# Verification report — OWR-14601-005 (Michałek, Oberwolfach Report 19/2016: the Gorenstein property of the affine cone over σ2((P^1)^5); lattice hulls of balls; the gradient map on a generic linear subspace)

Verification date: 2026-10-14.

**Verdict.** All three questions of the record are answered; only the answer to the first is new (to our knowledge).
- **(a) Settled: yes.** The homogeneous coordinate ring R of Y = σ2(P^1 × P^1 × P^1 × P^1 × P^1) ⊂ P^31 is a
  Gorenstein ring (Theorem A): Cohen–Macaulay of dimension 12, ω_R ≅ R(−4), Hilbert series
  (1 + 20t + 210t^2 + 772t^3 + 1250t^4 + 772t^5 + 210t^6 + 20t^7 + t^8)/(1 − t)^12, deg Y = 3256, a-invariant −4,
  regularity 8 (that of I(Y) is 9); Y is projectively normal, ω_Y ≅ O_Y(−4), and the intermediate cohomology of all
  twists O_Y(k) vanishes. Inputs: the toric chart of Michałek–Oeding–Zwiernik (IMRN 2015), Raicu's Theorem 4.1
  (Algebra Number Theory 2012), and standard theorems (Kawamata–Viehweg vanishing, Serre duality, the comparison of
  local and sheaf cohomology, Stanley's theorem). Base field C (characteristic 0).
- **(b) Answered: no.** By the companion note (A. Ferudun, Lattice Hulls of Euclidean Balls Need Not Be Normal,
  Zenodo, doi:10.5281/zenodo.23371741): a ball in R^6 whose lattice hull is not normal, and examples in every
  dimension n ≥ 6 (Theorem B of the note, quoted with its certificate). Dimensions 4 and 5 remain open.
- **(c) Stale: yes over C.** The source itself (p. 1001) records B. Teissier's observation that the answer is
  positive. The note gives a short self-contained proof (Theorem C) for the set-theoretic image, its closure and the
  total transform, and an example showing that the statement fails in characteristic p dividing deg f. Nothing is
  claimed as new for (c).

The note is unrefereed.

## Statement checked
- **Primary source.** M. Michałek, "Toric structures in nontoric varieties" (joint work with L. Oeding and
  P. Zwiernik), in: Toric Geometry, Oberwolfach Reports 13 (2016), no. 2, Report No. 19/2016, pp. 999–1001,
  doi:10.4171/OWR/2016/19 (issued 2017); the workshop was held from 27 March to 2 April 2016 (title page).
  - Read in the publisher's open PDF (ems.press; 1,234,272 bytes, sha256
    `daa7e7590a60a5bccc36c7532c260e7669b30893adc8c921ededf5cfc9d2b42e`), pages 999–1001 rendered and read in full.
  - p. 1000: Theorem 2 [MOZ]: σ2(P^{a1} × ⋯ × P^{an}) is (locally) Gorenstein iff it fills the ambient space or
    n = 2, a1 = a2; n = 3, (a_i) ∈ {(1,1,3), (1,3,3), (3,3,3)}; n = 5, (a_i) = (1,1,1,1,1). "The first family a1 = a2
    is classical"; affine cones in the second family were confirmed Gorenstein by Oeding. Conjecture 3: "Is the affine
    cone over σ2(P1 × P1 × P1 × P1 × P1) Gorenstein?"
  - p. 1000, Conjecture 5: (1) convex hull of the lattice points in any ball normal? (reference: Bruns–Gubeladze–
    Michałek, Question 7.2(b)); (2) for a homogeneous f in n+1 variables and a generic subspace L of P^n, is
    L^⊥ ∩ (closure of ∇f(L)) empty? (reference: Michałek–Sturmfels–Uhler–Zwiernik, cited by the source in the journal
    version as Conjecture 5.5; it is Conjecture 5.8 in arXiv versions 1 and 2, and the journal numbering was not
    compared). The source writes the closure (checked on the rendered page).
  - p. 1001: Teissier observed in Oberwolfach that the answer to (2) is positive, a special case of his results
    (Astérisque 7–8, 1973; LNM 961, 1982); new proof and application: Huh (JAMS 2012); (1) still awaits an answer.
- **Corpus record.** ulamai/UnsolvedMath, OWR-14601-005 (dataset version 1.6.0; upstream status `open`). The
  statement concatenates the three questions faithfully (modulo notation; in the third question it omits the
  closure that the source writes, and Theorem C covers both readings). The literature note says that the
  secant-cone question "is answered affirmatively for (P^1)^5" and lists Michałek–Oeding–Zwiernik as a source.
  What that paper proves is the local property: σ2((P^1)^5) is covered by charts A^5 × T_{2,5} (Theorem 4.5) and
  T_{2,5} is Gorenstein with terminal singularities (Corollary 5.7); the general classification (Theorem 7.18 = the
  source's Theorem 2) is local as well. The source poses the affine-cone question after quoting that result. The
  note records this neutrally (Section 1.1).

