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arxiv: 2605.03835 · v1 · submitted 2026-05-05 · 🧮 math.AG

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Birational Classification of Orbifold Compactified Jacobians

Jeremy Feusi , Sam Molcho
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Pith reviewed 2026-05-07 14:20 UTC · model grok-4.3

classification 🧮 math.AG
keywords orbifold compactificationsbirational classificationJacobianslogarithmic geometrytoroidal compactificationssemiabelian schemesalgebraic tori
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The pith

The birational classification of orbifold compactified Jacobians reduces to finding minimal logarithmic toroidal compactifications.

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

The paper reduces the equivariant orbifold birational classification of toroidal compactifications of groups G such as algebraic tori and semiabelian schemes over a toroidal base to the task of locating minimal orbifold toroidal compactifications of G inside logarithmic geometry. This reduction is shown to be purely combinatorial, and the authors carry it out explicitly for families of algebraic tori, for Jacobians of families of nodal curves, and for semiabelian schemes whose generic fiber is abelian. The remaining general semiabelian case is reduced to an open conjecture. A sympathetic reader would care because the result turns an abstract birational problem into a concrete combinatorial one that can be solved case by case.

Core claim

The equivariant orbifold birational classification problem for families of toroidal compactifications of a group G over a toroidal base reduces to the problem of finding the minimal orbifold toroidal compactifications of G in the world of logarithmic geometry, which is shown to be a combinatorial problem. We solve the problem for families of algebraic tori, Jacobians of families of nodal curves, and semiabelian schemes with abelian generic fiber. The general semiabelian case is reduced to an open conjecture. These results generalize and geometrically interpret recent results of Schmitt.

What carries the argument

The minimal orbifold toroidal compactification of G in logarithmic geometry, which carries the combinatorial data needed for the birational classification.

If this is right

  • Families of algebraic tori admit a complete combinatorial classification of their orbifold compactifications.
  • Jacobians of nodal curves have their birational types determined by explicit minimal logarithmic models.
  • Semiabelian schemes with abelian generic fiber are classified by the same combinatorial minimal compactifications.
  • The remaining general semiabelian case depends on resolving the open conjecture about minimal models.

Where Pith is reading between the lines

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

  • If the conjecture is settled, the same combinatorial method would classify compactifications for arbitrary semiabelian schemes.
  • Logarithmic geometry software could now be used to compute these minimal models and list the birational types in low-dimensional examples.
  • The approach suggests that other equivariant compactification problems in algebraic geometry may reduce to similar log-geometric combinatorics.

Load-bearing premise

The base of the family is toroidal and the group action is equivariant so that the classification reduces to a combinatorial problem in logarithmic geometry.

What would settle it

A concrete family of algebraic tori or a nodal curve Jacobian in which the predicted birational type fails to match the type coming from the minimal logarithmic toroidal compactification.

Figures

Figures reproduced from arXiv: 2605.03835 by Jeremy Feusi, Sam Molcho.

Figure 1
Figure 1. Figure 1: An example of a tropicalization of a compactified Jacobian (left), the minimal generalized tropical compactification it is birational to (center), and the tropical Jacobian (right). This is an equivariant analogue of the figure in the intro￾duction. where the tropical Jacobian is defined in [MW22, Definition 3.6.1] for the case in which S has a global chart. From the construction on page 1510 of [MW22] of … view at source ↗
read the original abstract

We study the equivariant orbifold birational classification problem for families of toroidal compactifications of a group $G$ over a toroidal base, in the cases where $G$ is an algebraic torus or a semiabelian scheme. The classification is reduced to the problem of finding the minimal orbifold toroidal compactifications of $G$ in the world of logarithmic geometry, which is shown to be a combinatorial problem. We solve the problem for families of algebraic tori, Jacobians of families of nodal curves, and semiabelian schemes with abelian generic fiber. The general semiabelian case is reduced to an open conjecture. These results generalize and geometrically interpret recent results of Schmitt.

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

0 major / 2 minor

Summary. The paper studies the equivariant orbifold birational classification problem for families of toroidal compactifications of a group G (an algebraic torus or semiabelian scheme) over a toroidal base. It reduces the classification to the problem of finding minimal orbifold toroidal compactifications of G in logarithmic geometry, which is asserted to be combinatorial, and provides explicit solutions for families of algebraic tori, Jacobians of families of nodal curves, and semiabelian schemes with abelian generic fiber. The general semiabelian case is reduced to an open conjecture. The results generalize and geometrically interpret recent work of Schmitt.

Significance. If the combinatorial reduction and explicit solutions hold, the work provides a valuable geometric framework for birational classification of orbifold compactifications in logarithmic geometry. The explicit combinatorial solutions for algebraic tori, nodal curve Jacobians, and abelian-generic semiabelian schemes, together with the reduction to combinatorics, constitute a clear strength that allows independent verification and extends Schmitt's results in a structured way.

minor comments (2)
  1. [Abstract] The abstract refers to 'recent results of Schmitt' without a specific citation; the introduction should include the precise reference to Schmitt's paper being generalized and interpreted.
  2. [Introduction] Notation for the group G, the toroidal base, and the equivariant setup should be introduced with explicit definitions in the first section to improve accessibility for readers working in logarithmic geometry.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of the manuscript, as well as for the recommendation of minor revision. We are pleased that the combinatorial reduction and explicit solutions for the listed cases are viewed as strengths.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper reduces the equivariant orbifold birational classification of toroidal compactifications of G to the problem of finding minimal orbifold toroidal compactifications in logarithmic geometry, which it shows is combinatorial. Explicit solutions are given for algebraic tori, Jacobians of nodal curves, and semiabelian schemes with abelian generic fiber, while the general semiabelian case is left as an open conjecture. This reduction and the combinatorial solutions are presented as independent of the target classification, generalizing Schmitt's prior results without self-referential definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The chain relies on external logarithmic geometry and toroidal compactification theory rather than reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on established frameworks of logarithmic geometry and toroidal embeddings without introducing new free parameters or postulated entities.

axioms (1)
  • standard math Standard axioms and properties of logarithmic schemes, toroidal embeddings, and orbifold structures as developed in prior literature.
    The reduction of the birational classification to a combinatorial problem invokes these background results from algebraic and logarithmic geometry.

pith-pipeline@v0.9.0 · 5401 in / 1380 out tokens · 74286 ms · 2026-05-07T14:20:36.670519+00:00 · methodology

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Reference graph

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