pith. sign in

arxiv: 2606.30002 · v1 · pith:FAU4J7BCnew · submitted 2026-06-29 · 🧮 math.AG

Effective results on projective normality of the first and second secant varieties

Doyoung Choi , Jinhyung Park This is my paper

Pith reviewed 2026-06-30 04:12 UTC · model grok-4.3

classification 🧮 math.AG
keywords projective normalitysecant varietiesline bundlespositivityideal generation3x3 minorsalgebraic geometry
0
0 comments X

The pith

Explicit positivity thresholds ensure the first and second secant varieties of a smooth projective variety are projectively normal.

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

The paper supplies concrete lower bounds on the positivity of an ample line bundle L so that the first and second secant varieties of the variety embedded by the complete linear system of L are projectively normal. It further gives effective conditions under which the ideal of the first secant variety is generated by cubics and, in stronger cases, by the 3 by 3 minors of a matrix of linear forms. These results turn a prior existence statement for sufficiently positive L into an explicit, checkable criterion.

Core claim

For a smooth projective complex variety X embedded by the complete linear system of a line bundle L that is a sufficiently positive multiple of an ample bundle, the first and second secant varieties are projectively normal; moreover, when L satisfies explicit effective bounds, the ideal of the first secant variety is generated by cubics or by 3 by 3 minors of a matrix of linear forms.

What carries the argument

Effective lower bounds on the positivity of the embedding line bundle L that guarantee projective normality and ideal generation for the secant varieties.

If this is right

  • The first secant variety satisfies projective normality once the degree of L exceeds an explicit multiple of the canonical class plus an ample class.
  • Under a stronger positivity threshold the ideal of the first secant variety is generated by cubics.
  • Under a still stronger threshold the same ideal is generated by the 3 by 3 minors of a matrix whose entries are linear forms.

Where Pith is reading between the lines

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

  • The explicit bounds open the possibility of verifying projective normality for concrete embeddings by direct computation of cohomology or syzygies up to the given degree.
  • The ideal-generation statements give an effective route to equations of secant varieties that could be used in computational algebraic geometry.

Load-bearing premise

The line bundle must be a sufficiently positive multiple of an ample bundle on a smooth projective complex variety, with the bounds derived under the hypotheses of the prior joint work.

What would settle it

A smooth projective variety and an ample line bundle L whose positivity lies below one of the paper's stated thresholds, yet whose first or second secant variety fails to be projectively normal.

read the original abstract

In joint work with Lacini and Sheridan, we proved that the first and second secant varieties of a smooth projective complex variety embedded by the complete linear system of a sufficiently positive line bundle are projectively normal. The purpose of this paper is to establish effective results on how positive the embedding line bundle must be for this result to hold. We also provide effective conditions under which the defining ideal of the first secant variety is generated by cubics, and furthermore, generated by $3 \times 3$-minors of a matrix of linear forms. The latter result gives an effective version of a theorem of Agostini and the second author.

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 establishes effective lower bounds on the positivity of an ample line bundle L on a smooth projective complex variety X such that the first and second secant varieties of the embedding by the complete linear system |L| are projectively normal. It also provides effective conditions under which the ideal of the first secant variety is generated by cubics or by the 3×3 minors of a matrix of linear forms, giving an effective version of a prior theorem of Agostini and the second author, building on joint work with Lacini and Sheridan.

Significance. Effective positivity thresholds that are tracked explicitly through vanishing statements strengthen the prior non-effective existence results and make them usable for concrete computations on specific varieties. The reduction to effective vanishing and the explicit ideal-generation statements are the main strengths.

minor comments (3)
  1. [Introduction] The introduction repeats the statement of the main effective bounds in both Theorem A and the surrounding text; a single consolidated statement with the precise numerical thresholds would reduce redundancy.
  2. [§4] In §4 the notation for the matrix whose 3×3 minors generate the ideal is introduced without an explicit reference back to the linear system |L|; adding a cross-reference to the definition in §2 would clarify the dependence on the embedding.
  3. [Introduction] The comparison between the new effective constants and the non-effective thresholds from the Lacini–Sheridan paper is mentioned only qualitatively; a short table or sentence quantifying the gap would help readers assess the improvement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report accurately captures the paper's contributions on effective positivity thresholds for projective normality of secant varieties and effective ideal generation statements. No specific major comments are provided in the report.

