^{-2}, 1, \ldots, 1), with fractional parts (5/6, 4/6, 0, \ldots, 0). After quotienting by the quasi-reflection g^3, the induced generator has two eigenvalues with fractional part 4/6, so its age is 4/3, not 2/3 as in Example 2.6. Thus Example 2.6, as stated, does not establish non-canonicity of the singularities at the general point of J_0. If the eigenvalue lists in Case 2 of Proposition 3.4 are intended to be eigenvalues on V^* rather than on V, this must be stated explicitly and the lists must be derived from the exact sequence (3.2) for the dual representation; as written, the derivation is for H^0(\Omega_C \otimes \omega_C(\sum p_i)).

minor comments (5)

[Section 5, definition of D^r_{g,n}] The displayed normalization n = (2r+1)(g+1) in the definition of D^r_{g,n} is inconsistent with the subsequent use of D^3_{3,14}: for g=3 and r=3, the displayed formula gives n=28, whereas the proof of Theorem 1.2 uses n=14. The computation is consistent with the normalization n = (2r+1)(g-1), which gives n=14; please correct the text and state the intended normalization explicitly.
[Section 3, Proposition 3.2] In the node-orbit contribution computation, the phrase 'the residue of l modulo m/N' should read 'modulo N/m'; as written the fraction is inverted.
[Lemma 3.1 and Proposition 3.2] The proof of Lemma 3.1 and the final paragraphs of Proposition 3.2 contain several terse 'one checks' and 'similarly' passages, for example in Cases B2–B5 of Lemma 3.1 and in the exclusion of cases (1), (3), (4) and (5) in Proposition 3.2. Since the completeness of these case splits is load-bearing for the singularity analysis, the checks should be expanded or relegated to a clearly verifiable appendix.
[References] References [AB] and [BMS] appear to list the same Agostini–Barros paper under two different keys; the bibliography should be corrected to avoid a duplicated entry.
[Corollary 3.5] The phrase 'has a non-canonical singularities' should be corrected to 'has a non-canonical singularity'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof combines external divisor-class results with internal casework, and the coefficient choices are a legitimate algebraic construction rather than a fitted prediction.

full rationale

The derivation chain is self-contained in the relevant sense. Theorem 1.1 is built from two independent ingredients: bigness of the canonical class (Theorem 1.2) and absence of adjunction conditions (Theorem 1.3). Bigness is proved by expressing K_{M3,n} as a positive linear combination of known effective classes: the pullback of the hyperelliptic divisor H from [HM], the symmetric pullback of Farkas' divisor D^3_{3,14} from [Fa], and boundary divisors. The constants t=n/56 and s=(13+73t)/9 are chosen by solving the two linear equations that make the lambda and psi coefficients of K_{M3,n} and s*pi^*H+t*D_n agree; the remaining coefficients in equation (5.3) are then explicitly positive for n>=15. This is a legitimate construction rather than a fit-to-conclusion. The singularity analysis in Propositions 3.2, 3.3, and 3.4 is internal casework whose external inputs are the classical age classification in [HM, Section 1], the Reid-Tai criterion [Ta, Re], and the lifting criterion [CCM]; none of these is a self-citation and none presupposes the target statement. Proposition 1.5 computes the rigid component 4m*Delta_{1,emptyset} from the intersection numbers of the test curve gamma with K and Delta_{1,emptyset} (Lemma 4.2), so the coefficient is derived rather than imposed. The paper contains no load-bearing self-citation. A possible concern that ages are computed on H^0(K^2(D)) while the Reid-Tai criterion is applied to the dual deformation space is a mathematical correctness issue, not a circular derivation, so it does not change the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

These are external mathematical results used as black boxes; none of them is equivalent to the target theorem. The constants t and s are forced algebraic choices, not fitted parameters.

