On quadratic persistence and Pythagoras numbers of totally real projective varieties

Euisung Park; Jaewoo Jung; Jong In Han

REVIEW 3 major objections 4 minor 15 references

On quadratic persistence and Pythagoras numbers of totally real projective varieties

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For totally real varieties in the covered ranges, $\mathrm{qp}(X)=c-1$, $\mathrm{py}(X)=n+2$, and being a divisor of a minimal-degree variety are equivalent.

desk verdict A worthwhile extension of the BSSV22 classifications with real new results, but the proof of the totally real K_{p,1}-theorem rests on an unjustified Betti number equality that the main theorem depends on. read the letter →

arxiv 2506.13247 v1 pith:QRJA7UAK submitted 2025-06-16 math.AG math.AC

Jong In Han , Jaewoo Jung , Euisung Park This is my paper · ORCID

classification math.AGmath.AC MSC 14P0514N0514Q3013D02

keywords quadraticpersistencePythagorasnumbertotallyrealvarietiessumsofsquaresminimaldegreeKoszulcohomologycurvesmaximalregularitygenus3

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper sets out to remove the arithmetic Cohen-Macaulay hypothesis from an earlier classification of real projective varieties. For a totally real nondegenerate variety $X \subset \mathbb{P}^r$, it aims to make three conditions equivalent: quadratic persistence $\mathrm{qp}(X)=c-1$, Pythagoras number $\mathrm{py}(X)=n+2$, and $X$ being a codimension-1 subvariety of a variety of minimal degree. The equivalence is proved when $c=3$ and $d \ge 6$, or when $c \ge 4$ and $d \ge 2c+3$, and the forward statements are proved for $d=c+2$. This matters because $\mathrm{py}(X)$ is the smallest number of squares of linear forms needed to represent every sum of squares in the real coordinate ring; the theorem turns that semialgebraic quantity into a geometric containment question. The paper also determines quadratic persistence and Pythagoras numbers for curves of maximal regularity and for linearly normal smooth genus-3 curves.

What carries the argument

The paper's central objects are quadratic persistence $\mathrm{qp}(X)$, defined as the smallest $k$ such that projection from $k$ linearly independent points of $X$ leaves an ideal with no quadrics, and the Pythagoras number $\mathrm{py}(X)$, defined as the smallest $t$ such that every sum of squares of linear forms in the real coordinate ring is a sum of at most $t$ squares. The inequality $r+1-\mathrm{qp}(X)\le\mathrm{py}(X)$ connects them, and the aim is to force equality. The new mechanism is the totally real $K_{p,1}$-theorem: for a divisor $X$ of a minimal-degree variety $Y$, the syzygies of $X$ force the defining equations of $Y$ to be real, so $Y$ is totally real and $\mathrm{py}(X)\le \mathrm{py}(Y)$. A second mechanism, the lifting theorem, uses Castelnuovo's bound on quadrics and partial elimination ideals to reconstruct the containing minimal-degree variety of $X$ from a general inner projection.

What would settle it

Look for a totally real nondegenerate surface $X\subset\mathbb{P}^5$ of codimension $3$ and degree $d\ge 6$ with $\mathrm{qp}(X)=2$ but with $X$ not contained in any minimal-degree threefold; such a surface would refute Theorem 1.1(2). Because the paper proves $\mathrm{qp}(X)=\ell(X)$ in codimension three, the same example would show that a quadratic strand of length $2$ does not force divisorial containment, contradicting the quoted criterion on which the proof leans.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: under the hypotheses $c=3, d\ge 6$ or $c\ge 4, d\ge 2c+3$, a totally real nondegenerate variety $X\subset\mathbb{P}^r$ of codimension $c$ and degree $d$ satisfies $\mathrm{qp}(X)=c-1$, $\mathrm{py}(X)=n+2$, and 'X is a codimension-$1$ subvariety of a variety of minimal degree' as three equivalent conditions; for $d=c+2$, the paper proves $\mathrm{qp}(X)=c-1$ and $\mathrm{py}(X)=n+2$ regardless of the Cohen-Macaulay property. The proof obtains equality in the bound $r+1-\mathrm{qp}(X)\le \mathrm{py}(X)$ by two new tools: a totally real $K_{p,1}$-theorem showing that a minimal-degree variety containing $X$ can be chosen to be defined by real polynomials, and a lifting theorem showing that for $d\ge 2c+3$, if a general inner projection of $X$ lies in a minimal-degree variety, then $X$ itself does. In codimension three the paper also proves $\mathrm{qp}(X)=\ell(X)$, the length of the quadratic strand of the minimal free resolution.

