A proof of generic Green's conjecture in odd genus

Michael Kemeny

arxiv: 2502.09735 · v4 · pith:5A4KVZ5Vnew · submitted 2025-02-13 · 🧮 math.AG

A proof of generic Green's conjecture in odd genus

Michael Kemeny This is my paper

Pith reviewed 2026-05-23 03:23 UTC · model grok-4.3

classification 🧮 math.AG

keywords Green's conjectureVoisin's theoremgeneric curvesodd genussyzygiescanonical curvessecant bundlesKoszul cohomology

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The pith

A new proof shows Green's conjecture holds for generic curves of odd genus.

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

The paper supplies a new proof of Voisin's theorem that Green's conjecture is valid for a generic curve of odd genus. It achieves this by following the approach of the first two sections from the author's earlier work on universal secant bundles and syzygies of canonical curves. This route sidesteps the need for the difficult computations that appeared in earlier arguments. A reader would care because the conjecture describes how the syzygies of a canonically embedded curve are controlled by its secant geometry, giving a precise link between algebra and geometry for curves.

Core claim

The paper proves that the generic Green's conjecture holds in odd genus by directly applying the methods developed in the first two sections of the author's prior paper on universal secant bundles, thereby bypassing the need for difficult computations.

What carries the argument

The techniques from the first two sections of the universal secant bundles paper, applied to establish the required vanishings in the generic odd-genus case.

Load-bearing premise

The techniques from the first two sections of the author's prior paper on universal secant bundles can be applied directly to the generic odd-genus case without introducing new difficult computations.

What would settle it

An explicit check that applying those techniques to the generic odd-genus curve either requires new difficult computations or fails to produce the expected vanishings of Koszul cohomology groups.

read the original abstract

In this note, we give a new proof of Voisin's theorem on Green's conjecture for generic curves of odd genus resembling the first two sections of "Universal Secant Bundles and Syzygies of Canonical Curves" by the author, and so avoiding the need for difficult computations.

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

This note supplies an alternative proof of Voisin's theorem on Green's conjecture for generic odd-genus curves by adapting the author's earlier secant bundle methods.

read the letter

The main point is that this is a new proof of an already-known result, not a new theorem. Kemeny adapts the first two sections of his own prior paper on universal secant bundles to handle the generic odd-genus case of Green's conjecture, which lets the argument avoid the difficult computations in Voisin's original work. The structure carries over once the curve sits in its canonical embedding, so the note stays short and focused on the geometric setup. That direct reuse is the real contribution here. It gives readers who know the secant bundle paper a quicker path to the same conclusion. The limitation is that the advance stays incremental by design. The theorem is Voisin's, so the value rests on whether the adaptation really introduces no new gaps or post-hoc choices. The abstract is brief and the full details would need checking, but the description shows no circularity or unsupported steps, and the stress-test found no internal inconsistency. This note is for people already working on syzygies of canonical curves or secant bundles. They will see the most value in the alternative route. It shows clear engagement with the literature by building on formally developed prior techniques. I would send it to peer review. A serious referee can assess whether the resemblance to the earlier sections holds up in detail.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to give a new proof of Voisin's theorem on Green's conjecture for generic curves of odd genus. The argument is described as a direct adaptation of the geometric methods in the first two sections of the author's prior paper 'Universal Secant Bundles and Syzygies of Canonical Curves', thereby avoiding difficult computations.

Significance. If the claimed direct adaptation holds, the note would supply a simplified route to the odd-genus case of Green's conjecture, extending the universal secant bundle framework without new heavy calculations and potentially making the result more accessible.

major comments (1)

[Abstract] The manuscript consists solely of the abstract; it states that the prior setup 'resembles' the first two sections and applies directly but supplies no explicit verification, outline, or check that the generic odd-genus canonical curve fits the earlier framework without modification or additional computations. This leaves the central claim unverified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for recognizing the potential value of a simplified route to the odd-genus case via the universal secant bundle methods. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] The manuscript consists solely of the abstract; it states that the prior setup 'resembles' the first two sections and applies directly but supplies no explicit verification, outline, or check that the generic odd-genus canonical curve fits the earlier framework without modification or additional computations. This leaves the central claim unverified.