assumptions (6)

standard math Pic_Q(M_{g,n}) has basis λ, ψ_i, δ_irr, and boundary classes δ_{i,S} (Arbarello-Cornalba).
Used in Section 2.1 to identify divisor classes on stack and coarse space; fundamental to all class computations.
standard math K_{M_{g,n}} = 13λ + ψ - 2δ - δ_{1,∅} (Harris-Mumford, ACG).
Equation (2.1); the starting point for the rigid component and bigness arguments.
domain assumption Reid-Shepherd-Barron-Tai criterion: quotient V/G is canonical iff G has no junior elements, and the [CCM] extension criterion (Proposition 2.4) for lifting pluri-canonical forms.
The singularity analysis in Sections 2.3 and 3 rests on these criteria; the [CCM] criterion is specialized and not proved in this paper.
domain assumption Harris-Mumford fixed-locus lifting criterion (Proposition 2.7): if η lifts around the fixed locus of every junior element, it lifts at the origin.
Used in Case 3 of Proposition 3.4 to handle the remaining codimension-3 locus.
domain assumption Farkas's divisor class D^r_{g,n} = -(6r^2+6r+1)λ + (r+1)ω + ... (Equation 5.2).
Provides the effective divisor D_n used in Theorem 1.2; cited from [Fa, Theorem 4.2].
domain assumption For a,b>0 the class aλ+bψ is big on M_{g,n} (Logan, Theorem 2.9).
Final step of Theorem 1.2: combines with effective boundary to conclude K is big.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The Kodaira classification of the moduli space of pointed curves in genus $3$." pith.science (2026). https://pith.science/paper/HSFYFDZT

@misc{pith2026250623812,
  author       = {Pith},
  title        = {Pith review of: The Kodaira classification of the moduli space of pointed curves in genus $3$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HSFYFDZT}},
  note         = {Machine review of arXiv:2506.23812}
}

abstract

We complete the Kodaira classification of the moduli spaces $\overline{\mathcal{M}}_{g,n}$ of curves with marked points in genus $g=3$, by proving that $\overline{\mathcal{M}}_{3,n}$ is of general type for $n \geq 15$. We prove that the singularities of $\overline{\mathcal{M}}_{3,n}$ impose no adjunction conditions for $n \geq 1$ and that the canonical class of $\overline{\mathcal{M}}_{3,n}$ is big for $n \geq 15$.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

FA-modules of holomorphic forms on $\overline{\mathcal{M}}_{g,n}$
math.AG 2025-09 accept novelty 8.0 of 10

The spaces of holomorphic forms on moduli spaces of stable curves, for degrees up to 18, are completely described as simple FA-modules; for degrees 19 and 20, the description is conditional on a genus-3 vanishing conjecture.

Reference graph

Works this paper leans on

17 extracted references · 16 canonical work pages · cited by 1 Pith paper

[1]

Agostini and I

D. Agostini and I. Barros, Pencils on surfaces with normal crossings and the Kodaira dimension of M _ g,n , Forum Math. Sigma 9 (2021), Paper No. e31, 22 pp

work page 2021
[2]

Arbarello and M

E. Arbarello and M. Cornalba, The Picard groups of the moduli space of curves, Topology 26 (1987), no. 2, 153--171

work page 1987
[3]

Arbarello, M

E. Arbarello, M. Cornalba and P. A. Griffiths, Geometry of algebraic curves. Volume II , Grundlehren der mathematischen Wissenschaften, 268, Springer, Heidelberg, 2011

work page 2011
[4]

Bini and C

G. Bini and C. Fontanari, Moduli of curves and spin structures via algebraic geometry, Trans. Amer. Math. Soc. 358 (2006), no. 7, 3207--3217

work page 2006
[5]

Agostini and I

D. Agostini and I. Barros, Pencils on surfaces with normal crossings and the Kodaira dimension of M _ g,n , Forum Math. Sigma 9 (2021), Paper No. e31, 22 pp.; MR4247637

work page 2021
[6]

D. Chen, M. Costantini and M. M\"oller, On the Kodaira dimension of moduli spaces of Abelian differentials, Camb. J. Math. 12 (2024), no. 3, 623--752

work page 2024
[7]

Casnati and C

G. Casnati and C. Fontanari, On the rationality of moduli spaces of pointed curves, J. Lond. Math. Soc. (2) 75 (2007), no. 3, 582--596

work page 2007
[8]

