Some local and global properties of secant varieties of nonsingular projective curves

Jinhyung Park; Lawrence Ein; Wenbo Niu

arxiv: 2604.26054 · v1 · submitted 2026-04-28 · 🧮 math.AG

Some local and global properties of secant varieties of nonsingular projective curves

Lawrence Ein , Wenbo Niu , Jinhyung Park This is my paper

Pith reviewed 2026-05-07 14:58 UTC · model grok-4.3

classification 🧮 math.AG

keywords secant varietiesprojective curvestangent conessymmetric productsHilbert polynomialscohomologyarithmetically Cohen-Macaulay

0 comments

The pith

Secant varieties of nonsingular projective curves have explicit tangent cone descriptions and recursive Hilbert polynomial formulas from symmetric product cohomology.

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

The paper establishes an explicit description of the tangent cones to secant varieties of nonsingular projective curves. It also computes the cohomology groups of the secant sheaves on the symmetric products of the curve. These computations answer a question left open in the authors' prior work and produce a recursive formula for the Hilbert polynomials of the secant varieties. The appendix supplies a cohomological proof of their arithmetical Cohen-Macaulay property. Such results clarify both the local geometry near points on these varieties and their global numerical invariants.

Core claim

The tangent cones of secant varieties of nonsingular projective curves admit an explicit description. The cohomology groups of secant sheaves on symmetric products of the curve are computed, which answers a question posed in earlier work and yields a recursive formula for the Hilbert polynomials of the secant varieties. A cohomological approach to arithmetical Cohen-Macaulayness of these varieties is given in the appendix.

What carries the argument

The secant sheaves on symmetric products of the curve, whose cohomology computation produces the recursive Hilbert polynomial formula and resolves the prior question.

Load-bearing premise

The curve is nonsingular and projective, with the secant varieties constructed in the same ambient space and notation as the authors' prior work.

What would settle it

Compute the Hilbert polynomial of the first secant variety of a smooth rational curve of degree 4 directly via resolution or Groebner basis, then check whether it matches the value obtained from the recursive formula derived from the cohomology groups.

read the original abstract

The main goal of this paper is to study some local and global properties of secant varieties of algebraic curves. These results complement our previous work [8] by addressing issues given therein and providing solutions to problems raised subsequently. Specifically, we show a description of tangent cones of secant varieties of curves, and compute the cohomology groups of secant sheaves on symmetric products of curves, which answers a question posed in [8] and leads to a recursive formula for Hilbert polynomials of secant varieties of curves. In the appendix, we present a cohomological approach to arithmetical Cohen--Macaulayness of secant varieties of curves, completing the proof in [8].

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

Ein, Niu, and Park describe the tangent cones to secant varieties of curves and derive a recursive Hilbert polynomial formula that settles a question from their previous paper.

read the letter

This paper from Ein, Niu, and Park focuses on local and global properties of secant varieties for nonsingular projective curves. The key new pieces are a description of the tangent cones and a computation of cohomology groups for secant sheaves on symmetric products of curves. Those calculations answer an open question from the authors' earlier work and produce a recursive formula for the Hilbert polynomials of the secant varieties. The appendix also completes the proof of arithmetical Cohen-Macaulayness using a cohomological approach. The paper does a good job staying concrete. It takes the setup from [8] and pushes it forward with explicit statements about tangent cones and the recursion. The logic flows from the cohomology to the polynomial formula without obvious gaps in the high-level argument. On the soft side, the results are limited to nonsingular curves in projective space, and they depend on the notation and construction from the prior paper. This makes the contribution incremental rather than a big shift. No major inconsistencies show up in the claims, but the specialized nature means it won't apply broadly outside this area of algebraic geometry. Readers who work on secant varieties, syzygies, or embedding problems for curves will get the most out of it. The recursive formula and tangent cone description provide tools that could be used in further calculations. The paper shows clear thinking and honest engagement with the literature it builds on. I think it deserves a serious referee. The concrete results and the resolution of the open question make it worth reviewing, even if revisions might be needed for clarity in the proofs. I would recommend sending it out for peer review.