## Readings
| Reading | Answer | Where |
|---|---|---|
| (a) "the affine cone is Gorenstein" = the homogeneous coordinate ring R = C[x]/I(Y) is a Gorenstein ring (equivalently, its localization at the vertex; Y arithmetically Gorenstein) | yes | Theorem A |
| (a) the local reading (all local rings of the projective variety Y Gorenstein) | yes, but this is Michałek–Oeding–Zwiernik's result, not the question asked | Corollary 2.4, Section 1.1 |
| (a) the section ring ⊕_k H^0(Y, O_Y(k)) | Gorenstein even without Raicu's theorem; equal to R | Remark 5.2, Section 5 |
| (b) closed or open balls with arbitrary centre and radius; "normal" = IDP (all conventions agree for spanning polytopes) | no (n = 6, and every n ≥ 6) | Theorem B (companion note) |
| (b) balls centred at lattice points; dimensions 4, 5 | not treated (open) | Section 6 |
| (c) ∇f(L) as the set-theoretic image ∇f(L∖B) | yes for generic L (char k ∤ deg f) | Theorem C |
| (c) its Zariski closure (the source's reading; the exponential variety of MSUZ) | yes for generic L | Theorem C |
| (c) the total transform under the closure of the graph | yes for generic L | Theorem C |
| (c) characteristic p dividing deg f | false (example f = x0^{p−1} x1) | Remark 7.1 |
| (c) all L instead of generic L | false (tangent line to a smooth conic) | Remark 7.1 |

