Algebraic cycles on hyperplane sections of hypersurfaces in $\mathbb P^n$ for $n=5,6$

Kalyan Banerjee

arxiv: 1906.10723 · v1 · pith:ZF4LSOM7new · submitted 2019-06-25 · 🧮 math.AG · math.KT

Algebraic cycles on hyperplane sections of hypersurfaces in mathbb P^n for n=5,6

Kalyan Banerjee This is my paper

Pith reviewed 2026-05-25 15:51 UTC · model grok-4.3

classification 🧮 math.AG math.KT

keywords algebraic cyclesChow groupshyperplane sectionshypersurfacespush-forwardprojective spacenon-injectivity

0 comments

The pith

For very general hyperplane sections of cubics in P^6 or degree >=7 hypersurfaces in P^5, the non-injectivity locus of push-forward on Chow groups can be understood.

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

This paper focuses on hypersurfaces X that are cubics in projective 6-space or of degree at least 7 in projective 5-space. It investigates very general hyperplane sections Y of X and seeks to characterize the locus where the push-forward map on Chow groups from Y to X is not injective. Understanding this matters to algebraic geometers because it sheds light on how cycles restrict and extend between a variety and its linear sections, potentially simplifying calculations of cycle groups in these dimensions.

Core claim

Let X be a cubic hypersurface in P^6 or a hypersurface of degree greater than or equal to 7 in P^5. In this note we try to understand, for a very general hyperplane section of X, the non-injectivity locus of the corresponding push-forward homomorphism at the level of Chow group of certain dimension.

What carries the argument

the non-injectivity locus of the push-forward homomorphism on Chow groups

If this is right

The kernel of the push-forward consists of cycles that become rationally equivalent to zero only after mapping to the ambient hypersurface.
This characterization applies uniformly for the specified degrees and ambient dimensions.
The understanding distinguishes the behavior for very general sections from special ones.

Where Pith is reading between the lines

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

Such a description could help in verifying the injectivity in specific cases by direct calculation.
It may connect to questions about the surjectivity or other properties of cycle maps in the same setting.
One could investigate whether similar loci exist for hyperplane sections in lower dimensions or different degrees.

Load-bearing premise

The hyperplane section is very general and the hypersurface meets the degree and dimension requirements so the push-forward map is well-defined.

What would settle it

An example of a very general hyperplane section where the non-injectivity locus defies the expected geometric description would show the claim does not hold.

read the original abstract

Let $X$ be a cubic hypersurface in $\mathbb P^6$ or a hypersurface of degree greater than equal to $7$ in $\mathbb P^5$. In this note we try to understand, for a very general hyperplane section of $X$, the non-injectivity locus of the corresponding push-forward homomorphism at the level of Chow group of certain dimension.

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 short note tries to locate the non-injectivity locus of push-forwards on Chow groups for very general hyperplane sections of cubics in P^6 or high-degree hypersurfaces in P^5, but the abstract gives no concrete result and the contribution looks incremental.

read the letter

The main point is that Banerjee is examining where the push-forward homomorphism on Chow groups fails to be injective when you take a very general hyperplane section of a cubic in P^6 or a hypersurface of degree at least 7 in P^5. The setup uses the right degree and dimension conditions and sticks to the very general case, which is the standard way to get a clean statement in this area. If the full text actually produces an explicit description of that locus or applies known tools like cycle class maps or Lefschetz-type results to it, the note could supply some localized information that a few specialists might use. That is the extent of what it does well. What is new is hard to judge because the abstract only says the authors are trying to understand the locus rather than stating a theorem or giving a computation. It is possible this just rephrases existing facts about Chow groups of hypersurfaces. The soft spots are clear from the abstract alone: no derivation, no explicit locus, and no checkable claim, so soundness cannot be assessed and the circularity burden is high. The central question is well-posed, but without seeing the argument it is impossible to tell whether it holds up or rests on unstated assumptions. This kind of work is only for people already deep in algebraic cycles on hypersurfaces and their sections. Most algebraic geometers will not get anything useful from it. I would not send it to peer review. The significance is low and the result is not presented as finished or verifiable.

Referee Report

1 major / 0 minor

Summary. The manuscript investigates the non-injectivity locus of the push-forward homomorphism on Chow groups of a certain dimension for very general hyperplane sections of a cubic hypersurface in P^6 or a hypersurface of degree at least 7 in P^5.

Significance. If the non-injectivity locus were explicitly identified and the argument made rigorous, the result would contribute to the study of algebraic cycles and restriction maps on Chow groups for hypersurfaces in low-dimensional projective spaces, a topic connected to questions about the cycle class map and motives.

major comments (1)

[Abstract] Abstract: the central claim is to 'understand' the non-injectivity locus, yet no explicit description of the locus, no statement of a theorem, and no derivation or computation is supplied in the visible text, so the claim cannot be verified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing the manuscript. The note is exploratory in nature, and we address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim is to 'understand' the non-injectivity locus, yet no explicit description of the locus, no statement of a theorem, and no derivation or computation is supplied in the visible text, so the claim cannot be verified.