Referee Report

0 major / 0 minor

Summary. The paper studies local and global properties of secant varieties of nonsingular projective curves. It provides a description of the tangent cones to these secant varieties, computes the cohomology groups of secant sheaves on symmetric products of curves (answering a question from the authors' prior work [8]), and derives a recursive formula for the Hilbert polynomials of the secant varieties. The appendix presents a cohomological approach to arithmetical Cohen-Macaulayness of these varieties, completing the proof from [8].

Significance. If the results hold, the work supplies a concrete recursive tool for computing Hilbert polynomials of secant varieties and resolves an open question from [8] via explicit cohomology calculations. The tangent-cone description adds local geometric information that complements the global invariants, and the appendix closes a prior gap. These contributions form a coherent extension of the setup in [8] with direct computational utility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of our results on tangent cones, cohomology of secant sheaves, the recursive formula for Hilbert polynomials, and the appendix on arithmetical Cohen-Macaulayness. The recommendation for minor revision is noted, but no specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper references prior work [8] for context, to answer an open question posed there, and to complete a proof in the appendix, but the core new results (tangent cone description of secant varieties and cohomology computations of secant sheaves on symmetric products) are presented as independent mathematical contributions that lead to the recursive Hilbert polynomial formula. No load-bearing step reduces by construction to a self-citation, fitted input, or self-definitional loop; the derivation chain relies on standard algebraic geometry techniques applied to the nonsingular projective curve setup without tautological restatement of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper operates inside standard algebraic geometry: nonsingular projective curves over an algebraically closed field, secant varieties defined via the usual incidence correspondence, and sheaf cohomology on symmetric products. No new free parameters, ad-hoc axioms, or invented entities are indicated in the abstract.

axioms (1)

domain assumption The base field is algebraically closed and the curve is nonsingular and projective.
Stated in the title and abstract; required for the classical theory of secant varieties.

pith-pipeline@v0.9.0 · 5409 in / 1210 out tokens · 89150 ms · 2026-05-07T14:58:08.338005+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Algebraic Geom.33(2024), 629–654

Daniele Agostini.The Martens–Mumford theorem and the Green–Lazarsfeld secant conjecture, J. Algebraic Geom.33(2024), 629–654

work page 2024
[2]

J.30(2013), 27–38

Paolo Cascini and Yoshinori Gongyo.On the anti-canonical ring and varieties of Fano type, Saitama Math. J.30(2013), 27–38

work page 2013
[3]

Junho Choe, Sijong Kwak, and Jinhyung Park.Syzygies of secant varieties of smooth projective curves and gonality sequences, preprint: arXiv:2307.03405

work page arXiv
[4]

Chih-Chi Chou and Lei Song.Singularities of secant varieties, Int. Math. Res. Not.2018no.9, 2844–2865

work page
[5]

Math.200(2006), 1–50

Ciro Ciliberto and Francesco Russo.Varieties with minimal secant degree and linear systems of maximal dimension on surfaces, Adv. Math.200(2006), 1–50

work page 2006
[6]

Gentiana Danila,Sections de la puissance tensorielle du fibr´ e tautologique sur le sch´ ema de Hilbert des points d’une surface, Bull. Lond. Math. Soc.39(2004), 311–316

work page 2004
[7]

Math.,116(1991), American Mathematical Society, Providence, RI, 9–18

Lawrence Ein.Normal sheaves of linear systems on curves, Contemp. Math.,116(1991), American Mathematical Society, Providence, RI, 9–18

work page 1991
[8]

Math.,222(2020), 615–665

Lawrence Ein, Wenbo Niu, and Jinhyung Park.Singularities and syzygies of secant varieties of nonsingular projective curves, Invent. Math.,222(2020), 615–665

work page 2020
[9]

Unione Mat

Lawrence Ein, Wenbo Niu, and Jinhyung Park.A remark on global sections of secant bundles of curves, Boll. Unione Mat. Ital.15(2022), 163–171

work page 2022
[10]

J´ anos Koll´ ar.Shafarevich maps and automorphic forms, M. B. Porter Lectures, Princeton Univ. Press, Cambridge, (1995)

work page 1995
[11]

J´ anos Koll´ ar.Singularities of the minimal model program, Cambridge Tracts in Mathematics,200, 2013

work page 2013
[12]

