# Verification report — OWR-13494-011 (Janssen: is there a non-ACM configuration of lines in P^3 with α(I^(2)) − α(I) = 1?)

Verification date: 2026-10-10.

**Verdict.** The note gives a **complete negative answer** to the question; every statement it makes is proved,
and the scope is as follows.
- **Settled.** Over an algebraically closed field of arbitrary characteristic, a finite union L of lines in P^3
  has α(I^(2)) = α(I) + 1 if and only if L is contained in a plane or is a pseudostar (Theorem 1.3). Both kinds
  of configurations are arithmetically Cohen–Macaulay (ACM). Hence there is no non-ACM configuration of lines
  with α(I^(2)) − α(I) = 1, over any field, for lines defined over that field (Corollary 1.4), and the ACM
  hypothesis in Janssen's theorem is not needed.
- **Also proved.** The same classification for reduced curves of pure dimension one: plane curves and
  pseudostars of lines (Theorem 1.5). The tool: the curves along which an irreducible surface of degree d in P^3
  is singular lie on a surface of degree d − 2; more precisely the conductor space W of forms of degree d − 2
  vanishes on them and has dimension at least d − 1 + π ≥ 1, for arbitrary singularities and in every
  characteristic (Theorem 1.6). The bound is attained (Proposition 7.4). A disjoint line added to a
  configuration with gap one gives a non-ACM configuration of type (d, d + 2) (Remark 7.5).
- **Not new, and credited.** Janssen's theorem (the ACM case), its converse part, and the ACM property of
  pseudostars (Janssen; Geramita–Harbourne–Migliore as cited by Janssen); the case of points in the plane
  (Bocci–Chiantini); the case of codimension two subspaces of P^N with Cohen–Macaulay ideal in characteristic 0
  (Fashami–Haghighi–Szemberg). **The case α(I) ≤ 2 of Theorem 1.3 already follows from Theorem B of Haghighi and
  Mosakhani (J. Algebra Appl. 20 (2021), no. 10, 2150178)**; this paper is known to the note only through its
  abstract and the review Zbl 1473.14102 (closed access; it could not be obtained), and its theorem was not
  verified. Theorem 1.6 is classical in spirit (adjoint surfaces; in characteristic 0 it follows from duality and
  a vanishing theorem for normal surfaces; the covering argument in its proof is Mumford's). The note claims only
  that no reference was found for Theorem 1.3 with α(I) ≥ 3, for Theorem 1.5, and for Theorem 1.6 in this
  generality. No priority is claimed.
- **Not treated.** Codimension two subspaces of P^N, N ≥ 4, without the Cohen–Macaulay hypothesis; configurations
  of type (d − 2, d); points in P^3; the question which configurations of lines are ACM.
- **Computations are corroboration only.** No proof depends on a computation.

The note is unrefereed.