Load-bearing premise

The proof needs a particular syzygy space of $X$—a space of quadratic relations among the defining equations—to have dimension $c-1$, matching the corresponding space of a minimal-degree variety containing $X$, and this equality is asserted rather than proved or cited; in codimension two it fails for a complete intersection of two quadrics, so the argument likely needs a codimension restriction.

Editorial extensions

If this is right

In the covered ranges, any totally real variety with $\mathrm{qp}(X)=c-1$ automatically has Pythagoras number $n+2$, with no Cohen-Macaulay assumption.
The semialgebraic condition $\mathrm{py}(X)=n+2$ becomes equivalent to the geometric inclusion: $X$ is a codimension-$1$ subvariety of a variety of minimal degree.
If the lifting theorem extends to all degrees $d\ge c+3$, the Cohen-Macaulay hypothesis disappears from the previous classification for every codimension.
In codimension three, $\mathrm{qp}(X)$ is read directly from the Betti table: $\mathrm{qp}(X)=\ell(X)$, so $\mathrm{qp}(X)=2$ exactly when $X$ is a divisor of a minimal-degree variety.
For totally real curves of maximal regularity, $\mathrm{py}(C)=r+1-\mathrm{qp}(C)$, with $\mathrm{qp}(C)\in\{r-3,r-2\}$, and $\mathrm{qp}(C)=r-2$ exactly when $C$ lies on a minimal-degree surface.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The remaining gap between the proved range and the full conjecture is the Lifting Problem for degrees $c+3\le d\le 2c+2$; solving that purely geometric statement would complete the removal of the Cohen-Macaulay assumption.
If $\mathrm{qp}(X)=\ell(X)$ were shown in higher codimension, quadratic persistence would become a free-resolution invariant, computable from Betti numbers instead of by searching over point configurations.
For genus-3 curves with $\mathrm{qp}(C)=r-3$, the paper leaves $\mathrm{py}(C)\in\{4,5\}$; a natural test is whether the minimal-degree threefold containing a totally real such curve is always totally real, which would force $\mathrm{py}(C)=4$.
Because Pythagoras numbers govern the number of squares needed in sum-of-squares representations, the geometric classification here suggests that in these ranges 'next-to-maximal' sum-of-squares complexity is controlled by containment in a single low-degree scroll, a condition that may be checkable in optimization practice.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does real work: it extends Blekherman–Sinn–Smith–Velasco's characterizations from aCM varieties to non-aCM ones in codimension three and in higher codimension with large degree, and it gives the first results for curves of maximal regularity and genus-3 curves. The lifting problem is a good framing, and the arguments in Sections 4–6 are concrete and largely convincing. The use of [BS07] and [KMP24] as black boxes is reasonable, and there are no fitted parameters or circular claims.

The soft spot is in Theorem 3.2, the "totally real K_{p,1}-theorem," which is load-bearing for the upper bound py(X) ≤ n+2 in Theorem 1.1. The proof asserts β_{c−1,1}(X) = c−1 without proof, and then uses that to identify Koszul cohomology groups of X and Y. Green's theorem does not give that equality, and it is not a formal consequence of X being a divisor of a minimal-degree variety Y. The stress-test example for c=2 — a complete intersection of two quadrics has β_{1,1}=2 while c−1=1 — shows the claim is not automatic, even if that example is excluded by the aCM hypothesis. The paper never explains how non-aCM-ness or the degree bound c≥3 supplies the missing equality. Since Theorem 3.2 is the only mechanism producing a totally real minimal-degree variety containing X, this gap directly threatens both Theorem 1.1(1) and the (iii)⇒(ii) direction of Theorem 1.1(2). The results may well be correct, but the text needs repair.

One note: the reader's concern about Proposition 3.3 is less serious. The proof there does run over all pairs of independent quadrics, so the passage from "no aQ1+bQ2" to "no quadrics at all" is actually handled, albeit tersely.

This paper is for specialists in real algebraic geometry and syzygies. It deserves a serious referee, not a desk reject, because the overall project is meaningful and most of the paper is solid. The referee should focus on Theorem 3.2 and ask for a proof or a citation for the Betti number equality. I would not cite the main theorem until that gap is closed.

Referee Report

3 major / 4 minor

Summary. The paper studies quadratic persistence qp(X) and the Pythagoras number py(X) for totally real projective varieties. Its main theorem, Theorem 1.1, proposes to extend the aCM classification of Blekherman--Sinn--Smith--Velasco to arbitrary totally real varieties: for codimension c=3 with d≥6, or c≥4 with d≥2c+3, the conditions qp(X)=c−1, py(X)=n+2, and X being a codimension-1 subvariety of a variety of minimal degree are equivalent; for d=c+2 it claims qp(X)=c−1 and py(X)=n+2. The proof is organized around a 'totally real K_{p,1}-theorem' (Theorem 3.2), a codimension-three computation (Theorem 3.1), and a lifting theorem (Theorem 4.1). The paper also gives results for curves of maximal regularity and for linearly normal smooth nonhyperelliptic genus-3 curves.