(eds) Commutative Algebra (2021), Springer, Cham, 689–721

Claudiu Raicu and Steven V Sam.Hermite Reciprocity and Schwarzenberger Bundles, in: Peeva, I. (eds) Commutative Algebra (2021), Springer, Cham, 689–721

work page 2021
[13]

Jessica Sidman and Peter Vermeire.Syzygies of the secant variety of a curve, Algebra Number Theory3 (2009), 445–465

work page 2009
[14]

Math.288(2016), 631–647

Brooke Ullery.On the normality of secant varieties, Adv. Math.288(2016), 631–647. Department of Mathematics, University Illinois at Chicago, 851 South Morgan St., Chicago, IL 60607, USA Email address:ein@uic.edu Department of Mathematical Sciences, University of Arkansas, F ayetteville, AR 72701, USA Email address:wenboniu@uark.edu Department of Mathemati...

work page 2016

[1] [1]

Algebraic Geom.33(2024), 629–654

Daniele Agostini.The Martens–Mumford theorem and the Green–Lazarsfeld secant conjecture, J. Algebraic Geom.33(2024), 629–654

work page 2024

[2] [2]

J.30(2013), 27–38

Paolo Cascini and Yoshinori Gongyo.On the anti-canonical ring and varieties of Fano type, Saitama Math. J.30(2013), 27–38

work page 2013

[3] [3]

Junho Choe, Sijong Kwak, and Jinhyung Park.Syzygies of secant varieties of smooth projective curves and gonality sequences, preprint: arXiv:2307.03405

work page arXiv

[4] [4]

Chih-Chi Chou and Lei Song.Singularities of secant varieties, Int. Math. Res. Not.2018no.9, 2844–2865

work page

[5] [5]

Math.200(2006), 1–50

Ciro Ciliberto and Francesco Russo.Varieties with minimal secant degree and linear systems of maximal dimension on surfaces, Adv. Math.200(2006), 1–50

work page 2006

[6] [6]

Gentiana Danila,Sections de la puissance tensorielle du fibr´ e tautologique sur le sch´ ema de Hilbert des points d’une surface, Bull. Lond. Math. Soc.39(2004), 311–316

work page 2004

[7] [7]

Math.,116(1991), American Mathematical Society, Providence, RI, 9–18

Lawrence Ein.Normal sheaves of linear systems on curves, Contemp. Math.,116(1991), American Mathematical Society, Providence, RI, 9–18

work page 1991

[8] [8]

Math.,222(2020), 615–665

Lawrence Ein, Wenbo Niu, and Jinhyung Park.Singularities and syzygies of secant varieties of nonsingular projective curves, Invent. Math.,222(2020), 615–665

work page 2020

[9] [9]

Unione Mat

Lawrence Ein, Wenbo Niu, and Jinhyung Park.A remark on global sections of secant bundles of curves, Boll. Unione Mat. Ital.15(2022), 163–171

work page 2022

[10] [10]

J´ anos Koll´ ar.Shafarevich maps and automorphic forms, M. B. Porter Lectures, Princeton Univ. Press, Cambridge, (1995)

work page 1995

[11] [11]

J´ anos Koll´ ar.Singularities of the minimal model program, Cambridge Tracts in Mathematics,200, 2013

work page 2013

[12] [12]

(eds) Commutative Algebra (2021), Springer, Cham, 689–721

Claudiu Raicu and Steven V Sam.Hermite Reciprocity and Schwarzenberger Bundles, in: Peeva, I. (eds) Commutative Algebra (2021), Springer, Cham, 689–721

work page 2021

[13] [13]

Jessica Sidman and Peter Vermeire.Syzygies of the secant variety of a curve, Algebra Number Theory3 (2009), 445–465

work page 2009

[14] [14]

Math.288(2016), 631–647

Brooke Ullery.On the normality of secant varieties, Adv. Math.288(2016), 631–647. Department of Mathematics, University Illinois at Chicago, 851 South Morgan St., Chicago, IL 60607, USA Email address:ein@uic.edu Department of Mathematical Sciences, University of Arkansas, F ayetteville, AR 72701, USA Email address:wenboniu@uark.edu Department of Mathemati...

work page 2016