## Results in the paper
- **Theorem A** (Section 1.2): R is Gorenstein; CM of dimension 12; ω_R ≅ R(−4); the Hilbert series above; deg Y =
  3256; a(R) = −4; reg R = 8, reg I(Y) = 9 (the "?" in Oeding's Table 1 is 9 in his convention); Y projectively
  normal; ω_Y ≅ O_Y(−4); H^i(Y, O_Y(k)) = 0 for 0 < i < 11 and all k. It confirms Oeding's conjecture (Remark 3.15
  of arXiv:1603.08980) that σ2((P^1)^5) is arithmetically Gorenstein, and its Cohen–Macaulay part is the case
  σ2((P^1)^5) of Oeding's Conjecture 1.1 (all secant varieties of Segre products are arithmetically Cohen–Macaulay),
  whose known cases listed in his Section 2 include the second secant varieties of at most four factors.
- **Theorem 2.1** (cited from MOZ, Lemma 3.1 and Theorem 4.5): in secant cumulant coordinates the chart
  Y ∩ {x_∅ ≠ 0} is A^5 × T, with T = T_{2,5} the affine toric variety of the monoid generated by the 26 points
  (1; χ_W), |W| ≥ 2.
- **Proposition 2.2** (proved in the note; this re-verifies MOZ Cor. 4.10/5.7 for T): the cone C = {0 ≤ x_i ≤ h,
  Σx_i ≥ 2h}; the monoid is saturated and generates Z^6; the interior lattice points are m0 + C ∩ M with
  m0 = (2; 1,1,1,1,1) (MOZ print the Gorenstein point with Σ_{i=1}^{4}, a misprint for Σ_{i=1}^{5}).
- **Corollaries 2.3, 2.4**: T normal, CM, Gorenstein, ω_T free on χ^{m0} dlog t; Y normal with rational
  singularities (MOZ Cor. 4.10), ω_Y invertible, generated on U_0 by s0 = dz1∧⋯∧dz5 ∧ t0 dt0 ∧ dt1∧⋯∧dt5.
- **Proposition 3.1**: HF(r) = P(r) := Σ h_i C(r − i + 11, 11) for all r ≥ 0, from Raicu's formula (2): closed form
  (4) of the multiplicities, telescoping (5), Lemma 3.2 (sums of polynomials and alternating sums over simplices),
  quasi-polynomiality of degree ≤ 11 on the classes {r odd, r ≥ 1}, {r ≡ 2 mod 4, r ≥ 2}, {r ≡ 0 mod 4, r ≥ 4}, and an
  exact evaluation for r ≤ 120 (12 values per class suffice, r ≤ 48); P(−4 − k) = −P(k).
- **Lemma 4.1** (proved): for smooth Z, every point of σ2(Z) lies on a secant line or on a line in an embedded
  tangent space. **Lemma 4.2**: H_0 = Y ∩ {x_∅ = 0} is irreducible and div(x_∅) = H_0 with multiplicity one (the
  DVR step written with x_∅/ℓ, ℓ = x_[5]).
- **Proposition 4.3** (first proof): ω_Y ≅ O_Y(a) and a = −4 by Serre duality and Proposition 3.1.
  **Lemma 4.4 / Proposition 4.5** (second proof, from verification run A): s_K = (x_∅/x_K)^4 s_0 on all 32 charts, so
  s_0 ⊗ x_∅^4 is a nowhere-vanishing section of ω_Y(4); independent of Raicu's theorem and of Lemma 4.2.
- **Lemma 5.1**: vanishing, from R^jπ_*O = 0 (rational singularities), Serre duality on a resolution,
  Kawamata–Viehweg, and Serre duality on Y. **Proof of Theorem A**: R ⊆ R' (saturation), dim R'_k = P(k) = dim R_k,
  so R = R'; local cohomology gives depth 12; Stanley's theorem gives Gorenstein; a(R) = −4.
  **Remark 5.2**: the section ring R' is Gorenstein without Raicu's theorem.
- **Theorem B** (Section 6, quoted from the companion note): the ball with centre (0,0,1/2,0,1/2,0) and squared radius
  317813/2 in R^6; n = (−36, −46, 50, −57, −41, 1); six lattice points on n·z = −41504; w = (413, 528, −572, 656,
  472, −12) ∈ 3P not a sum of three lattice points; padding gives every n ≥ 6.
- **Theorem C** (Section 7): a dimension count on the incidence variety; Remark 7.1: characteristic p and genericity.

