# Verification report — OWR-13494-009 (Harbourne's question on the Roulleau–Urzúa configurations of cubics: is I^(3) contained in I^2?)

Verification date: 2026-10-14.

**Verdict.** The note answers the **first clause** of the record, under a stated reading of "the ideal of the
singular points"; the second clause is **not addressed**.
- **Settled (first clause).** For every n = 3m (m ≥ 1), with I the ideal of the cluster of singular points of C_n —
  weight k = 3m² − 1 at each of the 12 vertices, weight 1 at each of the 12k infinitely near triple points and at
  each of the 9k(m² − 1) quadruple points — I^(3) ⊆ I^2 (Theorem 1.3). So the source's question "does I^(3) ⊄ I^2
  hold?" has the answer no, and the record's "does I^(3) ⊆ I^2 hold for all n?" the answer yes, for every n
  divisible by 3. Moreover α(I^(3)) = 36m² − 12 for all m, reg(I) = 12 for m = 1 and reg(I) ≤ 18m² − 8 for m ≥ 2
  (Theorem 1.4).
- **Reading.** For m = 1 this ideal is literally the one of the source's comment (the 12 points with multiplicity 2
  and the 24 infinitely near points with multiplicity 1). For m ≥ 2 the weight 3m² − 1 is the note's extension:
  locally, the ideal is spanned by the germs of curves passing through all proper and infinitely near singular points
  (Lemma 3.1(c)); it is the ideal of the consistent cluster with the smallest weights containing them. The reading with the reduced proper points already fails at n = 3
  (Dumnicki–Szemberg–Tutaj-Gasińska) and contradicts the source's statement on C_3; the reading with weight 2 at the
  vertices for n ≥ 6 is not studied.
- **Not new.** The case n = 3 (m = 1) is the source's own result, with the same numbers α(I^(3)) = 24, reg(I) = 12
  and the same criterion; the geometry of C_n is Bauer–Harbourne–Roé–Szemberg's; the criterion is Bocci–Harbourne's.
- **New, to our knowledge** (relative to the literature found): the containment for every m ≥ 2. No answer for
  m ≥ 2 was found in the accessible literature; a search that finds nothing is not a proof of novelty. Harbourne's
  2021 chapter (closed access) was not read.
- **Open / not addressed.** Szemberg's clause on analogous configurations of conics (open-ended); the weight-2
  reading for n ≥ 6; the exact value of reg(I) for m ≥ 2.
- **Computer-assisted step.** Only reg(I) ≤ 12 for m = 1 (Proposition 5.4), an exact rank computation over Q(ε).

The note is unrefereed.

## Statement checked
- **Primary source.** B. Harbourne, "H-constants and the containment problem", extended abstract in Oberwolfach
  Report 9/2015 (Mini-Workshop: Ideals of Linear Subspaces, Their Symbolic Powers and Waring Problems, organisers
  C. Bocci, E. Carlini, E. Guardo, B. Harbourne), Oberwolfach Rep. 12 (2015), no. 1, pp. 489–532, doi:10.4171/OWR/2015/9;
  Harbourne's abstract on pp. 510–513. Read in the publisher's open PDF (ems.press), downloaded anonymously on
  2026-10-14.
  - Ground field: the complex numbers (p. 510).
  - Example 3 (p. 512): for 3 | n, C_n is the union of 4(n² − 3)/3 cubics, members of four isotrivial pencils, each
    the pencil of cubics through 9 of the 12 triple points of the n = 3 Fermat (dual Hesse) arrangement; each pencil
    has 3 singular elements; deg C_n = 4(n² − 3); 12 points of multiplicity n² − 3; (n² − 3)(n² − 9)/3 quadruple
    points; 4(n² − 3) triple points, infinitely near the 12 points; no other singular points; H(C_n) = −4(n⁴ − 3n²)/(n⁴ + 27).
  - The question: for I the ideal of the singular points of C_n, does I^(3) ⊄ I^2 hold?
  - Interpolated comment: only C_3 was worked out; there I is the ideal of the 12 points with multiplicity 2 and the
    24 infinitely near points with multiplicity 1; α(I^(3)) = 24, reg(I) = 12, and the criterion of Bocci–Harbourne
    gives I^(3) ⊆ I^2.
  - Second question (Szemberg): "Are there interesting somehow analogous configurations of conics?"
- **Corpus record.** ulamai/UnsolvedMath, OWR-13494-009 (dataset version 1.6.0; upstream status `open`): it asks
  whether I^(3) ⊆ I^2 holds for all n, and whether there are interesting analogous configurations of conics. The polarity is flipped
  relative to the source; the decision problem is the same.
- **Range of n.** Source p. 512 ("In the case that 3 divides n"); Bauer–Harbourne–Roé–Szemberg, arXiv:1603.04480v1
  (n = 3m; Theorem 1 for n ∈ 3N); Roulleau, arXiv:1411.6996v3, §1 (n ∈ 3N); Roulleau–Urzúa, arXiv:1402.5801, §3
  (Eisenstein integers n divisible by 1 − ζ², the only case considered; for rational integers: 3 | n). The note
  concerns rational integers n = 3m.

