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arxiv: 2605.19353 · v1 · pith:HYNXWHL3new · submitted 2026-05-19 · 🧮 math.AG

Remarks on basepoint-freeness thresholds of polarized abelian surfaces

Atsushi Ito This is my paper

Pith reviewed 2026-05-20 02:51 UTC · model grok-4.3

classification 🧮 math.AG
keywords basepoint-freeness thresholdpolarized abelian surfacesmoduli spaceirrational thresholdsalgebraic geometrycomplex surfaceslinear systems
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The pith

The basepoint-freeness threshold of a very general polarized abelian surface is determined exactly and can be irrational.

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

The paper calculates the precise value of the basepoint-freeness threshold for polarized abelian surfaces that are very general in their moduli space over the complex numbers. This threshold marks the infimum at which multiples of the polarization yield basepoint-free linear systems. The work also constructs the first known polarized abelian surface where this threshold is an irrational number rather than a rational one. A sympathetic reader would care because the threshold controls when the surface can be embedded into projective space without basepoints and distinguishes general behavior from special cases in the moduli space.

Core claim

For a very general polarized abelian surface over the complex numbers the basepoint-freeness threshold is determined exactly. The paper also exhibits the first polarized abelian surface for which this threshold is an irrational number.

What carries the argument

basepoint-freeness threshold of a polarized abelian surface, the infimum of real numbers t such that the linear system given by t times the polarization is basepoint-free

If this is right

  • The threshold takes a definite numerical value for every very general polarized abelian surface.
  • Irrational values of the threshold occur for some polarized abelian surfaces.
  • The distinction between general and special loci in the moduli space controls whether the threshold is rational or irrational.

Where Pith is reading between the lines

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

  • The same threshold may take irrational values for polarized abelian varieties of higher dimension.
  • Explicit computation on non-general surfaces could reveal how the threshold jumps when crossing special loci in the moduli space.
  • The result may connect to questions about the minimal degree of projective embeddings of abelian surfaces.

Load-bearing premise

The polarized abelian surface is very general, lying outside a countable union of proper closed subvarieties in the moduli space.

What would settle it

An explicit computation of the threshold on a concrete very general polarized abelian surface that yields a value different from the one claimed or that is always rational would falsify the determination.

read the original abstract

We determine the basepoint-freeness threshold of a very general polarized abelian surface over the field of complex numbers. We also give the first example of a polarized abelian surface whose basepoint-freeness threshold is irrational.

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 / 3 minor

Summary. The manuscript determines the basepoint-freeness threshold of a very general polarized abelian surface over the complex numbers. It also constructs the first explicit example of a polarized abelian surface whose basepoint-freeness threshold is irrational.

Significance. If the results hold, the work advances the study of linear systems on abelian varieties by establishing constancy of the threshold outside a countable union of proper closed loci in the moduli space and by exhibiting an irrational value via explicit computation on a specific surface (likely using endomorphism rings or Fourier-Mukai transforms). The provision of a parameter-free determination for the very general case and a falsifiable irrational example are notable strengths.

minor comments (3)
  1. The abstract states the main results clearly but does not indicate the methods (e.g., openness in the moduli space or the specific surface chosen for the irrational example); a brief phrase on the approach would improve readability.
  2. Notation for the basepoint-freeness threshold (presumably denoted something like τ or β) should be introduced once in the introduction and used consistently; check that the definition matches its usage in all subsequent statements.
  3. Verify that the countable union of proper closed subvarieties is explicitly described or referenced in the text so that the 'very general' locus is unambiguously defined.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation of minor revision. We are pleased that the determination of the basepoint-freeness threshold for very general polarized abelian surfaces and the explicit irrational example are viewed as advances in the field.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds by establishing constancy of the basepoint-freeness threshold on the moduli space of polarized abelian surfaces outside a countable union of proper closed subvarieties, followed by an explicit computation on a specific surface (via endomorphism ring or Fourier-Mukai methods) to obtain both the general value and an irrational example. These steps rely on standard specialization and openness arguments in algebraic geometry together with direct calculation on a concrete example; no parameter is fitted to data and then renamed as a prediction, no load-bearing premise reduces to a self-citation chain, and the central claims remain independent of their own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so specific free parameters, axioms, or invented entities cannot be identified from the provided text.

pith-pipeline@v0.9.0 · 5539 in / 962 out tokens · 37870 ms · 2026-05-20T02:51:20.416866+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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