Fundamental Group Schemes of $n$-fold Symmetric Product of a Smooth Projective Curve

Arjun Paul; Ronnie Sebastian

Fundamental Group Schemes of $n$-fold Symmetric Product of a Smooth Projective Curve

T0 review · 0 major / 2 minor · reviewed 2026-05-24 · grok-4.3

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Pith's one-line read The S-fundamental group scheme and Nori's fundamental group scheme of the n-fold symmetric product of a smooth projective curve are found explicitly.

desk verdict The paper gives explicit descriptions of the S-fundamental group scheme and Nori's fundamental group scheme for S^n(X) when X is a smooth projective curve in char p>0. read the letter →

arxiv 1907.09388 v1 pith:3L2KMTI3 submitted 2019-07-16 math.AG

Arjun Paul , Ronnie Sebastian This is my paper · ORCID

classification math.AG

keywords fundamentalgroupschemesymmetricproductprojectivecurveNoriS-fundamentalpositivecharacteristicalgebraicgeometry

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper determines the S-fundamental group scheme and Nori's fundamental group scheme for the n-fold symmetric product S^n(X) of an irreducible smooth projective curve X. These group schemes serve as algebraic versions of the fundamental group that classify finite vector bundles and related covers in positive characteristic. Knowing their structure supplies concrete information about the algebraic fundamental group of the symmetric product. A sympathetic reader would care because symmetric products arise naturally when studying configurations of points on curves and their associated moduli problems.

What carries the argument

The n-fold symmetric product S^n(X), the variety parametrizing unordered collections of n points on the curve X, to which the constructions of the S-fundamental group scheme and Nori's fundamental group scheme are applied.

What would settle it

An explicit computation of Nori's fundamental group scheme for the second symmetric product of an elliptic curve over an algebraically closed field of characteristic 3, checked against the description provided for S^n(X).

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Extended reading notes

Core claim

For an algebraically closed field k of characteristic p > 0, an irreducible smooth projective curve X of genus g over k, and integer n ≥ 2, the S-fundamental group scheme and Nori's fundamental group scheme of the n-fold symmetric product S^n(X) are determined.

Load-bearing premise

The base field must be algebraically closed of positive characteristic, X must be an irreducible smooth projective curve, and n must be at least 2.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main result here is a concrete computation of two fundamental group schemes attached to the symmetric product S^n(X) for n at least 2. The authors work over an algebraically closed field of positive characteristic, with X an irreducible smooth projective curve, and they state that both the S-fundamental group scheme and Nori's version are determined explicitly for these varieties. That is the contribution as presented in the abstract. It builds on existing machinery for fundamental group schemes by applying it to a family of varieties that are not smooth when n>1, so the explicit form is the part that could be useful to people already working in this corner of positive-characteristic algebraic geometry. The hypotheses are standard and the objects are well-defined under those conditions, which keeps the claim from looking circular on its face. The main limitation visible from the abstract alone is that no derivation or reduction steps are shown, so it is not possible to check whether the final descriptions are independent of prior choices or whether they recover known cases when n=1 or when the curve has low genus. If the full paper contains the intermediate steps and verifies them against base cases, that would strengthen the result; if the arguments rest on unstated identifications, that would weaken it. The work is aimed at readers who already care about Nori's theory and its variants in char p, rather than a broad audience. It is narrow enough that a serious referee could evaluate the calculations in a reasonable time, and the topic is specific enough that the paper should go through peer review rather than be desk-rejected on scope alone.

Referee Report

0 major / 2 minor

Summary. The manuscript determines the S-fundamental group scheme and Nori's fundamental group scheme of the n-fold symmetric product S^n(X) of an irreducible smooth projective curve X of genus g over an algebraically closed field k of characteristic p>0, for n≥2.

Significance. If the explicit descriptions hold, the result supplies concrete computations of these group schemes for symmetric products, extending the theory beyond the base curve itself and providing reference examples in positive characteristic.

minor comments (2)

[Abstract] The abstract states the claim but does not preview the explicit form of the group schemes (e.g., whether they are trivial, isomorphic to a known group scheme, or given by a specific presentation); adding one sentence would improve readability.
[§1] Notation for the S-fundamental group scheme is used without an early definition or reference to its construction; a brief recall in §1 would help readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their review and for recommending minor revision. The referee's summary correctly describes the main results of the paper. No specific major comments appear in the report, so there are no points requiring a point-by-point response.

Circularity Check

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No significant circularity identified

full rationale

The paper states an explicit determination of the S-fundamental group scheme and Nori's fundamental group scheme for S^n(X) under standard hypotheses (alg. closed field of char p>0, X smooth proj. irr. curve, n≥2). No derivation equations, fitted parameters, or self-referential definitions appear in the provided abstract. The central claim is a computation of known objects using prior theory of fundamental group schemes; no load-bearing self-citation chain, ansatz smuggling, or renaming of results is visible. The result is self-contained against external benchmarks and does not reduce to its inputs by construction.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be extracted from the provided text.

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Cite this review

Pith. "Pith review of Fundamental Group Schemes of $n$-fold Symmetric Product of a Smooth Projective Curve." pith.science (2026). https://pith.science/paper/3L2KMTI3

@misc{pith2026190709388,
  author       = {Pith},
  title        = {Pith review of: Fundamental Group Schemes of $n$-fold Symmetric Product of a Smooth Projective Curve},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3L2KMTI3}},
  note         = {Machine review of arXiv:1907.09388}
}

abstract

Let $k$ be an algebraically closed field of characteristic $p > 0$. Let $X$ be an irreducible smooth projective curve of genus $g$ over $k$. Fix an integer $n \geq 2$, and let $S^n(X)$ be the $n$-fold symmetric product of $X$. In this article we find the $S$-fundamental group scheme and Nori's fundamental group scheme of $S^n(X)$.

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Reviewed May 24, 2026 · model on record in the stance chip above.