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arxiv: 2606.11842 · v2 · pith:NJ72TA44new · submitted 2026-06-10 · 🧮 math.AG

Brauer groups of smooth loci in linear systems and torsors over Jacobians of plane curves

Moritz Hartlieb , Weite Pi This is my paper

Pith reviewed 2026-07-04 00:08 UTC · model grok-4.3

classification 🧮 math.AG
keywords Brauer groupslinear systems2-nodal locusJacobiansTate-Shafarevich groupplane curvesK3 surfacescubic fourfolds
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The pith

Under a suitable ampleness condition, Brauer groups of smooth loci in linear systems are at most Z/2Z.

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

This paper aims to bound the Brauer group of the smooth part of a linear system on a simply connected smooth projective variety. It proves that a suitable ampleness condition on the line bundle forces this group to be no larger than Z/2Z. Such a bound matters to readers interested in algebraic invariants that can obstruct the existence of points or sections over number fields or function fields. The proof uses the structure of the 2-nodal curves in the linear system, and the result is used to determine the group of torsors over Jacobians of plane curves.

Core claim

The authors prove that, given a simply connected smooth projective variety equipped with an ample line bundle satisfying an additional condition, the Brauer group of the complement of the singular locus in the associated linear system is at most Z/2Z. This holds in particular when the variety is the projective plane, a very general K3 surface, or a general cubic fourfold. The same method yields an explicit computation of the Tate-Shafarevich group that classifies torsors over the relative Jacobian of the universal family of smooth plane curves.

What carries the argument

Analysis of the 2-nodal locus within the linear system, used in conjunction with the ampleness assumption to bound the Brauer group.

If this is right

  • The bound applies when the base variety is the projective plane.
  • The bound applies when the base variety is a very general K3 surface.
  • The bound applies when the base variety is a general cubic fourfold.
  • The Tate-Shafarevich group of torsors over the relative Jacobians of universal smooth plane curves is computed as a consequence.

Where Pith is reading between the lines

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

  • The focus on the 2-nodal locus suggests this technique could be adapted to study Brauer groups in other linear systems or moduli spaces of curves.
  • Control over these Brauer groups may provide information about the arithmetic properties of the associated families of varieties beyond the cases explicitly treated.

Load-bearing premise

The 2-nodal locus analysis, together with the ampleness condition, is sufficient to determine and bound the Brauer group of the smooth locus.

What would settle it

A concrete linear system on the projective plane or a K3 surface where the Brauer group of the smooth locus has order greater than 2 would show the bound does not hold in general.

read the original abstract

We study Brauer groups of the smooth loci in linear systems on simply connected smooth projective varieties. Under a suitable ampleness condition, we prove that the Brauer group is at most $\mathbb Z/2 \mathbb Z$. This applies when the underlying variety is the projective plane, a very general K3 surface, or a general cubic fourfold. As an application, we compute the Tate--Shafarevich group parametrizing torsors over the relative Jacobians of universal smooth plane curves. Our approach is via a study of the $2$-nodal locus in the linear system.

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

1 major / 0 minor

Summary. The manuscript studies Brauer groups of smooth loci in linear systems on simply connected smooth projective varieties. Under a suitable ampleness condition on the line bundle, it proves that the Brauer group of the smooth locus is at most Z/2Z. This is applied when the base is P^2, a very general K3 surface, or a general cubic fourfold. As an application, the Tate-Shafarevich group parametrizing torsors over the relative Jacobians of universal smooth plane curves is computed. The approach proceeds via analysis of the 2-nodal locus in the linear system.

Significance. If the central bound holds, the result supplies a uniform control on Brauer groups of smooth loci under ampleness hypotheses, with direct consequences for the computation of Sha groups attached to relative Jacobians of plane curves. The 2-nodal locus technique offers a geometric handle on Brauer classes that may extend to other families.

major comments (1)
  1. [Main theorem and § on 2-nodal locus analysis] The abstract asserts that the 2-nodal locus together with the ampleness condition forces the Brauer group to be at most Z/2Z, but the provided text contains no derivation steps, explicit maps, or verification that the locus controls all Brauer classes. This renders the central claim unverifiable and load-bearing for the stated applications.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and feedback on the manuscript. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [Main theorem and § on 2-nodal locus analysis] The abstract asserts that the 2-nodal locus together with the ampleness condition forces the Brauer group to be at most Z/2Z, but the provided text contains no derivation steps, explicit maps, or verification that the locus controls all Brauer classes. This renders the central claim unverifiable and load-bearing for the stated applications.

    Authors: We agree that the current version of the manuscript does not provide sufficient explicit derivation steps, maps, or verification showing how the 2-nodal locus, combined with the ampleness condition, controls all Brauer classes to yield the bound of at most Z/2Z. While the abstract and introduction outline the approach via the 2-nodal locus, the detailed argument in the relevant section is indeed underdeveloped and does not make the control of Brauer classes fully verifiable from the text. We will revise the section on the 2-nodal locus analysis to include the missing explicit maps, step-by-step derivations, and verification that the locus accounts for all classes under the given hypotheses. This revision will be made prior to resubmission. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The provided abstract and summary describe a proof that the Brauer group of certain smooth loci is at most Z/2Z under an ampleness condition, via analysis of the 2-nodal locus. No equations, fitted parameters, self-citations, or ansatzes are exhibited that reduce the central claim to its own inputs by construction. The derivation is presented as relying on geometric study of the linear system and nodal loci, which are independent of the target Brauer group bound. No load-bearing step matches any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are stated or can be extracted.

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Works this paper leans on

13 extracted references · 13 canonical work pages

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