## Readings
| Reading | Answer | Where |
|---|---|---|
| I = ideal of the cluster: weight 3m² − 1 at the vertices, 1 at the infinitely near triple points and at the quadruple points (the source's ideal for m = 1; our extension for m ≥ 2) | I^(3) ⊆ I^2 for every m ≥ 1 | Theorem 1.3 |
| Symbolic powers: Bocci–Harbourne's I^(r) = I^r R_A ∩ R | equals sat(I^r) for this I | Lemma 3.2 |
| I = ideal of the proper singular points, reduced | fails already for m = 1 (DST 2013); excluded by the source's statement on C_3 | Remark 1.2(c) |
| Weight 2 at the vertices for all m (copying the numbers of the comment) | for k ≥ 3 the cluster is not consistent and its ideal is that of the vertices with multiplicity 3 and of S; not studied | Remark 1.2(d) |
| Second clause (conics) | open-ended request for examples; not addressed | Remark 7.1 |

## Results in the paper
- **Theorem 1.3** (main): I^(3) ⊆ I^2 for every n = 3m.
- **Theorem 1.4**: α(I^(3)) = 36m² − 12 (all m); reg(I) = 12 (m = 1); reg(I) ≤ 18m² − 8 (m ≥ 2).
- **Section 2**: the configurations after Bauer–Harbourne–Roé–Szemberg (Theorem 2 and Theorem 1.5 of arXiv v1,
  quoted as Theorems 2.1, 2.2) and the consequences used (Lemma 2.3: smoothness and distinctness of the 4k cubics,
  the k tangent directions at each vertex with one cubic of each of the three pencils per direction, the quadruple
  points on one cubic of each pencil, the complete list of singular points, proper and infinitely near).
- **Section 3**: the blow-up Y, the classes of the strict transforms, Σ C~ = 12kL − 3kΣE_v − 3ΣE_q − 4ΣE_s, the
  cluster divisor Z, the local ideals I_{Z,v} = (Π l_{v,j}) + m^{k+1} (Lemma 3.1), symbolic powers (Lemma 3.2).
- **Proposition 4.1**: Bézout: C~·Z = 9k + 9 + 9(m² − 1) = 36m² − 9, so α(I^(3)) ≥ 36m² − 12; the product of the
  4k cubics lies in I^(3) (local grouping: at a vertex, the k cubics of each pencil have the k tangent lines), so
  equality.
- **Proposition 5.2** (m ≥ 2): N = tL − Z − K_Y − ΣẼ_v = (t + 3)L − (k + 2)ΣE_v − ΣE_q − 2ΣE_s, written as
  ½ΣC~ + aL + bΣẼ_v + (b + ½)ΣE_q with a = t + 3 − 6k ≥ 0, b = k/2 − 2 > 0; N·C~ = 3t − 45m² + 9 ≥ 9m² − 18,
  N·Ẽ_v = 2, N·E_q = 1; N² ≥ 12(k − 4)(k + 1) > 0; Kawamata–Viehweg, restriction to the disjoint Ẽ_v ≅ P¹ (degree
  −Z·Ẽ_v = 0), Leray: H¹(I_Z(t)) = 0 for t ≥ 18m² − 9.
- **Lemma 5.1**: dim(R/I)_t = ℓ − h¹(I_Z(t)); monotonicity; reg(I) = τ(I) + 1 (Bocci–Harbourne p. 3, after
  Geramita–Gimigliano–Pitteloud).
- **Proposition 5.4** (m = 1, computer-assisted): ranks 36, 45, 52, 57, 60, 60, 60 for t = 7..13; τ(I) = 11,
  reg(I) = 12.
- **Theorem 6.1** = Bocci–Harbourne, Lemma 2.3.4, quoted with its hypotheses ("homogeneous ideal (not necessarily
  saturated) defining a 0-dimensional subscheme", over an algebraically closed field, as in Bocci–Harbourne); our I
  satisfies them; r = 2, s = 3.
- **Proposition 7.2** (added at the writing): for m ≥ 2, α(I^(s)) ≥ (3k + 4)s for all s, hence by Bocci–Harbourne
  Corollary 2.3.5 the resurgence ρ(I) ≤ (6k − 2)/(3k + 4) < 2.
- **Remark 7.4**: two misprints in arXiv v1 of Bauer–Harbourne–Roé–Szemberg, stated neutrally: the class in
  Theorem 1.5 should be 12kL − 3kE (printed 12kL − 9kE); the six non-rational parameters of P¹[6] are the roots of
  (x² − x + 1)³ + 54x²(x − 1)² = x⁶ − 3x⁵ + 60x⁴ − 115x³ + 60x² − 3x + 1 (printed x⁶ − 3x⁵ + 5x³ − 3x + 1). Neither
  affects their theorems.