## Computations (sanity checks; programs and outputs in reproducibility/)
Three finite computations are inputs of proofs: the evaluation of Raicu's formula for r ≤ 48 (Step 4 of the proof of
Proposition 3.1), the rank of the Jacobian of φ at P_0 (Lemma 4.2), and the enumeration in Theorem B(ii) (companion
note). Everything else is a test.
- **The program of this note** (`check_note.py`, Python 3, standard library + numpy for part C, about 10 s; 27 checks,
  ALL PASS; recorded output `check_note_output.txt`):
  - A1–A10: Raicu's formula (2) for r ≤ 120 (term by term for r ≤ 14); closed form (4) against Raicu's case
    statement for r ≤ 120; telescoped form (5); the inclusion–exclusion identity of Step 3; HF(r) = P(r) for
    0 ≤ r ≤ 120; 12 values in each class with r ≤ 48; h-vector (1,20,210,772,1250,772,210,20,1) followed by zeros up
    to t^120; symmetry; degree 3256; P(−4 − k) = −P(k) for |k| ≤ 60; vanishing 12th differences on residue classes.
  - B1–B9: generators and lattice; facets and vertices (all 462 five-subsets of the 11 inequalities solved exactly);
    saturation in heights ≤ 6 (26, 192, 802, 2438, 6054, 13069 lattice points); m0 and the interior; the cumulant
    identity of Theorem 2.1(a) for all 26 sets |I| ≥ 2 by exact polynomial arithmetic; P_0 (x_∅(φ(P_0)) = 0,
    x_[5](φ(P_0)) = 274, Jacobian rank 12, rank 12 at random points); Lemma 4.4 (a), (b), (c) for all 32 sets K at
    random rational points, with the derivatives computed from the cumulant formulas by exact automatic
    differentiation.
  - C1–C4: HF(2), …, HF(5) = 528, 5216, 35539, 184096 certified from the parametrization and the 2240 cubic minors of
    flattenings in all weight orbits (dim I_3 = 768), independently of Raicu's theorem.
  - D1–D3: the certificate of Theorem B (Cauchy–Schwarz bound 61643/4 > 0, enumeration of the slice of squared radius
    270429/43372 in a box of 59049 points, affine independence, w ∉ sums of three F_j, spanning).
  - E1: f = x0^{p−1} x1 over F_p, p ≤ 13: ∇f(x) = (−b : a) and x lies on ∇f(x).
- **Programs with which the results were first obtained** (`original/`): `hilbert_raicu.py` (Raicu's formula, r ≤ 60,
  h-vector), `hilbert_cleanformula.py` (telescoped form; HF = P for r ≤ 120; 12th differences), `hf_cert.py`
  (certified HF(2..5)), `gorenstein_chart.py` (chart facts, m0, IDP ≤ 6, Jacobian at P_0), `check_ball_thmA.py`
  (certificate of Theorem B, eight steps).
- **Verification run A** (`verification_run_A/`): `vA_hilbert.py` (HF = P for r ≤ 124, closed form for r ≤ 200,
  telescoping, inclusion–exclusion), `vA_weightspaces.py` (Raicu's multiplicities against ranks of evaluation
  matrices in 42 weight spaces, degrees 6–14: all equal; lower bounds only, so a test), `vA_chart.py` (facets by brute
  force, m0, lattice, total unimodularity, IDP ≤ 5, interior, P_0), `vA_canonical.py` (MOZ Lemma 3.1 for n = 5; the
  gluing identities symbolically), `vA_degree.py` (double point formula: (14400 − 7888)/2 = 3256), `vA_ball.py`
  (certificate of Theorem B).
- **Verification run B** (`verification_run_B/`): `raicu_hf_B.py 300` (Raicu's formula, HF = P for r ≤ 300, sanity
  tests n = 1, 2, 3, a wrong-input test), `toric_chart_B.py` (chart, extreme rays, m0, IDP ≤ 6, cumulant identity
  n = 4, 5, P_0, wrong-input tests), `hf_linalg_B.py` (certified HF for (P^1)^4 in degrees 2–6 and for (P^1)^5 in
  degrees 2–6, with wrong-input tests; degree 6 takes about 160 s), `ball_check_B.py` (certificate with its own
  enumeration), `polar_map_test_B.py` (Gröbner tests of Theorem C on two examples; a third example did not finish).
- **Third verification run** (`independent_run_2/`, programs written for the written note before the package's
  programs were opened): `r2_hilbert.py 150` (Raicu's case rule literally; HF by Kronecker substitution, brute force
  for r ≤ 18; HF = P and the h-vector up to r = 150; P(−4−k) = −P(k) symbolically; Steps 1–3 separately; degree-11
  interpolation on each residue class), `r2_chart.py` (Theorem 2.1(a) symbolically for n = 5 from MOZ's definitions;
  facets of C from all 65780 five-subsets of generators; saturation up to height 7; interior up to height 8),
  `r2_canonical.py` (Lemma 4.4(a),(c) for all 32 K without the map τ_K: chart coordinates of σ_K(p)/x_K(p), exact
  derivatives; ε = −1), `r2_lemma42.py` (P_0 data; ranks 12 and 11), `r2_ball.py` (own enumeration of the slice with
  exact per-coordinate bounds; padding to R^7, R^8), `r2_gradient.py` (Remark 7.1; Theorem C for random lines in P^2 by
  exact gcd tests, with two non-generic controls), `r2_hfcert.py` (HF(2), …, HF(6) certified without Raicu's theorem in
  every weight space, all equal to Raicu's per-weight predictions). All checks pass.
- **Re-runs (14 October 2026).** All 19 program runs with a recorded output were repeated from a copy of the package
  (`reproducibility/RERUN_LOG.txt`): 17 outputs identical up to running times; `hf_linalg_B.py 4 … --wrong` identical
  except two wrong-input lines recorded with an earlier revision of that program; `polar_map_test_B.py` does not
  finish its third example, and its recorded partial output was reproduced before the run was stopped. The third
  verification run repeated all 19 runs from an extracted copy of the source archive with the same result
  (`reproducibility/independent_run_2/package_rerun.log`).