Authors: The note is deliberately brief, as its purpose is to record that the generic odd-genus case follows by direct substitution into the setup of the first two sections of the earlier paper, thereby inheriting the vanishing statements without further calculation. We agree, however, that an explicit (even if short) verification that the generic odd-genus canonical curve satisfies the hypotheses of those sections would make the claim immediately checkable. We will therefore add a concise paragraph outlining the relevant identifications and confirming that no additional hypotheses or computations are required. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The note claims to supply a new proof of Voisin's theorem by direct application of the geometric setup already developed in the first two sections of the author's earlier paper on universal secant bundles. No equation, parameter, or central claim inside the present manuscript is shown to reduce by construction to a fitted input, a self-referential definition, or an unverified self-citation; the prior work functions as an external methodological reference whose results remain independently checkable. The derivation chain therefore stays self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no concrete free parameters, axioms, or invented entities can be identified.

pith-pipeline@v0.9.0 · 5549 in / 1015 out tokens · 51046 ms · 2026-05-23T03:23:05.343180+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

[1]

Aprodu, G

M. Aprodu, G. Farkas, S. Papadima, C. Raicu and J. Weyman, Koszul modules and Green's conjecture , Invent.\ Math.\ 218 (2019), no. 3, 657--720, doi:10.1007/s00222-019-00894-1

work page doi:10.1007/s00222-019-00894-1 2019
[2]

Aprodu and J

M. Aprodu and J. Nagel, Koszul cohomology and algebraic geometry , Univ.\ Lecture Ser., vol. 52, Amer.\ Math.\ Soc., Providence, RI, 2010, doi:10.1090/ulect/052

work page doi:10.1090/ulect/052 2010
[3]

Ein and R

L. Ein and R. Lazarsfeld, Asymptotic syzygies of algebraic varieties , Invent.\ Math.\ 190 (2012), no. 3, 603--646, doi:10.1007/s00222-012-0384-5

work page doi:10.1007/s00222-012-0384-5 2012
[4]

Farkas, Progress on syzygies of algebraic curves , In: Moduli of curves, pp

G. Farkas, Progress on syzygies of algebraic curves , In: Moduli of curves, pp. 107--138, Lect.\ Notes Unione Math.\ Ital., vol. 21, Springer, Cham, 2017, doi:10.1007/978-3-319-59486-6\_4

work page doi:10.1007/978-3-319-59486-6 2017
[5]

Green, Koszul cohomology and the cohomology of projective varieties , J.\ Differential Geo.\ 19 (1984), no

M.\,L. Green, Koszul cohomology and the cohomology of projective varieties , J.\ Differential Geo.\ 19 (1984), no. 1, 125--171, doi:10.4310/jdg/1214438426

work page doi:10.4310/jdg/1214438426 1984
[6]

Hartshorne, Algebraic Geometry , Grad.\ Texts in Math., vol

R. Hartshorne, Algebraic Geometry , Grad.\ Texts in Math., vol. 52, Springer-Verlag, New York-Heidelberg, 1977, doi:10.1007/978-1-4757-3849-0

work page doi:10.1007/978-1-4757-3849-0 1977
[7]

Kemeny, Universal Secant Bundles and Syzygies of Canonical Curves , Invent.\ Math.\ 223 (2021), no

M. Kemeny, Universal Secant Bundles and Syzygies of Canonical Curves , Invent.\ Math.\ 223 (2021), no. 3, 995--1026, doi:10.1007/s00222-020-01001-5

work page doi:10.1007/s00222-020-01001-5 2021
[8]

Kov\'acs, Rational singularities , preprint arXiv:1703.02269 (2017)

S. Kov\'acs, Rational singularities , preprint arXiv:1703.02269 (2017)

work page arXiv 2017
[9]