## Computations (programs and outputs in reproducibility/)
Only Proposition 5.4 (m = 1, reg(I) ≤ 12) rests on a computation.
- **Certificate for m = 1.** Exact ranks of the 60 conditions over Q(ε), by five programs with their own
  constructions: `check_note.py` (36, 45, 52, 57, 60, 60, 60 for t = 7..13), `original/scripts/m1_exact.py`
  (52, 57, 60, 60 for t = 9..12), run A's `m1_verify.py` (45, 52, 57, 60, 60 for t = 8..12), run B's `b2_m1.py`
  (t = 7..13) and run 2's `r2_m1_exact.py` (t = 7..13). All agree.
- **Corroboration of BHRS Theorem 1.5** for m = 2 (modulo p = 1000039, 1000303, 1000171, 1000291; complex numerics
  with 50 digits, and with 80 digits in run 2) and m = 3 (modulo p = 2000521, 1500283; numerics with 50 and 80
  digits): 44 / 104 smooth cubics; 11 / 26 tangent
  directions at each vertex with 3 cubics from 3 pencils each; 297 / 1872 quadruple points, each on exactly 4 cubics
  (one per pencil); 27 / 72 on each cubic; no other intersections. With BHRS's printed sextic instead of the correct
  one: 1593 off-vertex points (both numerical programs), only 9 on four cubics (the test detects a wrong input).
- **m = 2 regularity modulo primes:** h¹(I_Z(t)) = 18, 3, 0 at t = 49, 50, 51 at each of six primes (the four above
  and, in run 2, 1002073 and 1002091), 0 at t = 52, 63 where computed (consistent with the proved vanishing from t = 63; would give reg(I) ≤ 52 if the points
  computed mod p are reductions of the characteristic-0 points, which is not proved).
- **Other checks:** α of the forms with div(π*F) ≥ 3Z equal to 24 for m = 1 (mod p, four programs); I^(3) ⊆ I^2 in
  degrees 24–26 (run B) and 24 (run 2) for m = 1, by showing that (I²)_t equals the space of forms with
  div(π*F) ≥ 2Z (mod p, valid in characteristic 0 by a dimension count); failure detection on the ideal of the 12
  vertices (DST); P¹[3], P¹[6], P¹[9] from division polynomials; symbolic check of all identities (runs B and 2).
- **Command lines** of every recorded output are in `reproducibility/README.md`. The recorded
  `m2_modp_output_p1000039.txt` was produced with the non-default first degree 46 (`m2_modp.py 1000039 46`).
- **Re-runs (2026-10-14):** all programs with a recorded output were run again from a copy of the package when the
  note was written, and again from the archive by run 2 (`reproducibility/independent_run_2/rerun_record.txt`);
  both times the outputs were identical up to timing fields. The programs of `original/` now write to `rerun_outputs/` (one-line change), so a
  re-run cannot overwrite the recorded outputs.
- **Field of definition (observation only):** at the four primes 1000039, 1000303, 1000171, 1000291 all
  intersection points of pairs of the 44 cubics were F_p-rational, and at the two primes of run 2 (where only the
  pairs from two of the pencils were intersected) the 297 points were F_p-rational as well; no claim is made that
  the quadruple points are defined over Q(ε, 2^{1/3}).

