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arxiv: 1907.06203 · v1 · pith:J7NKCS57new · submitted 2019-07-14 · 🧮 math.AG

Strict inclusions of high rank loci

Edoardo Ballico , Alessandra Bernardi , Emanuele Ventura This is my paper

Pith reviewed 2026-05-24 21:48 UTC · model grok-4.3

classification 🧮 math.AG
keywords high rank lociVeronese surfaceplane quarticsX-rankspace curvesstrict inclusionsmaximal rankprojective varieties
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The pith

The high rank loci of the Veronese surface of plane quartics and infinitely many curves in P^4 exhibit strict inclusions between consecutive levels.

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

The paper demonstrates examples where the closure of points with X-rank above the generic value is properly contained inside the closure for the next higher threshold. One such example is given explicitly for the Veronese surface that parametrizes plane quartics. The authors further produce infinitely many curves embedded in four-dimensional projective space that display the same strict nesting, extending an earlier single-curve case. For curves in three-dimensional space they supply two explicit criteria that decide when the set of points attaining the absolute maximal rank is finite or empty. These constructions show that the natural stratification by rank can be finer than the generic picture suggests.

Core claim

For a given projective variety X the high rank loci are the closures of the sets of points whose X-rank is higher than the generic one. We show examples of strict inclusion between two consecutive high rank loci. Our first example is for the Veronese surface of plane quartics. Although Piene had already shown an example when X is a curve, we construct infinitely many curves in P^4 for which such strict inclusion appears. For space curves, we give two criteria to check whether the locus of points of maximal rank 3 is finite (possibly empty).

What carries the argument

High rank loci, the closures of sets of points whose X-rank exceeds the generic rank.

If this is right

  • The high rank loci of the Veronese surface of plane quartics form a strictly nested chain.
  • Infinitely many curves in P^4 have consecutive high rank loci that are properly contained in one another.
  • Two algebraic criteria decide whether the maximal-rank-3 locus on a space curve is finite or empty.
  • The rank stratification of points relative to these varieties is strictly finer than the generic expectation at the first two steps above the generic rank.

Where Pith is reading between the lines

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

  • The same strict-inclusion phenomenon may appear for other Veronese embeddings of higher-degree forms once explicit equations are written down.
  • The finiteness criteria for space curves could be turned into an algorithm that enumerates all curves in P^3 whose maximal-rank locus is a single point.
  • If the strict nesting persists under small deformations, then the dimensions of the successive high-rank loci give new upper bounds on the dimensions of higher secant varieties.

Load-bearing premise

The standard notions of X-rank and the closure operation that produces the high rank loci apply without change to the chosen Veronese embedding and the selected curves in P^4 and P^3.

What would settle it

An explicit check that, for the Veronese surface of plane quartics, the closure of points of rank at least generic-plus-two equals the closure of points of rank at least generic-plus-one would show the claimed strict inclusion does not hold.

read the original abstract

For a given projective variety $X$, the high rank loci are the closures of the sets of points whose $X$-rank is higher than the generic one. We show examples of strict inclusion between two consecutive high rank loci. Our first example is for the Veronese surface of plane quartics. Although Piene had already shown an example when $X$ is a curve, we construct infinitely many curves in $\mathbb P^4$ for which such strict inclusion appears. For space curves, we give two criteria to check whether the locus of points of maximal rank 3 is finite (possibly empty).

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 paper defines the high rank loci of a projective variety X as the closures of the sets of points whose X-rank exceeds the generic rank. It exhibits examples of strict inclusions between consecutive high rank loci, including an explicit example on the Veronese surface parametrizing plane quartics and an infinite family of curves in P^4. For space curves it supplies two criteria that determine whether the locus of points of maximal rank 3 is finite (possibly empty).

Significance. The explicit constructions furnish concrete instances in which the high-rank stratification is strictly finer than the generic-rank stratification, extending Piene’s earlier example for curves. The infinite family in P^4 and the finiteness criteria for space curves supply testable, constructive content that can be used to probe the geometry of secant varieties and Waring problems.

minor comments (2)
  1. [Abstract] The abstract refers to “the Veronese surface of plane quartics” without specifying the ambient projective space or the precise Veronese embedding; a sentence clarifying that the surface lies in P^{14} (or whichever space is intended) would aid readability.
  2. [Abstract] The statement that the constructions yield “infinitely many curves in P^4” would benefit from an explicit reference to the parameter space or moduli component used to produce the family.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the paper and for acknowledging the significance of the explicit constructions and finiteness criteria. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central results consist of explicit geometric constructions establishing strict inclusions between high-rank loci for the Veronese surface and for infinitely many curves in P^4, together with finiteness criteria for space curves. These rest on the standard definitions of X-rank and high-rank loci (invoked without re-derivation) and on direct verification of the claimed inclusions, not on any fitted parameters, self-referential definitions, or load-bearing self-citations. The single external reference to Piene is to prior independent work on curves and does not serve as a uniqueness theorem or ansatz that the present constructions rely upon. No step in the argument reduces by construction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard definitions of rank and loci in projective algebraic geometry; no free parameters, ad-hoc axioms, or invented entities are indicated in the abstract.

axioms (1)
  • standard math Standard properties of X-rank and generic rank for projective varieties and their Veronese embeddings hold as defined in the literature.
    Invoked implicitly when defining high rank loci as closures of higher-than-generic rank sets.

pith-pipeline@v0.9.0 · 5617 in / 1215 out tokens · 28244 ms · 2026-05-24T21:48:36.165835+00:00 · methodology

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Reference graph

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