## Independent verification runs
The results were first obtained with proofs and the programs in `original/`. Two independent verification runs, both
AI-assisted, followed (2026-10-14); each wrote its own programs and read the sources. A third verification run, also
AI-assisted, then examined the written note (see the end of this section).

| Item | Run A | Run B |
|---|---|---|
| Statement, readings, record's literature note (local vs arithmetic Gorenstein) | CONFIRMED | CONFIRMED |
| MOZ Thm 4.5 / Lemma 3.1 (chart), Def. 4.1, Cor. 4.10, Cor. 5.7, Thm 7.18 | CONFIRMED (re-derived for n = 5; misprint Σ_{i=1}^{4} noted) | CONFIRMED (cumulant identity for n = 4, 5; misprint noted) |
| Raicu's Theorem 4.1 as transcribed | CONFIRMED (word for word) | CONFIRMED (from the TeX source) |
| Proposition 3.1 (closed form, telescoping, quasi-polynomiality, finite check) | PROVED (re-derived) | PROVED; HF = P for r ≤ 300 |
| HF in low degrees without Raicu | tested in degrees 6–14 (42 weight spaces, equal) | certified for r ≤ 6 |
| Lemma 4.2 (H_0 irreducible, multiplicity one) | CONFIRMED_WITH_FIXES (x_∅/ℓ) | CONFIRMED |
| ω_Y ≅ O_Y(a), a = −4 | CONFIRMED; second proof by gluing found and checked | CONFIRMED |
| Vanishing, R = R', CM, Gorenstein, a(R), reg | CONFIRMED (cosmetic fix: R ⊆ R' by saturation) | CONFIRMED |
| Theorem B certificate | CONFIRMED (own program) | CONFIRMED (own enumeration; padding argument) |
| Theorem C | CONFIRMED | CONFIRMED (Gröbner tests on examples) |
| Novelty of (a) | plausible; Daleo–Hauenstein (2016) to be read | no prior proof found; cite Daleo–Hauenstein |
| Bibliography | – | CHECKED (Crossref, zbMATH, OpenAlex) |

No run found a mathematical error.

