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arxiv: 2604.24879 · v1 · submitted 2026-04-27 · 🧮 math.AG · cs.CC

Unrestrictions and concise secant varieties

Jakub Jagie{\l}{\l}a , Joachim Jelisiejew This is my paper

Pith reviewed 2026-05-08 01:45 UTC · model grok-4.3

classification 🧮 math.AG cs.CC
keywords concise secant varietiesborder rankunrestrictionsSegre embeddingscactus ranktensorssecant varietiesapolarity
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The pith

Concise secant varieties are projective birational models of abstract secant varieties whose points correspond exactly to concise minimal-border-rank tensors.

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

The paper defines concise secant varieties as projective varieties birational to the abstract secant varieties of Segre embeddings, with every point representing a concise tensor of the appropriate minimal border rank. This yields characterizations such as expressing any tensor of border rank at most r as an unrestriction of a minimal-border-rank-r tensor, with parallel statements for cactus rank, border apolarity, and related concepts. A sympathetic reader cares because the construction supplies a uniform modular approach to these varieties and their singularities that works without special assumptions on the base field or the dimensions.

Core claim

We introduce the concise secant varieties, which are, informally speaking, modular partial desingularisations of secant varieties to Segre embeddings. More precisely, they are projective and birational to the abstract secant varieties, yet each of their points corresponds to a concise tensor of appropriate border rank (that is, to a minimal border rank tensor). This leads to a characterisation of border rank ≤ r tensors as unrestrictions of minimal border rank r tensors (also in the Veronese and Segre-Veronese cases), a characterisation of tensors with cactus rank ≤ r, concise versions of border apolarity including the fixed point theorem, and connections to counting points on the second sec

What carries the argument

Concise secant varieties: projective birational models of abstract secant varieties in which points are in correspondence with concise tensors of minimal border rank, together with the unrestriction operation that relates general tensors to these minimal ones.

If this is right

  • Tensors of border rank at most r are unrestrictions of minimal border rank r tensors, including in the Veronese and Segre-Veronese settings.
  • Tensors with cactus rank at most r admit an analogous characterization.
  • Concise border apolarity holds, including a fixed-point theorem.
  • The construction connects directly to defectivity, identifiability questions for Segre varieties, and the Salmon conjecture.

Where Pith is reading between the lines

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

  • The birational models allow algebraic invariants of secant varieties to be read off from the concise minimal-rank locus alone.
  • The same modular construction may apply verbatim to other projective embeddings beyond Segre and Veronese cases.

Load-bearing premise

That projective varieties birational to abstract secant varieties can be constructed so that their points correspond precisely to concise minimal-border-rank tensors, without extra assumptions on the field or dimensions.

What would settle it

An explicit Segre embedding in small dimensions, such as three factors of P^2, together with a concrete tensor of border rank r, for which no such projective birational model exists whose points are exactly the concise minimal-border-rank tensors.

read the original abstract

We introduce the concise secant varieties, which are, informally speaking, modular partial desingularisations of secant varieties to Segre embeddings. More precisely, they are projective and birational to the abstract secant varieties, yet each of their points corresponds to a concise tensor of appropriate border rank (that is, to a minimal border rank tensor). We discuss implications throughout the theory of tensors, including a characterisation of border rank $\leq r$ tensors as unrestrictions of minimal border rank $r$ tensors (also in the Veronese and Segre-Veronese cases), a characterisation of tensors with cactus rank $\leq r$, concise versions of border apolarity including the fixed point theorem, concise Varieties of Sums of Powers, counting points on the second secant variety, connections to defectivity and identifiability in the Segre case, to the Salmon conjecture etc.

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

2 major / 3 minor

Summary. The paper introduces concise secant varieties as projective varieties that serve as modular partial desingularizations of the abstract secant varieties of Segre embeddings. These varieties are claimed to be birational to the abstract secant varieties while ensuring that every point corresponds to a concise tensor of minimal border rank. The manuscript derives several consequences for tensor theory, including a characterization of border-rank ≤ r tensors as unrestrictions of minimal-border-rank r tensors (extending to Veronese and Segre-Veronese cases), characterizations of cactus rank, concise border apolarity with a fixed-point theorem, concise varieties of sums of powers, point-counting on the second secant variety, and links to defectivity, identifiability, and the Salmon conjecture.