Significance. If correct, the main theorem is a substantial and natural extension of [BSSV22, Thm 1.5], removing the arithmetic Cohen--Macaulay hypothesis in a significant range and tying next-to-maximal Pythagoras numbers to divisibility in a variety of minimal degree. The statement is precise about the parameter ranges, and the formulation of the remaining obstacle as a Lifting Problem is useful and honest. The curve results in Sections 5 and 6 provide concrete test cases for the inequality py(X)≥r+1−qp(X), and the paper gives a helpful overview of the surrounding literature. The main obstruction to accepting the paper is that the crucial Theorem 3.2, which supplies the totally real minimal-degree variety, is not proved at the level of detail needed; several load-bearing equalities are asserted without proof or citation.

major comments (3)

[§3, proof of Theorem 3.2] The proof asserts without proof or citation that 'For both cases (1) and (2), we have β_{c−1,1}(X)=c−1', and then uses this to conclude β_{c−1,1}(X)=β_{c−1,1}(Y) and hence K_{c−2,2}(I(X),V_C)=K_{c−2,2}(I(Y),V_C). Green's theorem and [KMP24, Thm 1.1] give nonvanishing of β_{c−1,1}(X) and the existence of a minimal-degree divisor Y, but not the asserted dimension. The equality is not a formal consequence of X being a divisor of Y: for c=2, a complete intersection of two quadrics is a divisor of a quadric surface with β_{1,1}(X)=2 while β_{1,1}(Y)=1=c−1. Although that example is outside the hypotheses of the theorem, it shows that the proof must explicitly use the non-aCM or degree assumptions to rule out such behavior. The same paragraph also jumps from i(γ)∈K_{c−2,2}(I(Y),V_C) to the conclusion that the syzygy ideal of i(γ) equals I(Y); no argument is given that the chosen nonzero class realizes all of I(Y). These steps are load-bearing, since Theorem 3.2 is the only mechanism producing a totally real minimal-degree Y, and it feeds Theorem 3.1, Theorem 1.1(1), and the implication (iii)⇒(ii) in Theorem 1.1(2).
[§3, Proposition 3.3] The proof is incomplete in its reduction to two quadrics. It states that qp(X)≥2 implies h^0(I_X(2))≥2, and later infers from the absence of quadratic combinations aQ1+bQ2 on the projected variety X_p that X_p satisfies no quadratic equation. The first assertion is not justified, and the second is only valid if I(X)_2 is spanned by the pair Q1,Q2, or if the pair can be chosen to contain any given quadric; the proof gives no such spanning or dimension statement. Since Proposition 3.3 is the bridge between qp(X) and ℓ(X) in Theorem 3.1, this gap affects the main classification.
[§4, proof of Theorem 4.1] The sentence 'Note that dim W = n or n+1' is used to define the locus U and to conclude that q∉U implies dim(K1(I(X),q))_1≥c−1, but it is not justified at that point in the proof. The facts that W_q is the variety of minimal degree Y and that q is not a vertex can be used to obtain this dimension dichotomy, but the text does not supply that argument. As written, the rank computation and the final equality dim I(X)_2 = binom(c,2) rest on an unproved dimension statement.

minor comments (4)

[Title and abstract] The title and abstract contain typographical errors ('TOTALL Y', 'toti all y'); the manuscript should be proofread.
[§2 and §3] The field is denoted K in some places and k in others (for example, Proposition 3.3 versus Lemma 3.4); please standardize the notation.
[§2, Example 2.8] The display involving the bound qp(C)≤(−1+√217)/2 is hard to parse; please clarify the derivation and the dimension of I(C)_2.
[§6, Remark 6.2] Remark 6.2 states a conditional conclusion about py(C)=4 without a definite theorem; it should be explicitly labeled as an open problem or supported by a proof.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central claims are derived from independent Betti-number and lifting arguments; the flagged issue in Theorem 3.2 is a proof gap, not a circular reduction.

