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arxiv: 2606.25576 · v1 · pith:UF34RBHCnew · submitted 2026-06-24 · 🧮 math.AG

Extremal Effective Cycles and Nef Line Bundles on \(overline{rm{M}}_(g,n)\)

Daebeom Choi This is my paper

Pith reviewed 2026-06-25 20:03 UTC · model grok-4.3

classification 🧮 math.AG
keywords effective conesmoduli of curvesextremal raysnef divisorsboundary stratakappa classesnon-polyhedral conesstable curves
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The pith

The effective cone of codimension-k cycles on the moduli space of stable pointed curves has infinitely many extremal rays and is non-polyhedral once the number of marked points meets explicit thresholds.

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

The paper establishes that the effective cones Eff^k of the compactified moduli space of genus-g curves with n marked points contain infinitely many extremal rays when k is at least 2, genus at least 3, and n at least 2k minus 2. It further shows these cones are non-polyhedral for k at least 2, any genus at least 1, and n at least k plus 5, while every rational tails boundary stratum is itself extremal. The proofs refine earlier arguments that used only semi-ample divisors by extending them to a larger class of nef divisors. This removes earlier genus-dependent restrictions on the number of marked points required for the statements to hold.

Core claim

Eff^k(ar M_{g,n}) has infinitely many extremal rays for k ≥ 2, g ≥ 3 and n ≥ 2k-2; it is non-polyhedral for k ≥ 2, g ≥ 1 and n ≥ k+5; and every rational tails boundary stratum spans an extremal ray. These conclusions follow from extending Chen-Coskun style tests based on morphisms to the setting of nef but non-semi-ample semigroup kappa divisors.

What carries the argument

semigroup kappa divisors, which are specific nef line bundles that certify extremal rays via intersection tests even when they do not arise from morphisms

If this is right

  • Adding marked points beyond the stated thresholds forces the effective cone to have infinitely many rays independent of further increases in genus.
  • Rational tails boundary classes remain extremal in every range covered by the theorems.
  • The non-polyhedrality statement applies already at genus 1 once n reaches k+5.
  • Previous genus-dependent bounds on n are replaced by uniform thresholds depending only on k.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same technique may produce infinitely many rays in Eff^k for other families of varieties once suitable nef classes are identified.
  • The results suggest that the effective cone stabilizes in its ray structure once n is large relative to k.
  • Explicit low-genus calculations for n just above k+5 could confirm or limit the non-polyhedral range.

Load-bearing premise

That the semigroup kappa divisors can be used to produce valid intersection tests for extremality without gaps or extra genus restrictions beyond those stated.

What would settle it

An explicit computation of Eff^2(ar M_{3,2}) showing only finitely many extremal rays would falsify the infinite-rays claim for the boundary case n=2k-2.

Figures

Figures reproduced from arXiv: 2606.25576 by Daebeom Choi.

Figure 1
Figure 1. Figure 1: Diagrammatic description. Blue: nodes. Red: marked points. Note that we draw only the degeneration of one of the genus 0 curves with 4 special points, since the degeneration of the other such curve cannot intersect ∆2,∅ because of the red dot (a marked point) on the right. This is a typical phenomenon: the only relevant degeneration occurs at the smoothed component. Strictly speaking, choosing a suitable r… view at source ↗
Figure 2
Figure 2. Figure 2: The case n = 1 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The case n = 2 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The case n = 3. The general case is obtained simply by increasing the genus and adding more marked points. More precisely, to increase the genus, we can keep adding two components for each additional genus, as in the figures above. To increase the number of marked points by one, we attach an additional rational component carrying the new marking to the lower-left component of fig. 4, in the same pattern. □… view at source ↗
read the original abstract

There has been a growing body of work devoted to the study of effective cones of codimension-\(k\) cycles \(\text{Eff}^k(\overline{\rm{M}}_{g,n})\) on \(\overline{\rm{M}}_{g,n}\), the moduli space of \(n\) pointed stable curves of genus \(g\). In this paper, we remove the genus-dependence present in previous bounds on the number of marked points, and prove the following results: (1) \(\text{Eff}^k(\overline{\rm{M}}_{g,n})\) has infinitely many extremal rays for \(k\ge 2\), \(g\ge 3\) and \(n\ge 2k-2\), and (2) \(\text{Eff}^k(\overline{\rm{M}}_{g,n})\) is non-polyhedral for \(k\ge 2\), \(g\ge 1\) and \(n\ge k+5\). Moreover, we show that (3) every rational tails boundary stratum spans an extremal ray. Our method refines that of Chen and Coskun by extending arguments based on morphisms, or equivalently semiample divisors, to a setting that also allows for the use of nef divisors. Certain non-semiample nef divisors on \(\overline{\rm{M}}_{g,n}\), namely so-called semigroup kappa divisors of a particular kind, play a crucial role.