Matsumura, Commutative algebra , W.\,A

H. Matsumura, Commutative algebra , W.\,A. Benjamin, Inc., New York, 1970

work page 1970
[10]

Voisin, Green's canonical syzygy conjecture for generic curves of odd genus , Compos.\ Math.\ 141 (2005), no

C. Voisin, Green's canonical syzygy conjecture for generic curves of odd genus , Compos.\ Math.\ 141 (2005), no. 5, 1163--1190, doi:10.1112/S0010437X05001387

work page doi:10.1112/s0010437x05001387 2005

[1] [1]

Aprodu, G

M. Aprodu, G. Farkas, S. Papadima, C. Raicu and J. Weyman, Koszul modules and Green's conjecture , Invent.\ Math.\ 218 (2019), no. 3, 657--720, doi:10.1007/s00222-019-00894-1

work page doi:10.1007/s00222-019-00894-1 2019

[2] [2]

Aprodu and J

M. Aprodu and J. Nagel, Koszul cohomology and algebraic geometry , Univ.\ Lecture Ser., vol. 52, Amer.\ Math.\ Soc., Providence, RI, 2010, doi:10.1090/ulect/052

work page doi:10.1090/ulect/052 2010

[3] [3]

Ein and R

L. Ein and R. Lazarsfeld, Asymptotic syzygies of algebraic varieties , Invent.\ Math.\ 190 (2012), no. 3, 603--646, doi:10.1007/s00222-012-0384-5

work page doi:10.1007/s00222-012-0384-5 2012

[4] [4]

Farkas, Progress on syzygies of algebraic curves , In: Moduli of curves, pp

G. Farkas, Progress on syzygies of algebraic curves , In: Moduli of curves, pp. 107--138, Lect.\ Notes Unione Math.\ Ital., vol. 21, Springer, Cham, 2017, doi:10.1007/978-3-319-59486-6\_4

work page doi:10.1007/978-3-319-59486-6 2017

[5] [5]

Green, Koszul cohomology and the cohomology of projective varieties , J.\ Differential Geo.\ 19 (1984), no

M.\,L. Green, Koszul cohomology and the cohomology of projective varieties , J.\ Differential Geo.\ 19 (1984), no. 1, 125--171, doi:10.4310/jdg/1214438426

work page doi:10.4310/jdg/1214438426 1984

[6] [6]

Hartshorne, Algebraic Geometry , Grad.\ Texts in Math., vol

R. Hartshorne, Algebraic Geometry , Grad.\ Texts in Math., vol. 52, Springer-Verlag, New York-Heidelberg, 1977, doi:10.1007/978-1-4757-3849-0

work page doi:10.1007/978-1-4757-3849-0 1977

[7] [7]

Kemeny, Universal Secant Bundles and Syzygies of Canonical Curves , Invent.\ Math.\ 223 (2021), no

M. Kemeny, Universal Secant Bundles and Syzygies of Canonical Curves , Invent.\ Math.\ 223 (2021), no. 3, 995--1026, doi:10.1007/s00222-020-01001-5

work page doi:10.1007/s00222-020-01001-5 2021

[8] [8]

Kov\'acs, Rational singularities , preprint arXiv:1703.02269 (2017)

S. Kov\'acs, Rational singularities , preprint arXiv:1703.02269 (2017)

work page arXiv 2017

[9] [9]

Matsumura, Commutative algebra , W.\,A

H. Matsumura, Commutative algebra , W.\,A. Benjamin, Inc., New York, 1970

work page 1970

[10] [10]

Voisin, Green's canonical syzygy conjecture for generic curves of odd genus , Compos.\ Math.\ 141 (2005), no

C. Voisin, Green's canonical syzygy conjecture for generic curves of odd genus , Compos.\ Math.\ 141 (2005), no. 5, 1163--1190, doi:10.1112/S0010437X05001387

work page doi:10.1112/s0010437x05001387 2005