## Statement checked
- **Primary source.** M. Janssen, "On the fattening of lines in P^3", extended abstract in: Mini-Workshop "Ideals
  of Linear Subspaces, Their Symbolic Powers and Waring Problems", Oberwolfach Reports 12 (2015), no. 1, 489–532,
  Report No. 9/2015, doi:10.4171/OWR/2015/9; the abstract is on pp. 513–515.
  - Read in the publisher's PDF (ems.press, 418 297 bytes), fetched on 2026-10-10 by the first stage, by the
    verification runs A and B, at the writing, and again by run 3.
  - The first of the three questions at the end of the abstract (p. 514) asks for a non-ACM configuration of
    lines L in P^3 with I = I(L) and α(I^(2)) − α(I) = 1. The second (p. 514) asks which configurations of lines
    are ACM; the third (p. 515) which reduced (possibly irreducible) curves have α(I^(2)) − α(I) = 1.
- **Underlying paper.** M. Janssen, "On the fattening of lines in P^3", J. Pure Appl. Algebra 219 (2015), no. 4,
  1055–1061, doi:10.1016/j.jpaa.2014.05.033.
  - Read completely in two versions: arXiv:1306.4387v2 of 29 June 2013 (runs A, B and 3, and at the writing; run
    A also compared v1) and the journal version (the publisher's file, provided by the author of the note; read
    by run 3). The statements carry the same numbers in both versions (Definitions 1.1, 1.2, 2.1, 2.3;
    Theorem 1.3; Lemmas 2.4, 2.5; Propositions 2.6, 2.7, 2.10; Corollaries 2.8, 2.9, 2.11; Notation 2.12;
    Theorem 2.13; Questions 3.1–3.4), and Definition 2.1, Theorem 2.13 and the four questions have the same
    wording. The note gives the page numbers of the journal version.
  - Field: algebraically closed, arbitrary characteristic (p. 1055). Symbolic power: Definition 1.1. α, type
    (d − t, d), t ≥ 1: p. 1056. ACM: Definition 2.3. Pseudostar: Definition 2.1, p. 1058 (pairwise intersections
    of planes no three of which meet in a line; no lower bound on the number of planes). Theorem 2.13, p. 1060:
    (a) ACM of type (d − 1, d) implies pseudostar or coplanar; (b) converse without ACM.
  - Question 3.1 (p. 1061) asks for a configuration of lines of type (d − 1, d) which is not ACM, and is followed
    by the remark that a negative answer makes the ACM hypothesis of Theorem 2.13 unnecessary. Question 3.2
    (reduced curves) is said there to have been suggested by Juan Migliore.
  - Only in the journal version: the introduction says that all configurations of lines with gap one known at
    the time were ACM and that the existence of others was open (p. 1057); before Question 3.1 the hope is
    expressed that the ACM assumption can be relaxed or removed; after Question 3.1 computed examples are
    reported in which a disjoint line added to an ACM configuration of type (d − 1, d) gives a non-ACM
    configuration of type (d, d + 2) (p. 1061). The last statement is proved in general in Remark 7.5 of the
    note.
- **Corpus record.** ulamai/UnsolvedMath, OWR-13494-011 (dataset version 1.6.0; upstream status open). Its
  statement is a faithful rewording of the first question of the abstract.

## Readings
| Reading | Answer | Where |
|---|---|---|
| The question as posed (k algebraically closed, any characteristic; finite unions of lines with reduced structure; symbolic square) | no such configuration exists | Theorem 1.3, Corollary 1.4 |
| Ground field not algebraically closed; lines defined over the field | no such configuration exists; the planes are defined over the field | Corollary 1.4 |
| Curves over a non-closed field that split into conjugate lines only over an extension | not treated | — |
| Pseudostars with two planes admitted (literal Definition 2.1 of Janssen) | the class "coplanar or pseudostar" is the same | Section 1.1, convention (vi) |
| Ordinary square I^2 instead of I^(2) | not the question (the difference would be α(I)) | Section 1.1, convention (i) |
| Third question of the abstract (Question 3.2 of the paper), with "reduced curve" read as reduced of pure dimension one (the note's reading) | exactly the plane curves and the pseudostars of lines | Theorem 1.5 |
| The same with isolated points admitted | not treated; the list would be longer (a line with two points not in a plane with it has type (2,3)) | Section 1.2 |
| A line added to a configuration with gap one that meets none of its lines (remark after Question 3.1 in the journal version) | type (d, d + 2), not ACM | Remark 7.5 |
| Second question of the abstract (which configurations are ACM) | not treated | Section 1.5 |
| Codimension two subspaces of P^N, N ≥ 4, without the Cohen–Macaulay hypothesis | not proved | Section 1.5 |

