Measures of association between algebraic varieties, II: self-correspondences
Pith reviewed 2026-05-24 09:28 UTC · model grok-4.3
The pith
Self-correspondences of algebraic varieties receive measures of complexity that resolve questions about embedded curves.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We study measures of complexity for self-correspondences of some classes of varieties. We also answer a question concerning curves sitting in the square of a very general hyperelliptic curve.
What carries the argument
Measures of complexity for self-correspondences of algebraic varieties
Load-bearing premise
The classes of varieties under consideration admit well-defined and useful measures of complexity for their self-correspondences.
What would settle it
An explicit example of a curve in the square of a very general hyperelliptic curve whose existence or properties would contradict the predictions of the complexity measures.
read the original abstract
Following a suggestion of Jordan Ellenberg, we study measures of complexity for self-correspondences of some classes of varieties. We also answer a question of Rhyd concerning curves sitting in the square of a very general hyperelliptic curve.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. Following a suggestion of Jordan Ellenberg, the paper studies measures of complexity for self-correspondences of certain classes of algebraic varieties. It also resolves a question of Rhyd on the existence and properties of curves embedded in the square of a very general hyperelliptic curve.
Significance. If the proposed measures are shown to be well-defined, functorial, and independent of auxiliary choices, and if the resolution of Rhyd's question is established by a rigorous argument under the very-general hypothesis, the work would supply concrete tools for quantifying complexity of correspondences and add a concrete result to the literature on hyperelliptic curves and their self-products. The absence of free parameters and the purely existential character of the claims are consistent with a theoretical contribution in algebraic geometry.
minor comments (3)
- [Abstract] The abstract is extremely terse; it does not name the classes of varieties under consideration nor indicate whether the measures are defined via cohomology, Chow rings, or another invariant. A single sentence clarifying the scope would improve readability.
- [Introduction] The introduction should explicitly state the main theorems (e.g., Theorem A on the measure, Theorem B answering Rhyd) with precise references to the sections where the proofs appear.
- Notation for the self-correspondence and the complexity measure should be introduced once and used consistently; currently the same symbol appears to be overloaded in different contexts.
Simulated Author's Rebuttal
We thank the referee for their positive assessment of the manuscript and for recommending minor revision. No major comments were provided in the report.
Circularity Check
No significant circularity detected
full rationale
This is a pure existence/proof paper in algebraic geometry. The abstract and description indicate the authors define measures of complexity for self-correspondences and prove a result about curves on hyperelliptic varieties under a standard 'very general' hypothesis. No equations, parameters, or claims reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations. The derivation chain consists of independent mathematical arguments with no evidence of the enumerated circularity patterns.
discussion (0)
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