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arxiv: 2403.07315 · v3 · pith:4J44472Bnew · submitted 2024-03-12 · 🧮 math.AG

Secant variety and syzygies of Hilbert scheme of two points

Chiwon Yoon , Haesong Seo This is my paper

Pith reviewed 2026-05-24 03:21 UTC · model grok-4.3

classification 🧮 math.AG
keywords secant varietyHilbert scheme of two pointsGrothendieck-Plücker embeddingGreen's condition (N_p)identifiabilitysyzygiessingular locusresolution of singularities
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The pith

When X is embedded by a 4-very ample line bundle, the secant variety of its Hilbert scheme of two points has singular locus exactly equal to the Hilbert scheme itself.

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

The paper proves that Sec(X^[2]) is identifiable under the Grothendieck-Plücker embedding of the Hilbert scheme of two points whenever X is embedded by a 4-very ample line bundle. It also shows that the same embedding satisfies Green's condition (N_p) once the original embedding of X is positive enough. These two facts together imply that the singular locus of Sec(X^[2]) coincides exactly with X^[2]. A sympathetic reader cares because the results give a concrete description of the geometry of these secant varieties and of a resolution of singularities when X is a surface.

Core claim

We prove that Sec (X^[2]) features the identifiability under the Grothendieck-Plücker embedding X^[2] ↪ P^N when X is embedded by a 4-very ample line bundle. We also prove that the embedding X^[2] ↪ P^N satisfies Green's condition (N_p) when the embedding of X is positive enough. Accordingly, the singular locus of Sec (X^[2]) is exactly X^[2] when the embedding of X is positive enough. As an application, we describe the geometry of a resolution of singularities from the secant bundle to Sec(X^[2]) when X is a surface.

What carries the argument

The Grothendieck-Plücker embedding of X^[2] together with the secant variety Sec(X^[2]) and Green's condition (N_p) on its syzygies.

If this is right

  • The singular locus of Sec(X^[2]) equals X^[2] exactly when the embedding of X is positive enough.
  • The embedding of X^[2] satisfies Green's condition (N_p) for sufficiently positive embeddings of X.
  • Identifiability of Sec(X^[2]) holds under the Grothendieck-Plücker embedding once the line bundle on X is 4-very ample.
  • The geometry of the resolution of singularities from the secant bundle to Sec(X^[2]) admits an explicit description when X is a surface.

Where Pith is reading between the lines

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

  • The threshold of 4-very ampleness may be improvable for specific classes of varieties such as curves or abelian varieties.
  • The identifiability statement could be used to study uniqueness of decompositions in related secant varieties of higher-order Hilbert schemes.
  • The link between Green's condition and the singular locus may extend to other embeddings or to higher secant varieties.

Load-bearing premise

The line bundle used to embed X must be at least 4-very ample (or sufficiently positive) for the induced embedding of X^[2] to satisfy identifiability and Green's (N_p).

What would settle it

A concrete counterexample would be a variety X embedded by a 4-very ample line bundle for which either Sec(X^[2]) fails to be identifiable or contains a singular point lying outside X^[2].

read the original abstract

In this paper, we prove that $\mathrm{Sec} (X^{[2]})$ features the identifiability under the Grothendieck-Pl\"ucker embedding $X^{[2]} \hookrightarrow \PP^N$ when $X$ is embedded by a $4$-very ample line bundle. We also prove that the embedding $X^{[2]} \hookrightarrow \PP^N$ satisfies Green's condition $(N_p)$ when the embedding of $X$ is positive enough. Accordingly, the singular locus of $\mathrm{Sec} (X^{[2]})$ is exactly $X^{[2]}$ when the embedding of $X$ is positive enough. As an application, we describe the geometry of a resolution of singularities from the secant bundle to $\mathrm{Sec}(X^{[2]})$ when $X$ is a surface.

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 / 2 minor

Summary. The manuscript proves that Sec(X^[2]) is identifiable under the Grothendieck-Plücker embedding X^[2] ↪ ℙ^N when X is embedded by a 4-very ample line bundle. It further shows that the embedding X^[2] ↪ ℙ^N satisfies Green's condition (N_p) for sufficiently positive embeddings of X, implying that the singular locus of Sec(X^[2]) is exactly X^[2]. As an application, the geometry of a resolution of singularities of Sec(X^[2]) via the secant bundle is described when X is a surface.

Significance. If the claims hold, the results link identifiability, syzygies via Green's condition, and the geometry of singular loci for secant varieties of Hilbert schemes of two points. The explicit positivity hypotheses (4-very ample or stronger) and the application to surfaces provide concrete geometric consequences in algebraic geometry.

minor comments (2)
  1. [Introduction / §1] The abstract and introduction state the main theorems clearly, but the transition from (N_p) to the singular-locus statement in the body would benefit from an explicit reference to the relevant lemma or proposition establishing the implication.
  2. [Application to surfaces] Notation for the Grothendieck-Plücker embedding and the secant bundle in the application section could be cross-referenced to earlier definitions for readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the manuscript and for recommending minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; theorems rest on independent positivity hypotheses

full rationale

The paper states its main results as theorems conditioned on explicit external assumptions (X embedded by a 4-very ample line bundle, or sufficiently positive embeddings) that are standard in algebraic geometry and not derived from the paper's own outputs. The claims about identifiability of Sec(X^[2]), satisfaction of Green's (N_p), and the resulting singular locus being exactly X^[2] are presented as consequences of these hypotheses via presumably standard syzygy and secant-variety techniques; no equations or steps reduce by construction to fitted parameters, self-definitions, or self-citation chains. The application to resolutions is listed separately as a downstream consequence. This is the normal case of a self-contained mathematical argument.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper is a proof-based work in pure mathematics relying on standard definitions of very ample line bundles, Hilbert schemes, and Grothendieck-Plücker embeddings; no fitted parameters or new postulated entities appear in the abstract.

axioms (2)
  • standard math Standard properties of very ample and k-very ample line bundles on projective varieties
    Invoked to guarantee the embedding of X^[2] and the applicability of Green's condition (N_p).
  • standard math Existence and basic properties of the Hilbert scheme X^[2] and its Grothendieck-Plücker embedding
    Background structure used throughout the statements.

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