full rationale

The derivation chain is not circular. Quadratic persistence and Pythagoras number are defined independently, and the paper's main equalities are proved via Green-Lazarsfeld Betti numbers, inner projections, and lifting arguments rather than by renaming, fitting, or importing the target conclusion. The only apparent self-citation, [KMP24, Theorem 1.1] in Notation 2.1(3), is used as an external published classification of the condition ℓ(X)=c−1; although one of the current authors is a coauthor of [KMP24], its content is not the same as the target result about Pythagoras numbers and quadratic persistence, so it is not load-bearing circularity. The skeptical concern about Theorem 3.2 is a genuine gap, not a circular step: the proof states without proof 'For both cases (1) and (2), we have β_{c−1,1}(X)=c−1. Hence β_{c−1,1}(X)=β_{c−1,1}(Y), so K_{c−2,2}(I(X),V_C)=K_{c−2,2}(I(Y),V_C)'; neither equality follows from the quoted Green theorem alone, and the c=2 complete-intersection example shows it is not automatic. But an unsupported intermediate assertion is a correctness risk, not a case of the conclusion being built into the premise. No fitted parameter is later called a prediction, and no claimed result is definitionally equivalent to its input.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The proof rests on standard theorems from algebraic geometry and on the prior framework of Blekherman et al. No free parameters are fit to data, and no new mathematical entities are postulated.

assumptions (4)

domain assumption Varieties are integral and nondegenerate; totally real means real points are Zariski dense.
Stated in Section 2 and used throughout; it is the standing setup of the paper.
standard math Green's Kp,1 theorem and [KMP24, Theorem 1.1]: ℓ(X)=c−1 iff d=c+2 or X is a divisor of a minimal degree variety.
Invoked in Remark 2.1(3) and in the proof of Theorem 3.2.
standard math [BS07]: non-aCM varieties of degree c+2 are divisors of varieties of minimal degree, with an exceptional Veronese cone case.
Used in the proof of Theorem 3.2, case (2), and in Theorem 1.1(1).
standard math [BSSV22] results on quadratic persistence: r+1−qp(X) ≤ py(X), monotonicity of py and qp under containment, qp(X) ≥ ℓ(X), and the minimal-degree classification.
Foundational results used repeatedly in Sections 3 through 6.

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Cite this review

Pith. "Pith review of On quadratic persistence and Pythagoras numbers of totally real projective varieties." pith.science (2026). https://pith.science/paper/QRJA7UAK

@misc{pith2026250613247,
  author       = {Pith},
  title        = {Pith review of: On quadratic persistence and Pythagoras numbers of totally real projective varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QRJA7UAK}},
  note         = {Machine review of arXiv:2506.13247}
}

abstract

In this paper, we study the relationship between quadratic persistence and the Pythagoras number of totally real projective varieties. Building upon the foundational work of Blekherman et al. in arXiv:1902.02754, we extend their characterizations of arithmetically Cohen-Macaulay varieties with next-to-maximal quadratic persistence to arbitrary case. Our main result classifies totally real non-aCM varieties of codimension $c$ and degree $d$ that exhibit next-to-maximal quadratic persistence in the cases where $c=3$ and $d \geq 6$ or $c \geq 4$ and $d \geq 2c+3$. We further investigate the quadratic persistence and Pythagoras number in the context of curves of maximal regularity and linearly normal smooth curves of genus 3.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Diego Ar \'e valo Ovalle, Minimizaci \'o n de rango y el n \'u mero de pit \'a goras

work page

[2] [2]

Grigoriy Blekherman, Alex Dunbar, and Rainer Sinn, Pythagoras numbers for ternary forms, arXiv preprint arXiv:2410.17123 (2024)

work page Pith review arXiv 2024

[3] [3]

959, Springer, Berlin-New York, 1982, pp

Eberhard Becker, Valuations and real places in the theory of formally real fields, Real algebraic geometry and quadratic forms ( R ennes, 1981), Lecture Notes in Math., vol. 959, Springer, Berlin-New York, 1982, pp. 1--40. 683127

work page 1981

[4] [4]

Markus Brodmann and Peter Schenzel, On projective curves of maximal regularity, Math. Z. 244 (2003), no. 2, 271--289. 1992539

work page 2003

[5] [5]

Algebraic Geom

, Arithmetic properties of projective varieties of almost minimal degree, J. Algebraic Geom. 16 (2007), no. 2, 347--400. 2274517

work page 2007

[6] [6]

Smith, and Mauricio Velasco, Sums of squares: a real projective story, Notices Amer

Grigoriy Blekherman, Rainer Sinn, Gregory G. Smith, and Mauricio Velasco, Sums of squares: a real projective story, Notices Amer. Math. Soc. 68 (2021), no. 5, 734--747. 4249429

work page 2021

[7] [7]

, Sums of squares and quadratic persistence on real projective varieties, J. Eur. Math. Soc. (JEMS) 24 (2022), no. 3, 925--965. 4397034

work page 2022

[8] [8]

Grigoriy Blekherman, Rainer Sinn, Mauricio Velasco, and Shixuan Zhang, Spurious local minima in nonconvex sum-of-squares optimization, arXiv preprint arXiv:2411.02208 (2024)

work page Pith review arXiv 2024