Complete Families of Curves in the Moduli Space of Genus g Curves

Chloe Stewart

Complete Families of Curves in the Moduli Space of Genus g Curves

T0 review · reviewed 2026-06-28 · grok-4.3

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

Pith's one-line read A generalized construction produces new complete curves in the moduli space of genus g curves along with a genus formula.

desk verdict This extends González Díez–Harvey with new complete curves in M_g plus a Galois genus formula, but the abstract alone leaves the actual derivations uncheckable. read the letter →

arxiv 2606.03779 v1 pith:3A4TW5SD submitted 2026-06-02 math.AG math.NT

Chloe Stewart This is my paper · ORCID

classification math.AGmath.NT

keywords modulispaceofcurvescompletegenusgGaloistheoryfunctionfieldsalgebraicgeometrycurvefamiliesM_g

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

The paper sets out to generalize a construction that yields complete curves inside the moduli space of smooth curves of genus g. This generalization creates additional such curves for each g at least 3. It also derives a formula for the genus of the source curve in each family by using Galois theory on function fields. A reader would care because these complete curves serve as explicit examples that help map out the structure of the moduli space, which encodes all possible smooth curves of a given genus. If the claims hold, researchers gain more concrete objects to analyze within this space.

What carries the argument

The generalized construction of families of curves mapping to M_g, with Galois theory applied to compute the genus of the parameter curve T.

What would settle it

For a specific small genus g, compute one of the constructed curves T and check if it is complete in M_g or if its genus matches the formula.

Watch

Extended reading notes

Core claim

Generalizing the construction produces new complete curves in the moduli space M_g. Galois theory for function fields supplies a formula for the genus of each such curve T.

Load-bearing premise

The generalized construction still produces complete curves when applied in the new settings described.

Editorial extensions

If this is right

Additional complete curves become available in M_g beyond earlier examples.
The genus of T is given by an explicit formula in each case.
These new curves can be studied to understand properties of the moduli space.
Galois theory provides a systematic way to determine the genera.

Reading between the lines

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

This method could potentially be adapted to construct complete surfaces or higher-dimensional subvarieties in M_g.
The genus formula might relate to other invariants such as the degree of the map or ramification data.
Exploring the geometry of these specific curves T could reveal patterns in the moduli space that apply more broadly.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper generalizes the González Díez–Harvey construction to produce additional complete curves in M_g for g ≥ 3 and supplies a genus formula for the parameter curve T derived from Galois theory on function fields.

What is new is the claim of further families beyond the original ones together with an explicit genus computation. The abstract frames this as a direct extension rather than a restatement, and the Galois-theoretic approach is a standard tool for extracting genus information in this setting.

The work stays within established methods and cites the relevant prior result cleanly. That keeps the contribution focused and avoids unnecessary machinery.

The main limitation is that only the abstract is available, so there is no way to inspect the actual generalization step or verify that the Galois formula applies without gaps to the new families. The soundness score stays low until the body is checked.

This is aimed at specialists who track explicit complete subvarieties in moduli spaces of curves. Someone compiling lists of examples or studying their geometry could find the additional curves and the genus formula useful.

It deserves peer review so the details can be examined by people who work in this corner of algebraic geometry.

Referee Report

0 major / 0 minor

Summary. The paper generalizes the González Díez–Harvey construction to produce new complete curves in the moduli space ℳ_g of smooth genus-g curves for every g ≥ 3. It also derives an explicit formula for the genus of each such parameter curve T by applying Galois theory to the associated function fields.

Significance. If the generalization is valid, the work supplies additional explicit examples of complete curves in ℳ_g, a topic of ongoing interest because such curves are scarce and constrain the geometry of the moduli space. The Galois-theoretic genus formula offers a concrete computational tool that could be applied to other families.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary recognizing the potential interest of our generalization of the González Díez–Harvey construction and the Galois-theoretic genus formula. No major comments were provided in the report, so we have no specific points to address point-by-point. The recommendation is listed as uncertain, but absent any concrete concerns we maintain that the results as stated are correct.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation relies on external construction and standard Galois theory

full rationale

The paper generalizes the González Díez–Harvey construction (external citation) to produce complete curves in M_g and derives a genus formula for T via Galois theory on function fields. Both steps are standard techniques in algebraic geometry and function field theory; the abstract and description give no indication that any prediction reduces to a fitted input, self-definition, or load-bearing self-citation chain. The result is self-contained against external benchmarks and does not invoke uniqueness theorems or ansatzes from the authors' prior work.