## Results in the paper
- **Lemma 2.1.** Multiplicity along a line is additive; I^(2) = ∩ I(l_i)^2; α(I^(2)) ≥ α(I) + 1 in every
  characteristic; restriction to a plane through the line. **Lemma 2.2.** A quotient S/J with a free resolution
  of length two, all minimal primes of J of height two, is Cohen–Macaulay (Auslander–Buchsbaum).
- **Lemma 3.1 (coplanar).** (α(I), α(I^(2))) = (1, 2); complete intersection, ACM; for at least two lines the
  forms of degree 2 in I^(2) are the multiples of h^2; for one line they are products of two planes through it.
- **Lemma 3.2 (pseudostar of m ≥ 3 planes).** (α(I), α(I^(2))) = (m − 1, m); the forms of degree m in I^(2) are
  the multiples of h_1 ⋯ h_m; I is generated by the m products of m − 1 of the equations, with an explicit
  resolution of length two; ACM.
- **Theorem 1.6 (Section 4).** F irreducible of degree d ≥ 2, X = V(F), ν the normalisation, W = forms G of
  degree d − 2 with G·ν_*O ⊂ O_X(d − 2). (a) W vanishes on every reduced curve along which F has multiplicity at
  least two. (b) dim W = h^2(X̄, O(−2)) ≥ max{p_g, d − 1 + 2π − q + p_g} (duality for the hypersurface from Serre
  duality on P^3; Euler characteristics). (c) q − p_g ≤ π, hence dim W ≥ d − 1 + π ≥ 1. The proof of W ≠ 0
  (Corollary 4.6) uses: a general plane section pulls back to a reduced connected curve C̄ with h^1(O) = π
  (Lemma 4.4); dim Pic_X̄ ≤ π because the restriction to C̄ of an abelian variety of invertible sheaves has finite
  kernel (Proposition 4.5, Mumford's covering argument with a point of prime order different from the
  characteristic); and the smoothness of Pic_X̄ if p_g = 0, while for p_g ≥ 1 one has dim W ≥ p_g directly. The
  general inequality in (c) uses in addition dim Pic ≥ h^1(O) − h^2(O) (Lemma 4.7, proved in the text).
- **Theorem 1.3 (Section 5).** Induction on d = α(I^(2)) and a case distinction on a form F of degree d in I^(2):
  not squarefree (L is coplanar, d = 2); irreducible (impossible by Theorem 1.6); squarefree and reducible (every
  factor is linear, and L is the set of all pairwise intersections of the d planes).
- **Corollary 1.4.** Extension of the ground field does not change α(I), α(I^(2)); the planes are defined over
  the ground field; Lemmas 3.1 and 3.2 give the Cohen–Macaulay property over that field.
- **Theorem 1.5 (Section 6).** The same proof with the order of vanishing along a component (Lemma 6.1).
- **Remarks and further statements.** 7.1: the deduction of the case α(I) ≤ 2 from Theorem B of
  Haghighi–Mosakhani as stated in Zbl 1473.14102. 7.2: for α(I^(2)) ≤ 4 a count of parameters replaces
  Theorem 1.6; from d = 5 on the count no longer suffices; it is not asserted that Theorem 1.6 is necessary there.
  Proposition 7.3: five singular lines of an irreducible quintic, one of them skew to the others, lie on a
  quadric. 4.10: in characteristic 0, dim W = d − 1 + 2π − q + p_g. Proposition 7.4: for a surface of bidegree
  (a, b) over a smooth curve E in P^1 × P^1, W = k[x,y]_(a−1) ⊗ k[z,w]_(b−1) and dim W = ab = d − 1 + π (stated by
  run B, proved in the note in a form checked by run 3). Remark 7.5: a line added to a configuration with gap
  one that meets none of its lines gives type (d, d + 2) and a non-ACM configuration.

