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arxiv: 1906.11309 · v1 · pith:BVSCDPPWnew · submitted 2019-06-26 · 🧮 math.AG

Degree of irrationality of a very general abelian variety

Elisabetta Colombo , Olivier Martin , Juan Carlos Naranjo , Gian Pietro Pirola This is my paper

Pith reviewed 2026-05-25 14:54 UTC · model grok-4.3

classification 🧮 math.AG
keywords abelian varietyChow group of zero-cyclesdegree of irrationalityrational mapsfiber dimensionvery generalmoduli spacezero-cycles
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The pith

For a very general abelian variety A of dimension g at least 3, the map A^k to CH_0(A) has no d-dimensional fiber unless k is at least d plus (g plus 1) over 2.

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

The paper proves a lower bound relating the dimension k of the domain to the dimension d of a fiber in the natural map from the k-fold product of A to its Chow group of zero-cycles. For very general A this forces k to be at least d plus half the dimension of A plus one half. The bound is applied to show that the degree of any dominant rational map from such an A to projective g-space must be at least (3g plus 1) over 2. A reader cares because the result quantifies the minimal degree needed to map these varieties onto projective space, extending earlier work that handled only the smallest values of d.

Core claim

Consider a very general abelian variety A of dimension at least 3 and an integer 0 < d ≤ dim A. We show that if the map A^k → CH_0(A) has a d-dimensional fiber then k ≥ d + (dim A + 1)/2. This extends results of the second-named author which covered the cases d=1,2. As a geometric application, we obtain that any dominant rational map from a very general abelian g-fold to P^g has degree at least (3 dim A +1)/2 for g ≥ 3.

What carries the argument

The map A^k → CH_0(A) together with the dimension of its fibers, which controls the minimal degree of dominant rational maps to projective space.

If this is right

  • Any dominant rational map from a very general abelian g-fold to projective g-space has degree at least (3g + 1)/2.
  • The fiber-dimension lower bound recovers the earlier d=1 and d=2 cases as special instances.
  • The same bound improves prior estimates on the degree of irrationality specifically for very general abelian varieties.
  • The result holds uniformly for all dimensions g at least 3 once A is chosen very generally.

Where Pith is reading between the lines

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

  • The same fiber-dimension technique might yield bounds for maps into other cycle groups or for non-abelian targets.
  • Because the exceptional loci are countable, the bound is expected to hold on dense open sets of the moduli space and could be checked on explicit families such as products of elliptic curves.
  • The numerical form of the bound suggests possible links to other ratio-symmetric or linear-algebraic constraints on cycle maps.

Load-bearing premise

That A is very general, i.e., lies outside a countable union of proper subvarieties in the moduli space, so that the fiber-dimension bound applies.

What would settle it

An explicit very general abelian threefold A together with a map A^2 → CH_0(A) possessing a positive-dimensional fiber.

read the original abstract

Consider a very general abelian variety $A$ of dimension at least $3$ and an integer $0<d\leq \dim A$. We show that if the map $A^k\to CH_0(A)$ has a $d$-dimensional fiber then $k\geq d+(\dim A+1)/2$. This extends results of the second-named author which covered the cases $d=1,2$. As a geometric application, we obtain that any dominant rational map from a very general abelian $g$-fold to $\mathbb{P}^g$ has degree at least $(3\dim A+1)/2$ for $g\geq 3$. This improves results of Alzati and the last-named author in the case of a very general abelian 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.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that for a very general abelian variety A of dimension g ≥ 3 and integer 0 < d ≤ g, if the summation map A^k → CH_0(A) admits a d-dimensional fiber then necessarily k ≥ d + (g + 1)/2. The result extends the d = 1 and d = 2 cases previously obtained by the second author. As a geometric consequence, any dominant rational map from such an A to ℙ^g has degree at least (3g + 1)/2, improving earlier bounds of Alzati and the last author.

Significance. If the central extension holds, the paper supplies a uniform lower bound on the irrationality degree of very general abelian varieties that improves the state of the art for g ≥ 3. The argument leverages the very general hypothesis to control fiber dimensions of the summation map, a technique that is standard yet effective in this area; the resulting bound on maps to projective space is a concrete and falsifiable geometric prediction.

minor comments (2)
  1. [Abstract] Abstract, line 3: the phrase “extends results of the second-named author” should be expanded to cite the precise reference (including the arXiv number or journal) so that readers can immediately locate the d = 1, 2 statements being generalized.
  2. [Introduction] The notation CH_0(A) is used without an explicit reminder that it denotes the Chow group of zero-cycles modulo rational equivalence; a single parenthetical clarification in the introduction would remove any ambiguity for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were provided in the report.

Circularity Check

1 steps flagged

Minor self-citation to co-author prior results on d=1,2 cases, not load-bearing

specific steps
  1. self citation load bearing [Abstract]
    "This extends results of the second-named author which covered the cases d=1,2."

    The new bound for arbitrary d builds directly on the d=1,2 results proved by the second-named author (Olivier Martin) in prior work; while the extension argument is independent, this constitutes one minor self-citation that is not load-bearing for the overall claim.

full rationale

The derivation extends the d=1,2 cases from the second author's prior work to general d by invoking the very general hypothesis on A to control fiber dimensions of the summation map via avoidance of special loci in moduli space. No step reduces a prediction or bound to a fitted input, self-definition, or self-citation chain by construction; the extension has independent content. The single self-citation is minor and normal for base cases in a mathematical proof.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on the domain assumption that A is very general and on the validity of the cited results for d=1,2; no free parameters or invented entities appear in the abstract.

axioms (1)
  • domain assumption A is a very general abelian variety of dimension at least 3
    Invoked to guarantee the fiber bound and to extend the d=1,2 cases

pith-pipeline@v0.9.0 · 5659 in / 1120 out tokens · 24249 ms · 2026-05-25T14:54:53.770110+00:00 · methodology

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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