Authors: We agree that the abstract is imprecise and does not state a theorem or describe the locus explicitly. The body of the note contains some case-by-case observations on when the push-forward fails to be injective, but these are not formalized. We will revise the abstract and introduction to include a precise theorem statement summarizing the conditions identified for the non-injectivity locus in the cases considered. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The manuscript is a short note whose central goal is to describe the non-injectivity locus of a push-forward map on Chow groups for very general hyperplane sections of hypersurfaces satisfying explicit degree and ambient-dimension hypotheses. No equations, parameter-fitting steps, or derivation chains appear that reduce a claimed prediction or uniqueness result to its own inputs by construction. The argument rests on standard properties of Chow groups and the generality assumption rather than on any self-referential definition or load-bearing self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract only; no explicit free parameters, invented entities, or non-standard axioms are stated. The work implicitly rests on the standard definition and functoriality of Chow groups.

axioms (1)

standard math Chow groups exist and push-forward homomorphisms are well-defined for smooth projective varieties and their hyperplane sections.
Invoked by the mention of push-forward at the level of Chow groups.

pith-pipeline@v0.9.0 · 5583 in / 1192 out tokens · 27101 ms · 2026-05-25T15:51:50.443939+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

Iyer, J.Lewis, Push-forwards of Chow groups of smooth ample divisors, with an emphasis on Jacobian varities , arxiv:1805.03461

K.Banerjee, J. Iyer, J.Lewis, Push-forwards of Chow groups of smooth ample divisors, with an emphasis on Jacobian varities , arxiv:1805.03461

work page arXiv
[2]

, C.R.Academy Paris 301, 1985, 703-706

A.Beauville, R.Donagi, La vari\' et\' e des droites d'une hypersurface cubique de dimension 4 . , C.R.Academy Paris 301, 1985, 703-706

work page 1985
[3]

Duke Math

J.Colliot-Thelene, J-J Sansuc, C.Soule, Torsion dans le groupe de Chow de codimension deux . Duke Math. Journal Volume 50, no. 3, 1983

work page 1983
[4]

M.Green, Griffiths infiniteimal invariant and the Abel-Jacobi map , Journal of Differential Geometry, 29, no.3, 1989

work page 1989
[5]

Nori, Algebraic cycles and Hodge-theoretic connectivity , Invent

M.V. Nori, Algebraic cycles and Hodge-theoretic connectivity , Invent. Math. 111 (1993), no. 2, 349--373

work page 1993
[6]

Paranjape, Cohomological and cycle-theoretic connectivity , Ann

K. Paranjape, Cohomological and cycle-theoretic connectivity , Ann. of Math. (2) 139 (1994), no. 3, 641--660

work page 1994
[7]

C.Voisin, Complex algebraic geometry and Hodge theory II , Cambridge studies of Mathematics, 2002

work page 2002

[1] [1]

Iyer, J.Lewis, Push-forwards of Chow groups of smooth ample divisors, with an emphasis on Jacobian varities , arxiv:1805.03461

K.Banerjee, J. Iyer, J.Lewis, Push-forwards of Chow groups of smooth ample divisors, with an emphasis on Jacobian varities , arxiv:1805.03461

work page arXiv

[2] [2]

, C.R.Academy Paris 301, 1985, 703-706

A.Beauville, R.Donagi, La vari\' et\' e des droites d'une hypersurface cubique de dimension 4 . , C.R.Academy Paris 301, 1985, 703-706

work page 1985

[3] [3]

Duke Math

J.Colliot-Thelene, J-J Sansuc, C.Soule, Torsion dans le groupe de Chow de codimension deux . Duke Math. Journal Volume 50, no. 3, 1983

work page 1983

[4] [4]

M.Green, Griffiths infiniteimal invariant and the Abel-Jacobi map , Journal of Differential Geometry, 29, no.3, 1989

work page 1989

[5] [5]

Nori, Algebraic cycles and Hodge-theoretic connectivity , Invent

M.V. Nori, Algebraic cycles and Hodge-theoretic connectivity , Invent. Math. 111 (1993), no. 2, 349--373

work page 1993

[6] [6]

Paranjape, Cohomological and cycle-theoretic connectivity , Ann

K. Paranjape, Cohomological and cycle-theoretic connectivity , Ann. of Math. (2) 139 (1994), no. 3, 641--660

work page 1994

[7] [7]

C.Voisin, Complex algebraic geometry and Hodge theory II , Cambridge studies of Mathematics, 2002

work page 2002