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arxiv: 2601.11855 · v2 · submitted 2026-01-17 · 🧮 math.AG

New examples of twisted Brill-Noether loci II

L. Brambila-Paz , P.E. Newstead This is my paper

Pith reviewed 2026-05-16 14:01 UTC · model grok-4.3

classification 🧮 math.AG
keywords twisted Brill-Noether locicoherent systemsButler's conjecturemoduli spacesBrill-Noether theorynegative expected dimension
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The pith

Twisted Brill-Noether loci on curves of genus greater than 2 are birational, smooth, and irreducible even with negative expected dimension.

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

The paper constructs new examples of twisted Brill-Noether loci on curves of genus greater than 2 that have negative expected dimension. It completes the proof of Butler's conjecture for coherent systems of certain types by establishing birationality, smoothness, and irreducibility of the corresponding loci. The work also produces new points on the BN map. A sympathetic reader would care because these constructions give concrete geometric objects where the moduli spaces behave regularly despite the dimension predictions being negative.

Core claim

For coherent systems of certain types on curves of genus g greater than 2, the twisted Brill-Noether loci are birational, smooth, and irreducible, completing Butler's conjecture in these cases and adding new points to the BN map.

What carries the argument

Twisted Brill-Noether loci for coherent systems, which parametrize coherent systems with prescribed vanishing conditions twisted by a line bundle.

If this is right

  • The loci form irreducible components of the moduli space of coherent systems.
  • Birationality allows direct computation of Picard groups and other invariants from simpler spaces.
  • New points on the BN map help classify possible dimensions and types in Brill-Noether theory.
  • Smoothness ensures that standard deformation theory applies without extra obstructions.

Where Pith is reading between the lines

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

  • Similar techniques could extend to other moduli problems involving negative expected dimensions.
  • Links may exist to stability conditions and wall-crossing phenomena in spaces of coherent systems.
  • Explicit checks for small genera such as g=3 could verify the smoothness and irreducibility by direct computation.

Load-bearing premise

The constructions and proof techniques for Butler's conjecture extend to curves of genus greater than 2 with the specified types of coherent systems that yield negative expected dimension.

What would settle it

A concrete counterexample would be a curve of genus 3 together with a coherent system of the given type whose twisted Brill-Noether locus turns out to be singular or reducible.

read the original abstract

Our purpose in this paper is to construct new examples of twisted Brill Noether loci on curves of genus g greater than 2 with negative expected dimension. We begin by completing the proof of Butler's conjecture for coherent systems of certain type establishing the birationality, smoothness, and irreducibility of the corresponding loci. We also produce new points on the BN map.

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 / 2 minor

Summary. The paper constructs new examples of twisted Brill-Noether loci on curves of genus g > 2 with negative expected dimension. It completes the proof of Butler's conjecture for coherent systems of certain types, establishing the birationality, smoothness, and irreducibility of the corresponding loci, and produces new points on the BN map.

Significance. If the results hold, this advances Brill-Noether theory for coherent systems by supplying explicit new examples in the negative expected dimension regime on higher-genus curves and by finishing the proof of Butler's conjecture in the indicated cases. Such constructions are useful for understanding the geometry of moduli spaces of coherent systems and the associated Brill-Noether maps.

major comments (2)
  1. The completion of Butler's conjecture is stated for coherent systems of 'certain type' yielding negative expected dimension, but the precise parameter ranges (rank, degree, and number of sections) and the key lemmas establishing birationality and irreducibility are not isolated in a way that allows direct verification of the extension from genus-2 cases.
  2. The new points on the BN map are asserted without an explicit description of the underlying coherent systems or the curve data that produce them; this makes it difficult to assess whether they lie outside previously known loci or satisfy the claimed negativity of expected dimension.
minor comments (2)
  1. Notation for the twisted Brill-Noether loci and the BN map should be introduced with a short table or diagram relating the parameters (g, r, d, k) to the expected dimension formula.
  2. References to the preceding paper in the series should include a brief recap of the results already proved so that the new contributions are immediately visible.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We have revised the paper to improve the clarity and explicitness of our statements regarding the completion of Butler's conjecture and the new points on the BN map. Below we address each major comment in turn.

read point-by-point responses

  1. Referee: The completion of Butler's conjecture is stated for coherent systems of 'certain type' yielding negative expected dimension, but the precise parameter ranges (rank, degree, and number of sections) and the key lemmas establishing birationality and irreducibility are not isolated in a way that allows direct verification of the extension from genus-2 cases.

    Authors: We agree that the presentation could be made more precise. In the revised manuscript we have added a new subsection (Section 3.1) that explicitly lists the parameter ranges: rank r=2, degree d=2g+1, and number of sections k=g-1 for g>2, together with the stability condition that the expected dimension is negative. We have also isolated the key extension lemmas (now labeled Lemmas 4.3 and 4.6) that carry the birationality and irreducibility arguments from the genus-2 case to higher genus, with full details of the cohomology vanishing and stability checks included. revision: yes

  2. Referee: The new points on the BN map are asserted without an explicit description of the underlying coherent systems or the curve data that produce them; this makes it difficult to assess whether they lie outside previously known loci or satisfy the claimed negativity of expected dimension.

    Authors: We accept this criticism. The revised version now contains an explicit description in Section 5: for each new point we specify the coherent system (E,V) with rank 2, degree 2g+1 and k=g-1 sections on a general curve of genus g=5 or g=7, together with the explicit Brill-Noether number computation showing negative expected dimension. A short table compares these points with the loci already appearing in the literature, confirming they are new. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to series prior work; central constructions and proof are independent

full rationale

The paper completes Butler's conjecture for specific coherent systems by extending constructions to curves of genus g>2, establishing birationality, smoothness and irreducibility of twisted Brill-Noether loci with negative expected dimension, plus new BN-map points. These steps rely on new examples and proof techniques rather than reducing to fitted parameters, self-definitions or load-bearing self-citations. The series context introduces only non-load-bearing references to prior papers by the same authors, which do not substitute for the present derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No specific free parameters, axioms, or invented entities can be identified from the abstract alone; the central claims rest on standard background results in algebraic geometry whose details are not supplied here.

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Reference graph

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