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation. All computations are exact linear algebra over Q or over prime fields.
- **The projective space over F_2** (35 lines, 2^35 − 1 = 34 359 738 367 non-empty sets of lines).
  - `original/scripts/exhaust_F2_full.py`: 1 771 439 orbits under GL_4(F_2); exactly 17 orbits have gap one: 9
    orbits of coplanar sets (1 835 sets) and 8 orbits of pseudostars (2 928 sets); 0 violations of Theorem 1.3 in
    either direction.
  - Run B, `f2_full.py` with `f2_group.py`: the same enumeration with its own program; the number of orbits at
    each of the 35 levels equals the number given by Burnside's lemma (total 1 771 439); the distribution of α(I)
    and the gap-1 orbits agree level by level with the first program (`compare_claim_f2.py`).
  - Direct counts of coplanar sets (1 835) and pseudostars (2 928): `f2_count_families.py` (run B),
    `capcount_F2.py` (run A).
  - Run A, `exhaust_small_F2.py`: 19 554 065 sets representing all sets of at most ten lines; 0 violations.
  - ACM over F_2: not tested by the first program (its test needs a rational plane section). The 17 gap-1 orbits
    have zero Hartshorne–Rao module (run B, `f2_validate.py`) and are ACM by the test of run A
    (`crosscheck_claim_reps.py`). Among the 6 440 orbits with at most nine lines, 5 612 are not ACM, with least
    gap two (run B, `f2_small.py`).
  - Run 3, `independent_run_2/r2_f2_orbits.py` (its own orbit enumeration; I^(2) from the generators of the
    ideals; ACM by the length criterion with two cubic forms): all 16 672 orbits of sets of at most ten lines;
    exactly 14 with gap one (9 coplanar, 5 pseudostars with 3, 4, 5 planes; 1 835 + 2 373 sets), all of them ACM;
    14 804 not ACM, least gap two; for at most nine lines again 6 440 orbits, of which 5 612 are not ACM.
- **Characteristic zero.** `survey_Q.py` (original): 16 376 configurations over Q; gap one exactly for the 919
  coplanar ones and the 358 pseudostars; 9 647 non-ACM configurations (4 452 connected), none with gap one.
  Run B, `survey.py cube`: all 4 791 322 sets of at most eight of the 28 lines through two vertices of a cube
  (102 817 orbits): 65 gap-1 orbits, all coplanar or pseudostars; 90 214 non-ACM orbits, none with gap one.
  Run B, `survey.py pstar`: 101 510 sub-configurations of ten pseudostars; `survey2.py`: classical
  configurations. Run A, `candidates.py`: 88 configurations over Q with reductions (302 computations).
  Run 3, `independent_run_2/r2_survey_Q.py`: 541 configurations, exact over Q (random lines, pseudostars with
  3 to 6 planes and with one line removed or added, skew lines, grids on a quadric, the 27 lines of a cubic
  surface, a double six, cones): gap one exactly for the 56 coplanar ones and the 42 pseudostars; 246 not ACM
  (82 connected), gaps 2 to 5; the 42 configurations of Remark 7.5 among them (27 pseudostars and 15 coplanar
  sets with a disjoint line) have type (d, d + 2).
- **Odd characteristic.** `survey_Fp.py` (original): 71 882, 14 398, 4 800 sets over F_3, F_5, F_7. Run A,
  `survey.py`: 391 173 configurations over F_2, F_3, F_5, F_7 (90 850 with gap one, all coplanar or pseudostars
  and ACM); `survey_alpha.py`: 294 540 more, 75 503 of them with α(I) ≥ 4. Run B: 11 998 random and
  incidence-rich configurations over F_3, F_5, F_7 (in `survey2.py`).
- **Theorem 1.6.** W computed from its definition: run B on 125 parametrised rational surfaces (dim W =
  d − 1 + 2π in all cases; `lemmaA_toric.py`) and on 20 ruled surfaces of bidegree (a, b) (dim W = ab = d − 1 + π;
  `lemmaA_bigraded.py`; smoothness of the random curves not certified); run A on 98 surfaces in six
  characteristics (`conductor.py`). `lemmaA_examples.py` (original) computes dim I(L)_(d−2), which is only an
  upper bound for dim W. Run 3, `independent_run_2/r2_conductor.py`: 33 computations (18 over Q, 15 over F_p,
  p ≤ 11) on surfaces of degree 3 to 6 — cones over singular rational plane curves, ruled surfaces with a
  (d − 1)-fold line, Steiner's surface, two monomial surfaces, the surfaces x^p w = z^p y in characteristic
  p = 2, 3, 5 (inseparable p-fold line), and seven surfaces of bidegree (2, b), b = 2, 3, 4, with certified
  smooth curve E: W vanishes on the singular lines and dim W ≥ d − 1 + π in all cases (equality in 29, among
  them the seven surfaces of Proposition 7.4, where W = ⟨x, y⟩ ⊗ k[z,w]_(b−1)).
- **Theorem 1.5.** Run A, `curves.py`: 30 reduced curves over F_32003; gap one exactly for the plane curves and
  the two pseudostars.
- **Re-runs of 2026-10-10.** `run_quick.sh` (one process; every program that finishes within a few minutes, the
  enumerations up to their first levels) was run twice from copies of the package when it was assembled, and a
  third time by run 3 from the extracted archive: each time 37 comparisons with the recorded outputs, none
  failed (20 byte-identical, 12 identical up to the fields recording running times, 5 partial runs agreeing on
  the levels computed); 23 min 25 s, 29 min 23 s and 38 min 54 s (the last two while other computations were
  running on the same machine). `run_long.sh` (six processes: the two complete enumerations over F_2, the
  enumeration of run A up to ten lines, the cube survey up to eight lines) was run once, and its commands once
  before; all recorded results were reproduced (1 771 439 orbits and the same 17 gap-1 orbits in both
  enumerations; identical SHA-256 values of the 36 files of orbit representatives; byte-identical records of the
  cube survey); 21 min 35 s and 19 min 9 s. The orbit representatives chosen by `f2_full.py`, and the order of two
  entries of its summary line, depend on the scheduling of its processes; a literal comparison of that line
  reported a difference in the long run and was replaced by a comparison of the sorted list, which agrees
  (README, "Re-runs" and "Notes"). Not re-run: `survey.py` and `survey_alpha.py` of run A, which run for a fixed
  time, so that their counts depend on the machine; and the long runs were not repeated by run 3. Complete
  output: `reproducibility/RERUN_LOG.txt`.