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.

Referee Report

2 major / 1 minor

Summary. This paper proves three results on the effective cones Eff^k(ar M_{g,n}) of codimension-k cycles: it has infinitely many extremal rays for k≥2, g≥3, n≥2k-2; it is non-polyhedral for k≥2, g≥1, n≥k+5; and every rational tails boundary stratum spans an extremal ray. The proofs refine Chen and Coskun's method by extending it from semi-ample divisors to nef but not semi-ample semigroup kappa divisors.

Significance. If the results hold, they advance the field by removing genus dependence from previous bounds and showing greater complexity in the effective cones. The methodological extension to kappa divisors is notable and could have wider applications. The work is a proof-based contribution building directly on cited prior results.

major comments (2)
  1. [The refinement of Chen-Coskun arguments (in the proofs of the main theorems)] The central claims depend on the intersection positivity and supporting hyperplane properties transferring to the semigroup kappa divisors. Since these are not semi-ample and do not come from morphisms, the manuscript needs to supply explicit verification that the kappa divisors intersect the test families non-negatively and vanish only on the claimed rays, without introducing extra restrictions on g or n. This verification is load-bearing for both the infinite extremal rays and non-polyhedrality statements.
  2. [Statement of the ranges in the main theorems] The paper asserts the results hold for the given ranges without genus-dependent restrictions, but the extension may require additional checks. The authors should confirm in the text that the arguments apply verbatim in the stated ranges for g and n.
minor comments (1)
  1. [Abstract] Ensure consistent use of mathematical notation for the effective cone throughout the paper and abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. Below we address the major comments point by point.

read point-by-point responses

  1. Referee: [The refinement of Chen-Coskun arguments (in the proofs of the main theorems)] The central claims depend on the intersection positivity and supporting hyperplane properties transferring to the semigroup kappa divisors. Since these are not semi-ample and do not come from morphisms, the manuscript needs to supply explicit verification that the kappa divisors intersect the test families non-negatively and vanish only on the claimed rays, without introducing extra restrictions on g or n. This verification is load-bearing for both the infinite extremal rays and non-polyhedrality statements.

    Authors: The intersection computations with the relevant test families (including rational tails strata and other standard test curves) are carried out explicitly in the proofs of Theorems 1.1 and 1.2, relying on the combinatorial definition of the semigroup kappa divisors. These calculations establish non-negativity and vanishing precisely on the claimed rays, and they are independent of further restrictions on g and n beyond those stated in the theorems. To make this verification more prominent, we will add a dedicated subsection isolating the intersection numbers in the revised manuscript. revision: yes

  2. Referee: [Statement of the ranges in the main theorems] The paper asserts the results hold for the given ranges without genus-dependent restrictions, but the extension may require additional checks. The authors should confirm in the text that the arguments apply verbatim in the stated ranges for g and n.

    Authors: The proofs are combinatorial and use only the nefness and intersection properties of the kappa divisors, which hold for all g and n satisfying the hypotheses of the theorems. We will insert an explicit remark at the beginning of Section 3 confirming that the arguments apply verbatim in the stated ranges without additional genus-dependent restrictions. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation is self-contained proof extension

full rationale

The paper derives its claims on extremal rays and non-polyhedrality of Eff^k via explicit intersection-theoretic constructions using semigroup kappa divisors, extending Chen-Coskun arguments to nef (non-semiample) cases. No step reduces a claimed result to a fitted parameter, self-definition, or load-bearing self-citation; the cited Chen-Coskun work is by non-overlapping authors and supplies independent base morphisms. The derivation chain consists of direct positivity and supporting-hyperplane verifications on constructed families of cycles, remaining independent of the target statements.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on standard facts about the moduli space of stable curves and the existence of certain nef kappa divisors; no free parameters or invented entities are visible in the abstract.

axioms (2)
  • standard math Standard properties of the moduli space ar M_{g,n}, its boundary strata, and the effective cone Eff^k.
    Invoked throughout the study of extremal rays and polyhedrality.
  • domain assumption Existence and nefness properties of semigroup kappa divisors of the required kind.
    Central to extending the method beyond semi-ample divisors.

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Reference graph

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