## Independent verification runs
The results were first obtained, with proofs and the programs in `original/`. Two independent verification runs,
both AI-assisted, followed; each wrote its own programs and neither saw the other's report. After the note was
written, a further independent AI-assisted run (run 2) examined its final text (last item below).
- **Run A** (statement and proofs): compared the statement with the source (and re-derived Harbourne's H-constant
  formula from the counts); analysed the readings of the ideal; re-derived every step of Sections 3–6 with all
  intersection numbers; recomputed the m = 1 certificate exactly with its own construction (ranks 45, 52, 57, 60, 60);
  corroborated BHRS Theorem 1.5 modulo new primes for m = 2 (1000171, 1000291) and m = 3 (1500283); confirmed the
  sextic misprint; checked the sentences about sources. Verdict: confirmed with fixes (wording and caveats; no
  mathematical error).
- **Run B** (independent computation; literature and novelty): re-derived the proofs and checked all identities
  symbolically in m and t; exact m = 1 ranks for t = 7..13 and exact intersection multiplicities; mod-p checks of
  α = 24 and of I^(3) ⊆ I^2 in degrees 24–26; complex numerics for m = 2, 3; mod-p regularity for m = 2 at two new
  primes; failure detection (DST ideal, wrong sextic); re-ran the original programs; literature search (arXiv,
  zbMATH, OpenAlex, Semantic Scholar, three web searches). Found the misprinted class 12kL − 9kE in BHRS Theorem 1.5
  and the direct argument that the product of the cubics lies in I^(3). Verdict: confirmed with minor fixes.
- **Required fixes and where they were applied:** naming of the ideal and of the extension (abstract, Definition 1.1, Remark 1.2, "Scope and priority"); status of
  the two clauses (abstract, Sections 1.1, 1.5, Remark 7.1); "attained" only with a proof (Proposition 4.1(b), via
  the direct argument, no use of Zariski's theorem); field-of-definition remark as an observation (Section 8);
  caveat on Harbourne's 2021 chapter (Section 1.5, "Scope and priority"); range of n with references (Section 1.2);
  class 12kL − 3kE (Remark 7.4(a)); command lines (Section 8, README); completed bibliography with DOIs checked
  against Crossref; status wording.
- **Added at the writing:** the monotonicity argument in Lemma 5.1, Proposition 7.2, and `check_note.py`. Run 2
  examined the first two line by line and executed the third (its results agree with run 2's own programs; its code
  was not reviewed line by line).
- **Run 2 (on the final text of the note; programs and outputs in `reproducibility/independent_run_2/`).** Read the
  source, Bauer–Harbourne–Roé–Szemberg (arXiv v1), Bocci–Harbourne (arXiv v2), Dumnicki–Szemberg–Tutaj-Gasińska,
  Roulleau and Roulleau–Urzúa again, and the passages of Harbourne 2018 (arXiv v1) on containment and on the
  configurations of cubics (no statement there on C_n beyond pointing to the examples); re-derived every proof of
  Sections 2–7 line by line and checked all identities
  symbolically in m and t; wrote its own programs: the configuration for m = 1 from scratch with the exact ranks
  36, 45, 52, 57, 60, 60, 60 (t = 7..13); for m = 1 modulo p = 1000003 the 3Z dimensions (0 up to degree 23, 1 in
  degree 24) and a direct test of I^(3) ⊆ I² in degree 24 (the products of a basis of I_12 span the 133-dimensional
  space of forms of degree 24 with div(π*F) ≥ 2Z, and the product of the eight cubics lies in it), with the DST
  ideal as control; P¹[3], P¹[6], P¹[9] from division polynomials and both misprints of Remark 7.4; Theorem 1.5 of
  Bauer–Harbourne–Roé–Szemberg for m = 2 and 3 by complex numerics with 80 digits (all cross-pencil pairs
  intersected completely; the printed sextic as control); m = 2 modulo 1002073 and 1002091 (h¹ = 186, 143, 102, 63,
  35, 18, 3, 0, 0, 0 for t = 44..52, 63). It re-ran all programs of the package from the archive (all outputs
  reproduced up to timing fields) and repeated the literature search (Crossref for all DOIs, OpenAlex, arXiv API,
  zbMATH, one web search). Verdict: confirmed, no mathematical error. Required corrections, all made: the ground
  field in Theorem 6.1 (algebraically closed, as in Bocci–Harbourne); in Section 2.1 the origin of the elliptic curve
  at a ramification point in the definition of P¹[3m], and the uniqueness of f stated as in BHRS (Lemma 1.1, which is
  on p. 5, and p. 6; the earlier text cited "Lemma 1.1, p. 6" and said only "up to automorphisms"); Remark 1.2(b) in
  the local form proved by Lemma 3.1(c) (the global form was checked for m = 1 exactly and
  for m = 2 modulo p, but is not proved in general); the record of this run in the note, in this report and in the
  package.