**Corrections required by runs A and B**, all applied in the note:
1. (A) Daleo–Hauenstein, MACIS 2015, LNCS 9582 (2016) 137–142, read in the authors' extended abstract: its three
   examples are σ3((P^1)^4), σ3(P^1 × P^1 × P^1 × P^2) and σ3(ν4(P^2)); it is cited as related numerical work
   (Section 1.3), and "to our knowledge" is kept.
2. (A) Khadam–Michałek–Zwiernik are quoted as stating that the classification of arithmetically Gorenstein secant
   varieties remains open even for Segre products (Question 4.7); that σ2((P^1)^5) was the only undecided Segre case
   is stated as an inference from MOZ Theorem 7.18 and Oeding's Theorem 3.14.
3. (A) Lemma 4.2: the multiplicity-one step is written with x_∅/ℓ.
4. (A, cosmetic) R ⊆ R' because I(Y) is saturated; the MOZ misprint Σ_{i=1}^{4} is pointed out; "Theorem 1" of the
   source is attributed to MOZ; BGM Question 7.2(b) is described as a question on ellipsoids.
5. (A, recommended) The gluing proof of ω_Y ≅ O_Y(−4) (Section 4.3) and the double point degree check
   (Section 8) were added.
6. (B, recommended) Daleo–Hauenstein cited; the toric facts are proved completely (Proposition 2.2 replaces the
   lattice-point check in heights ≤ 6 by a proof); the misprint mentioned; bibliographic data completed (KMZ:
   Linear Algebra Appl. 588 (2020) 428–457; OWR issue data).

**Changes made when the note was written, after runs A and B.** Proposition 2.2 and Lemma 4.1 are proved in the note
(the first version cited these facts); Remark 5.2 and `check_note.py` were added; the proof of Lemma 5.1 uses only
R^jπ_*O = 0, Serre duality and Kawamata–Viehweg vanishing (the first version also used Grauert–Riemenschneider
vanishing and Kempf's criterion). None of these changes a statement.

**Third verification run (the written note).** It examined the text line by line as if nothing had been checked,
read the sources again (the whole report 19/2016 including its title page and the rendered p. 1000; the TeX sources
of MOZ v2, Raicu v2, Oeding v2, KMZ v1, BGM v2 and MSUZ v1 and v2; Oeding's and KMZ's printed numbers in the PDFs;
OWR 44/2017 p. 2684; the extended abstract of Daleo–Hauenstein; the companion note as published on Zenodo), re-derived
every proof, wrote its own programs (`independent_run_2/`) before opening those of the package, re-ran the package
from the source archive and repeated the literature search (one web search; arXiv, OpenAlex, zbMATH, Crossref).

| Item | Third run |
|---|---|
| Statement, readings, record (incl. the closure in (c)) | CONFIRMED; the workshop dates were wrong in the note (corrected) |
| Section 2: chart, Proposition 2.2, Corollaries 2.3–2.4 | CONFIRMED (re-derived; facets from the generators; saturation ≤ height 7; Theorem 2.1(a) symbolically) |
| Raicu's Theorem 4.1 as transcribed; Proposition 3.1, Lemma 3.2 | CONFIRMED (word for word; Steps 1–4 re-derived; HF = P for r ≤ 150 by another summation) |
| HF in low degrees without Raicu | certified for r ≤ 6 with its own program, all weight spaces equal to Raicu's predictions |
| Lemmas 4.1, 4.2, Proposition 4.3 | CONFIRMED (observation: the Jacobian bound in Lemma 4.2 is not needed for the irreducibility) |
| Lemma 4.4, Proposition 4.5 | CONFIRMED (re-derived; checked for all 32 K without τ_K; ε = −1) |
| Lemma 5.1, proof of Theorem A, Remark 5.2 | CONFIRMED |
| Theorem B | CONFIRMED (own enumeration; padding); one sentence of the sketch of (iii) made precise |
| Theorem C, Remark 7.1 | CONFIRMED (proof re-derived; exact tests with controls) |
| Cited statements and numbers | CONFIRMED; the numbering of the MSUZ conjecture in the journal version is not known (wording corrected) |
| Novelty, credit | no earlier proof found; credit to Oeding's Conjecture 1.1 added |
| Package re-run from the source archive | 17 of 19 identical, the two documented exceptions reproduced |