## Independent verification runs
The results were first obtained with proofs and with the programs in `original/`. Three independent verification
runs, all AI-assisted, followed on 2026-10-10; each wrote its own programs. Runs A and B examined the first
written version of the results and did not read each other's reports: run A the statement and the proofs line by
line; run B the computations, Theorem 1.6 from a second angle, the elementary range, and the literature. Run 3
examined the final text of the note (its programs are in `reproducibility/independent_run_2/`).

| Item | Run A | Run B | Run 3 (final text) |
|---|---|---|---|
| Statement, conventions, Janssen's theorem against the sources | CONFIRMED (abstract; arXiv v1, v2) | sources read; consistent | CONFIRMED (abstract; arXiv v2; journal version read completely for the first time) |
| Lemma 2.1, Lemmas 3.1 and 3.2(a)–(c) | CONFIRMED (re-derived) | read; no gap found | CONFIRMED |
| Lemma 2.2; Lemma 3.2(d) in its final form (induction) | not examined (written later); syzygies CONFIRMED | not examined | CONFIRMED |
| Theorem 1.6(a): W vanishes on D | CONFIRMED | CONFIRMED | CONFIRMED |
| Theorem 1.6(b): duality, Euler characteristics | CONFIRMED (re-derived from Serre duality on P^3) | CONFIRMED | CONFIRMED (the derivation from Serre duality on P^3 is complete) |
| General plane section (Lemma 4.4) | CONFIRMED | CONFIRMED | CONFIRMED |
| Picard schemes and the covering argument | CONFIRMED in the first form (Pic smooth); Kleiman's statements read | CONFIRMED_WITH_FIXES (attribution to Mumford) | CONFIRMED in the final form (reduced identity component; lemma of Enriques–Severi–Zariski); every cited statement of Kleiman read in the arXiv source |
| W ≠ 0 (Corollary 4.6) | CONFIRMED | CONFIRMED | CONFIRMED |
| q − p_g ≤ π without hypothesis (Lemma 4.7, Theorem 1.6(c)) | not examined (written later) | pointed out as a possible strengthening | CONFIRMED (the proof of Lemma 4.7 re-derived step by step) |
| Theorem 1.3 (case distinction, induction) | CONFIRMED | read; no gap found | CONFIRMED |
| Corollary 1.4 | CONFIRMED (first form) | not its part | CONFIRMED (final form: Cohen–Macaulay property over the ground field) |
| Theorem 1.5 | CONFIRMED_WITH_FIXES (two implicit facts to be written) | read only | CONFIRMED (final form) |
| Proposition 7.4 | not its part | stated and proved in its report | CONFIRMED_WITH_FIXES (the proof is now in the note) |
| Stacks Project citations | not examined (inserted later) | not examined | CONFIRMED with two precisions (all thirteen tags opened) |
| Computations | own computations, no violation | CONFIRMED (independent reproduction of all numbers of the F_2 enumeration) | CONFIRMED (own programs with other methods; quick script of the package reproduced) |
| Remark on the elementary range, Proposition 7.3 | count confirmed | CONFIRMED_WITH_FIXES (wording); Proposition 7.3 proved by it | CONFIRMED |
| Literature, novelty, credit | not its part | CONFIRMED_WITH_FIXES (credit to Haghighi–Mosakhani for α(I) ≤ 2) | CONFIRMED_WITH_FIXES (status of that source in the abstract; reservation in the novelty statements) |