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

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the central claim rests on the validity of the cited prior construction and on standard facts from Galois theory of function fields.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Complete Families of Curves in the Moduli Space of Genus g Curves." pith.science (2026). https://pith.science/paper/3A4TW5SD

@misc{pith2026260603779,
  author       = {Pith},
  title        = {Pith review of: Complete Families of Curves in the Moduli Space of Genus g Curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3A4TW5SD}},
  note         = {Machine review of arXiv:2606.03779}
}

abstract

Let $\mathcal{M}_g$ be the moduli space of smooth curves of genus $g$. The image of a non-constant morphism from a curve $T$ to $\mathcal{M}_g$ is a curve in $\mathcal{M}_g$. By work of Gonz\'alez D\'iez and Harvey, for every integer $g \geq 3$, there exists a complete curve in $\mathcal{M}_g$. Here we generalize the construction to produce new complete curves in $\mathcal{M}_g$. We also find a formula for the genus of each curve $T$ using Galois theory for function fields.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 1 canonical work pages

[1]

, journal=

González Díez, Gabino and Harvey, William J. , journal=. On complete curves in moduli space. 1991 , publisher=

1991
[2]

Geom Dedicata 56 , pages=

Explicit complete curves in the moduli space of curves of genus three , author=. Geom Dedicata 56 , pages=. 1995 , publisher=

1995
[3]

Recent Work on

Harris, Joe , month=. Recent Work on. Proceedings of International Congress of Mathematicians, Warsaw , pages=. 1983 , publisher=

1983
[4]

Complete Subvarieties of

Choi, Daebeom , year=. Complete Subvarieties of. 2304.08568 , archivePrefix=

work page arXiv
[5]

A Bound on the Dimensions of Complete Subvarieties of Complete Subvarieties of

Diaz, Steven , month=. A Bound on the Dimensions of Complete Subvarieties of Complete Subvarieties of. Duke Mathematical Journal, vol. 51 , pages=. 1984 , publisher=

1984
[6]

Complete subvarieties of moduli spaces of algebraic curves , author=
[7]

On the tautological ring of

Looijenga, Eduard , journal=. On the tautological ring of. 1995 , volume=

1995
[8]

Journal für die reine und angewandte Mathematik , doi =

Covers of elliptic curves and the moduli space of stable curves , author =. Journal für die reine und angewandte Mathematik , doi =

Show all 11 references

[9]

Serdica Mathematical Journal , publisher =

Subvarieties of the hyperelliptic moduli determined by group actions , author =. Serdica Mathematical Journal , publisher =
[10]

Transformation Groups , doi =

On non-normal subvarieties of the moduli space of riemann surfaces , author =. Transformation Groups , doi =
[11]

Journal of Algebra and Its Applications , volume =

Shaska, Tanush , title =. Journal of Algebra and Its Applications , volume =. 2004 , doi =

2004

Pith tools

Reviewed June 28, 2026 · model on record in the stance chip above.

[1] [1]

, journal=

González Díez, Gabino and Harvey, William J. , journal=. On complete curves in moduli space. 1991 , publisher=

1991

[2] [2]

Geom Dedicata 56 , pages=

Explicit complete curves in the moduli space of curves of genus three , author=. Geom Dedicata 56 , pages=. 1995 , publisher=

1995

[3] [3]

Recent Work on

Harris, Joe , month=. Recent Work on. Proceedings of International Congress of Mathematicians, Warsaw , pages=. 1983 , publisher=

1983

[4] [4]

Complete Subvarieties of

Choi, Daebeom , year=. Complete Subvarieties of. 2304.08568 , archivePrefix=

work page arXiv

[5] [5]

A Bound on the Dimensions of Complete Subvarieties of Complete Subvarieties of

Diaz, Steven , month=. A Bound on the Dimensions of Complete Subvarieties of Complete Subvarieties of. Duke Mathematical Journal, vol. 51 , pages=. 1984 , publisher=

1984

[6] [6]

Complete subvarieties of moduli spaces of algebraic curves , author=

[7] [7]

On the tautological ring of

Looijenga, Eduard , journal=. On the tautological ring of. 1995 , volume=

1995

[8] [8]

Journal für die reine und angewandte Mathematik , doi =

Covers of elliptic curves and the moduli space of stable curves , author =. Journal für die reine und angewandte Mathematik , doi =