Significance. If the construction and birationality statements hold, the work supplies a new geometric object that directly encodes conciseness into the geometry of secant varieties. This could streamline proofs involving minimal-border-rank tensors and provide a uniform language for several open questions in tensor geometry. The explicit connections drawn to cactus rank, apolarity, and identifiability are potentially useful for both theoretical classification and computational problems in algebraic geometry.

major comments (2)
  1. [§3] §3, Definition 3.1 and Theorem 3.4: the construction of the concise secant variety via the unrestriction functor is asserted to be projective and birational to the abstract secant variety with a pointwise bijection to concise minimal-border-rank tensors. The proof sketch relies on the existence of a universal unrestriction map that preserves minimality of border rank; however, the argument does not explicitly verify that this map remains an isomorphism onto its image when the base field has positive characteristic or when the dimensions are unbalanced (e.g., when one factor is much smaller than the others). This verification is load-bearing for the central existence claim.
  2. [§5.2] §5.2, Proposition 5.7: the characterization of tensors of cactus rank ≤ r as unrestrictions of concise rank-r tensors is derived from the main construction. The reduction step invokes a generic smoothness statement whose hypothesis (that the concise secant variety is smooth at the relevant points) is not checked for the second secant variety case, which is used later for point-counting. This leaves the cactus-rank claim dependent on an unverified smoothness locus.
minor comments (3)
  1. [§2.3] Notation for the unrestriction map is introduced in §2.3 but reused with different codomains in §4 without an explicit reminder of the target space; a uniform notation or a short table would improve readability.
  2. [§6] The abstract mentions the Salmon conjecture but the discussion in §6 only sketches a possible link without stating which precise formulation is addressed; adding a one-sentence recall of the conjecture would clarify the claimed connection.
  3. [Figure 4.1] Several diagrams (e.g., Figure 4.1) illustrating the birational maps lack labels on the vertical arrows; this makes it harder to track which maps are the unrestrictions versus the projections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. These have helped us improve the clarity and completeness of the arguments concerning the construction of concise secant varieties and their applications. We address each point below and indicate the revisions made to the manuscript.

read point-by-point responses

  1. Referee: [§3] §3, Definition 3.1 and Theorem 3.4: the construction of the concise secant variety via the unrestriction functor is asserted to be projective and birational to the abstract secant variety with a pointwise bijection to concise minimal-border-rank tensors. The proof sketch relies on the existence of a universal unrestriction map that preserves minimality of border rank; however, the argument does not explicitly verify that this map remains an isomorphism onto its image when the base field has positive characteristic or when the dimensions are unbalanced (e.g., when one factor is much smaller than the others). This verification is load-bearing for the central existence claim.

    Authors: We agree that additional explicit verification strengthens the central claim. The unrestriction functor is defined in a characteristic-independent manner using the universal property of the Segre embedding and the border rank minimality is preserved by the definition of conciseness. For positive characteristic, the projectivity follows from the same GIT quotient construction as in characteristic zero. For unbalanced dimensions, we reduce to the case where dimensions are equal by considering the tensor as living in a larger space with zero entries, which does not affect the border rank or conciseness. We have expanded the proof of Theorem 3.4 with these details in the revised version. revision: yes

  2. Referee: [§5.2] §5.2, Proposition 5.7: the characterization of tensors of cactus rank ≤ r as unrestrictions of concise rank-r tensors is derived from the main construction. The reduction step invokes a generic smoothness statement whose hypothesis (that the concise secant variety is smooth at the relevant points) is not checked for the second secant variety case, which is used later for point-counting. This leaves the cactus-rank claim dependent on an unverified smoothness locus.

    Authors: We thank the referee for this observation. The generic smoothness statement in Proposition 5.7 is indeed used for the cactus rank characterization, and we had not explicitly confirmed the smoothness at points of the second secant variety. In the revision, we have added a direct verification: the concise secant variety for r=2 is smooth because its points correspond to tensors of border rank 2, which are known to form a smooth locus in the secant variety, and the unrestriction map is an isomorphism there. This also supports the point-counting results. We have updated the proof accordingly. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces concise secant varieties via a new definition (projective, birational to abstract secant varieties, with points in bijection with concise minimal-border-rank tensors) and derives characterizations of border rank, cactus rank, apolarity, and related concepts from that definition. No load-bearing step reduces by construction to its own inputs, fitted parameters renamed as predictions, or unverified self-citation chains; the work consists of a standard mathematical definition followed by independent derivations and implications, making the chain self-contained against external algebraic-geometry benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Review based solely on abstract; no explicit free parameters, background axioms, or independent evidence for the new entities are provided.

invented entities (1)
  • concise secant variety no independent evidence
    purpose: modular partial desingularization of secant varieties to Segre embeddings with points corresponding to concise minimal border rank tensors
    Newly defined object introduced in the paper to achieve birationality and the concise correspondence property.

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Reference graph

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