Eisenbud and J

D. Eisenbud and J. D. Harris, The Kodaira dimension of the moduli space of curves of genus 23 , Invent. Math. 90 (1987), no. 2, 359--387

work page 1987

Show all 17 references

[9]

Farkas, Koszul divisors on moduli spaces of curves, Amer

G. Farkas, Koszul divisors on moduli spaces of curves, Amer. J. Math. 131 (2009), no. 3, 819--867

2009
[10]

Farkas, D

G. Farkas, D. Jensen, and S. Payne, The Kodaira dimensions of _ 22 and _ 23 , preprint (2025), arXiv:2005.00622

2025 arXiv
[11]

J. D. Harris, On the Kodaira dimension of the moduli space of curves. II. The even-genus case, Invent. Math. 75 (1984), no. 3, 437--466

1984
[12]

J. D. Harris and D. B. Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), no. 1, 23--88

1982
[13]

Logan, The Kodaira dimension of moduli spaces of curves with marked points, Amer

A. Logan, The Kodaira dimension of moduli spaces of curves with marked points, Amer. J. Math. 125 (2003), no. 1, 105--138

2003
[14]

D. B. Mumford, Towards an enumerative geometry of the moduli space of curves, in Arithmetic and geometry, Vol. II , 271--328, Progr. Math., 36, Birkh\"auser Boston, Boston, MA
[15]

Reid, Canonical 3 -folds, in Journ\'ees de G\'eometrie Alg\'ebrique d'Angers, Juillet 1979/Algebraic Geometry, Angers, 1979 , pp

M. Reid, Canonical 3 -folds, in Journ\'ees de G\'eometrie Alg\'ebrique d'Angers, Juillet 1979/Algebraic Geometry, Angers, 1979 , pp. 273--310, Sijthoff & Noordhoff, Alphen aan den Rijn---Germantown, Md

1979
[16]

Severi, Sulla classificazione delle curve algebriche e sul teorema d'esistenza di Riemann , Rendiconti della R

F. Severi, Sulla classificazione delle curve algebriche e sul teorema d'esistenza di Riemann , Rendiconti della R. Accad. Naz. Lincei (1915), 877--888

1915
[17]

Tai, On the Kodaira dimension of the moduli space of abelian varieties, Invent

Y.-S. Tai, On the Kodaira dimension of the moduli space of abelian varieties, Invent. Math. 68 (1982), no. 3, 425--439

1982

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Agostini and I

D. Agostini and I. Barros, Pencils on surfaces with normal crossings and the Kodaira dimension of M _ g,n , Forum Math. Sigma 9 (2021), Paper No. e31, 22 pp

work page 2021

[2] [2]

Arbarello and M

E. Arbarello and M. Cornalba, The Picard groups of the moduli space of curves, Topology 26 (1987), no. 2, 153--171

work page 1987

[3] [3]

Arbarello, M

E. Arbarello, M. Cornalba and P. A. Griffiths, Geometry of algebraic curves. Volume II , Grundlehren der mathematischen Wissenschaften, 268, Springer, Heidelberg, 2011

work page 2011

[4] [4]

Bini and C

G. Bini and C. Fontanari, Moduli of curves and spin structures via algebraic geometry, Trans. Amer. Math. Soc. 358 (2006), no. 7, 3207--3217

work page 2006

[5] [5]

Agostini and I

D. Agostini and I. Barros, Pencils on surfaces with normal crossings and the Kodaira dimension of M _ g,n , Forum Math. Sigma 9 (2021), Paper No. e31, 22 pp.; MR4247637

work page 2021

[6] [6]

D. Chen, M. Costantini and M. M\"oller, On the Kodaira dimension of moduli spaces of Abelian differentials, Camb. J. Math. 12 (2024), no. 3, 623--752

work page 2024

[7] [7]

Casnati and C

G. Casnati and C. Fontanari, On the rationality of moduli spaces of pointed curves, J. Lond. Math. Soc. (2) 75 (2007), no. 3, 582--596

work page 2007

[8] [8]

Eisenbud and J

D. Eisenbud and J. D. Harris, The Kodaira dimension of the moduli space of curves of genus 23 , Invent. Math. 90 (1987), no. 2, 359--387

work page 1987