On positive cones of finite quotients of a normal variety

Ashima Bansal; Indranil Biswas; Souradeep Majumder

arxiv: 2606.29803 · v1 · pith:YIAUAG2Lnew · submitted 2026-06-29 · 🧮 math.AG

On positive cones of finite quotients of a normal variety

Ashima Bansal , Indranil Biswas , Souradeep Majumder This is my paper

Pith reviewed 2026-06-30 04:36 UTC · model grok-4.3

classification 🧮 math.AG

keywords positive conesfinite quotientsnormal projective varietiesnumerical groupspositivity propertiesfinite flat morphisms

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The pith

Finite flat quotients of a normal projective variety have numerical groups and positive cones related to those of the original variety.

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

The paper examines how positivity properties behave under finite flat quotients of a normal projective variety. It establishes explicit relations between the numerical groups of the quotient and those of the original variety, and likewise for their positive cones. A sympathetic reader cares because these relations let one transfer information about numerical equivalence and positivity from the base variety to its quotients without separate computation. The setting is restricted to finite flat morphisms from normal projective varieties.

Core claim

The numerical groups and the positive cones of these quotient varieties are related to those of the original variety.

What carries the argument

The finite flat quotient morphism, which induces corresponding maps between numerical groups and between positive cones.

Load-bearing premise

The quotients are finite and flat, and the original variety is normal and projective.

What would settle it

An explicit finite flat quotient in which the positive cone of the quotient fails to match the image of the original positive cone under the induced map on numerical classes.

read the original abstract

We study the positivity properties of finite flat quotients of a normal projective variety. The numerical groups and the positive cones of these quotient varieties are related to those of the original variety.

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

The paper relates numerical groups and positivity cones between a normal projective variety and its finite flat quotients, but the abstract gives no sign of new mechanisms or detailed proofs.

read the letter

The core claim is that numerical groups and positive cones on finite flat quotients of a normal projective variety relate back to those on the original variety. This is the one thing to take away.

The work studies positivity properties in this quotient setting. It may give explicit relations via the quotient map, which can be convenient when handling group actions or symmetric cases.

Nothing in the abstract points to new tools, first-principles derivations, or broad applications. The topic is a standard one in algebraic geometry, so the result looks like a routine extension rather than a shift in approach.

The clearest limitation is that only the abstract is visible here. Without proofs, examples, or explicit statements of the relations, it is impossible to judge whether the arguments are solid or whether they add anything beyond what follows from pushforward of classes under finite flat maps. The setting itself is the usual one where such relations are expected, so no internal contradiction shows up.

This is for people already working on positivity cones and numerical groups in algebraic geometry. A specialist might pull a reference or calculation from it if the full proofs are clean, but a general reader will not find much.

It is worth sending to a referee to check the details and see whether the relations are stated and proved in a useful way.

Referee Report

1 major / 0 minor

Summary. The manuscript studies positivity properties of finite flat quotients of a normal projective variety, claiming that the numerical groups and positive cones of the quotient varieties are related to those of the original variety.

Significance. The topic addresses a standard question in algebraic geometry concerning descent of numerical classes and positivity under finite flat morphisms. If the relations were stated precisely with proofs, the work could supply useful comparison maps between N^1 or N_1 groups and their cones. However, the provided text contains only the abstract and no derivations, theorems, or examples, so significance cannot be evaluated.

major comments (1)

No theorems, propositions, or proofs are present in the manuscript. The central claim that numerical groups and positive cones are related cannot be checked for correctness or even stated precisely, rendering the paper unverifiable.

Simulated Author's Rebuttal

1 responses · 1 unresolved

We thank the referee for their report. We acknowledge that the version under review contains only the abstract and no theorems or proofs, which prevents verification of the claims.

read point-by-point responses

Referee: No theorems, propositions, or proofs are present in the manuscript. The central claim that numerical groups and positive cones are related cannot be checked for correctness or even stated precisely, rendering the paper unverifiable.

Authors: The referee is correct: the submitted manuscript consists solely of the abstract and provides no derivations, theorems, or examples. Without these, the precise statements relating N^1, N_1, and the positive cones under finite flat quotients cannot be evaluated. We will prepare a revised version that includes the full statements, proofs, and any necessary examples. revision: yes

standing simulated objections not resolved

The current manuscript text contains no theorems or proofs, so the central claims remain unverifiable until a complete version is supplied.

Circularity Check

0 steps flagged

No significant circularity; derivation chain self-contained against external benchmarks

full rationale

The paper studies relations between numerical groups and positive cones of a normal projective variety and its finite flat quotients. No equations, fitted parameters, self-citations, or ansatzes are visible in the abstract or described setting. The claimed relations follow from standard pushforward and descent properties of the quotient map in algebraic geometry, which are independent of the present work and externally verifiable. No load-bearing step reduces to a definition, fit, or self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be identified.

pith-pipeline@v0.9.1-grok · 5544 in / 1026 out tokens · 49351 ms · 2026-06-30T04:36:39.452286+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 1 canonical work pages · 1 internal anchor

[1]

Degrees of Iterates of Rational Maps on Normal Projective Varieties

N.B. Dang, Degrees of iterates of rational maps on normal projective varieties, Proc. Lond. Math. Soc. 121 (2020), 1268--1310. We use the result from the arXiv version: arXiv:1701.07760

work page internal anchor Pith review Pith/arXiv arXiv 2020
[2]

Fulton, Intersection theory, Second edition, Springer-Verlag, Berlin, 1998

W. Fulton, Intersection theory, Second edition, Springer-Verlag, Berlin, 1998

1998
[3]

Fulger and B

M. Fulger and B. Lehmann, Morphisms and faces of pseudo-effective cones, Proc. London Math. Soc. 112 (2016), 651--676

2016
[4]

Fulger and B

M. Fulger and B. Lehmann, Positive cones of dual cycle classes, Algebraic Geometry 4 (2007), 1--28

2007
[5]

Fulger, The cones of effective cycles on projective bundles over curves, Math

M. Fulger, The cones of effective cycles on projective bundles over curves, Math. Zeit. 269 (2011), 449--459

2011
[6]

The Stacks Project Authors, Stacks Project, https://stacks.math.columbia.edu (2018)

2018

[1] [1]

Degrees of Iterates of Rational Maps on Normal Projective Varieties

N.B. Dang, Degrees of iterates of rational maps on normal projective varieties, Proc. Lond. Math. Soc. 121 (2020), 1268--1310. We use the result from the arXiv version: arXiv:1701.07760

work page internal anchor Pith review Pith/arXiv arXiv 2020

[2] [2]

Fulton, Intersection theory, Second edition, Springer-Verlag, Berlin, 1998

W. Fulton, Intersection theory, Second edition, Springer-Verlag, Berlin, 1998

1998

[3] [3]

Fulger and B

M. Fulger and B. Lehmann, Morphisms and faces of pseudo-effective cones, Proc. London Math. Soc. 112 (2016), 651--676

2016

[4] [4]

Fulger and B

M. Fulger and B. Lehmann, Positive cones of dual cycle classes, Algebraic Geometry 4 (2007), 1--28

2007

[5] [5]

Fulger, The cones of effective cycles on projective bundles over curves, Math

M. Fulger, The cones of effective cycles on projective bundles over curves, Math. Zeit. 269 (2011), 449--459

2011

[6] [6]

The Stacks Project Authors, Stacks Project, https://stacks.math.columbia.edu (2018)

2018