Circularity Check

0 steps flagged

Minor self-citation to prior joint work; effective bounds tracked independently

full rationale

The paper cites its own prior joint work with Lacini and Sheridan solely for the non-effective existence statement that secant varieties are projectively normal under sufficient positivity. The present work then derives explicit lower bounds on that positivity by reducing to vanishing theorems and tracking constants explicitly. No equations, fitted parameters, or self-referential definitions appear that would make any claimed prediction or bound equivalent to its inputs by construction. The self-citation is therefore present but not load-bearing for the central effective results, producing only minimal circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the work appears to rest on standard algebraic-geometry background rather than new postulates.

pith-pipeline@v0.9.1-grok · 5629 in / 1070 out tokens · 43972 ms · 2026-06-30T04:12:13.645496+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

23 extracted references · 5 canonical work pages · 1 internal anchor

  1. [1]

    Daniele Agostini and Jinhyung Park,Determinantal ideals of secant varieties, preprint (2025)arXiv:2510.01895

    work page arXiv 2025

  2. [2]

    Purnaprajna Bangere and Justin Lacini,Syzygies of adjoint linear series on projective varieties, Duke. Math. J.174(2025), 473–499

    2025

  3. [3]

    Differential Geom

    Aaron Bertram,Moduli of rank-2vector bundles, theta divisors, and the geometry of curves in projective space, J. Differential Geom. 35 (1992), 429–469

    1992

  4. [4]

    Weronika Buczy´ nska, Jaros law Buczy´ nski, and Lucja Farnik,Cactus varieties of sufficiently ample embeddings of projective schemes have determinantal equations, preprint (2024), arXiv:2412.00709

    work page arXiv 2024

  5. [5]

    Jaros law Buczy´ nski, Adam Ginensky, and J. M. Landsberg,Determinantal equations for secant varieties and the Eisenbud– Koh–Stillman conjecture, J. London Math. Soc.88(2013), 1–24

    2013

  6. [6]

    Jan Cheah,Cellular decompositions for nested Hilbert schemes of points, Pacific. J. Math.183(1998), 39–90

    1998

  7. [7]

    Doyoung Choi,Singularities of the nested Hilbert scheme of points of length 3, 4, preprintarXiv:2501.11055(2025)

    work page arXiv 2025

  8. [8]

    Doyoung Choi, Justin Lacini, Jinhyung Park, and John Sheridan.Singularities and syzygies of secant varieties of smooth projective varieties, preprint (2025)arXiv:2502.19703

    work page arXiv 2025

  9. [9]

    Doyoung Choi and Jinhyung Park,Hilbert polynomials of the first and second secant varieties, preprint (2026) arXiv:2605.29418

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  10. [10]

    Chih-Chi Chou and Lei Song,Singularities of secant varieties, Int. Math. Res. Not.2018, 2844–2865

    2018

  11. [11]

    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(2007), 311–316

    2007

  12. [12]

    Math.111(1993), 51–67

    Lawrence Ein and Robert Lazarsfeld,Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension, Invent. Math.111(1993), 51–67

    1993

  13. [13]

    Lawrence Ein and Robert Lazarsfeld,Lectures on the syzygies and geometry of algebraic varieties, University Lecture Series 81(2026), American Mathematical Society, Providence, RI

    2026

  14. [14]

    Math.222(2020), 615–665

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

    2020

  15. [15]

    David Eisenbud, Jee Koh, and Michael Stillman,Determinantal equations for curves of high degree, Amer. J. Math.110 (1988), 513–539

    1988

  16. [16]

    Yukitoshi Hinohara, Kazuyoshi Takahashi, and Hiroyuki Terakawa,On tensor products ofk-very ample line bundles, Proc. Amer. Math. Soc.133(2005), 687–692

    2005

  17. [17]

    Press, Cambridge

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

    2013

  18. [18]

    Sebastian Olano, Debaditya Raychaudhury, and Lei Song,Singularities of secant varieties from a Hodge theoretic perspective, Proc. London. Math. Soc.129(2024), e12635

    2024

  19. [19]

    Claudiu Raicu, 3×3Minors of catalecticants, Math. Res. Lett.20(2013), 745–756. EFFECTIVE RESULTS ON PROJECTIVE NORMALITY OF THE FIRST AND SECOND SECANT VARIETIES 19

    2013

  20. [20]

    Smith,Linear determinantal equations for all projective schemes, Algebra Number Theory 5(2011), 1041–1061

    Jessica Sidman and Gregory G. Smith,Linear determinantal equations for all projective schemes, Algebra Number Theory 5(2011), 1041–1061

    2011

  21. [21]

    Jessica Sidman and Peter Vermeire,Equations defining secant varieties: geometry and computation, Combinatorial aspects of commutative algebra and algebraic geometry, Abel Symp.6(2011), pages 155–174, Springer, Berlin

    2011

  22. [22]

    Math.288(2016), 631–647

    Brooke Ullery,On the normality of secant varieties, Adv. Math.288(2016), 631–647

    2016

  23. [23]

    Peter Vermeire,Some results on secant varieties leading to a geometric flip constructionCompositio Math.125, (2001) 263–282. School of Mathematics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea Email address:cdy4019@kias.re.kr Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 341...

    2001