It found no mathematical error and no gap. **Corrections required by the third run**, all applied in the note:
1. The workshop dates: 27 March – 2 April 2016 (not "April 2016").
2. The source cites the journal version of MSUZ as Conjecture 5.5; in arXiv versions 1 and 2 it is Conjecture 5.8
   (the note no longer says that the source cites "the journal numbering").
3. Sketch of Theorem B(iii): the F_j are linearly independent (affinely independent on a hyperplane not through 0),
   so w = ½(F_0 + ⋯ + F_5) is the only linear combination of them equal to w (the earlier wording spoke of
   barycentric coordinates).
4. Credit: the Cohen–Macaulay part of Theorem A is the case σ2((P^1)^5) of Oeding's Conjecture 1.1; Oeding's
   Section 2 lists second secant varieties of at most four factors among the known cases.
5. The record of this run in the Verification paragraph, in Section 8, in "What was read", in the search paragraph,
   in this report and in `reproducibility/independent_run_2/`.

## Relation to the literature, novelty and scope
- **Searches (14 October 2026, at the first writing and by all three runs).** arXiv API (secant + Gorenstein + Segre,
  arithmetically Gorenstein + secant, Cohen–Macaulay, Hilbert series, border rank, binary tensors, canonical module,
  secant cumulants, the authors of MOZ), OpenAlex citation lists (MOZ: 23, Oeding 2016: 4, KMZ 2020: 5, Raicu 2012:
  66), zbMATH Open, Crossref, Semantic Scholar, five web searches in all.
  - Latest statements found on (a): Oeding (arXiv:1603.08980v2, 2016), Table 1 ("?") and Remark 3.15 (conjectured,
    "not able to prove"); KMZ (LAA 2020), end of Section 4 and Question 4.7 (classification open even for Segre
    products).
  - Related numerical work: Daleo–Hauenstein (2016), three other secant varieties.
  - No proof (or disproof) of the arithmetic Gorenstein or Cohen–Macaulay property of σ2((P^1)^5) was found; Oeding
    (2016, Conjecture 1.1 and Section 2) lists the aCM property of second secant varieties as known for at most four
    factors. A search that finds nothing is not a proof of novelty; no priority is claimed.
- **What was read.** The source pp. 999–1001 (publisher's PDF); MOZ arXiv v2 (Sections 2–5, 7; numbering used;
  journal version not compared); Raicu arXiv v2 (Theorem 4.1); Oeding arXiv v2 (Sections 1–3, Table 1); KMZ arXiv v1
  (Section 4, Question 4.7); MSUZ arXiv v2 (Section 5: Theorem 5.5, Conjecture 5.8, Proposition 5.9,
  Corollary 5.10) and, by the third run, v1 (same numbering); Daleo–Hauenstein (authors' extended abstract, not the
  published chapter); the companion note (from which the statements on BGM, Gubeladze and the 2017 question were
  taken; the third run read BGM arXiv v2, Theorem 6.5 and Question 7.2, OWR 44/2017 p. 2684 and Gubeladze's abstract).
  Teissier's papers and Huh's paper are cited only as in the source and were not used. Textbook statements
  (Hartshorne, Eisenbud, Bruns–Herzog, Cox–Little–Schenck, Lazarsfeld, Mumford, KKMS) are cited without re-checking
  their proofs; their theorem and exercise numbers were not compared with the books (CLS Theorem 11.4.2 is cited as
  in MOZ), and where a number was uncertain a chapter or section is cited instead.
- **Bibliography.** All DOIs checked with Crossref on 2026-10-14 when the note was written (20 DOIs; that of
  Teissier 1982 found by a bibliographic query), the companion note's DOI with DataCite; all 21 checked again by the
  third run (metadata matching the entries).
- **Scope.** Characteristic 0 for (a); higher secant varieties and other Segre(–Veronese) varieties are not treated;
  for (b) the dimensions 4 and 5 and lattice-point-centred balls remain open; (c) is not new.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/