## Relation to the literature, novelty and scope
- **Searches** (2026-10-14: first results, runs A and B, partly again at the writing, and run 2): arXiv API (Halphen cubics,
  Roulleau, elliptic curve configurations, infinitely near points, Hesse configuration, each with symbolic powers or
  containment; cubics/conics with containment), zbMATH ("Halphen cubics": 7 hits, none on containment), Semantic
  Scholar, OpenAlex (6 works citing BHRS or the OWR report: Harbourne 2018 (Banach Center Publ. 116), Dimca–
  Harbourne–Sticlaru 2022, Bauer–Harbourne–Küronya–Nickel (IMRN), Harbourne 2021 (chapter, closed access), Haghighi–
  Mosakhani 2018, Guardo–Van Tuyl 2015 (book chapter); none of the titles or available abstracts announces an
  answer; run 2 also listed the 14 works citing Roulleau's paper and the works citing Bocci–Harbourne found with the
  words "cubics" (23) or "Roulleau" (1): no title announces an answer), Crossref, six general web searches in all.
  No answer for m ≥ 2 was found.
- **What was read.** The source (pp. 510–513); BHRS arXiv v1, pp. 1–10 (journal version not read); Bocci–Harbourne
  arXiv v2, §§1.1, 1.2, 2.3; DST arXiv v3, §§1–2; Roulleau arXiv v3, introduction and §1; Roulleau–Urzúa arXiv, §3;
  passages of Harbourne 2018 (arXiv v1) and Szemberg–Szpond (arXiv v1); abstract of Pokora–Tutaj-Gasińska.
- **Not read.** Harbourne, "Algebraic Geometry, Commutative Algebra and Combinatorics: Interactions and Open
  Problems", in Commutative Algebra (I. Peeva, ed.), Springer 2021, pp. 445–473, doi:10.1007/978-3-030-89694-2_14
  (closed access; cites BHRS; could restate or settle the question — a reader with access should check);
  Geramita–Gimigliano–Pitteloud (cited after Bocci–Harbourne); Lazarsfeld, Positivity I (standard statements; theorem
  numbers as usually cited).
- **Scope.** First clause answered under the stated reading; second clause not addressed; no priority claimed.

## 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/