No run found a wrong theorem, proposition or lemma, or a gap.

**Corrections required by runs A and B**, all applied:
1. (A) Theorem 1.5: the order of a linear form along a component is exactly one; an irreducible curve contained
   in H_i ∩ H_j is that line; for curves I^(2) = ∩ P_i^(2) is the definition: Lemma 6.1 and the proof of 1.5.
2. (A) Theorem 1.6 for d = 2 forces D = ∅: Remark 4.8.
3. (A) The remark on adjoints of a cone: Remark 4.11.
4. (A) The inequality dim Pic ≥ h^1(O) − h^2(O): proved as Lemma 4.7.
5. (A) Textbook citations quoted from memory: replaced by direct arguments (Lemma 3.2(d), Lemma 4.2, Lemma 4.4)
   or by Stacks Project tags that were read; the remaining ones are listed under "Scope and priority".
6. (A, optional) Corollary 1.4 states that the lines are defined over the field.
7. (B) Credit to Haghighi–Mosakhani: abstract, Section 1.3, Remark 7.1, "Scope and priority".
8. (B) Attribution of the covering argument to Mumford, via Kleiman's Remark 5.8: Remark 4.9.
9. (B) References for the vanishing theorem and the regularity of the adjoint system, marked as taken from
   Kleiman: Remark 4.10.
10. (B) "The count no longer suffices" for d ≥ 5, without asserting necessity; Proposition 7.3: Remark 7.2.
11. (B) `lemmaA_examples.py` computes an upper bound for dim W; the ACM test of the first program is not
    available over F_2: Section 8 and the README.
12. (B, optional) The uniform inequality q − p_g ≤ π: Theorem 1.6(c) with Lemma 4.7.

**Changes made when the note was written, after runs A and B** (all examined by run 3, see the table):
- Lemma 4.7 and with it Theorem 1.6(c) without hypothesis. Theorems 1.3 and 1.5 and Corollary 1.4 do not depend
  on it: they use only Corollary 4.6.
- In the proof of Proposition 4.5 the connectedness of the divisor on the covering is derived from the lemma of
  Enriques–Severi–Zariski on the normal surface itself, and the abelian variety is the reduced identity component
  also when the Picard scheme is not smooth.
- Lemma 3.2(d): I = (Φ_1, …, Φ_m) is proved by induction on m; Lemma 2.2 was formulated.
- The last step of the proof of Theorem 1.5 refers to Theorem 1.3; in Corollary 1.4 the Cohen–Macaulay property
  is obtained over the ground field directly.
- Citations of the Stacks Project replace citations of textbooks.

**Run 3: what it examined, and the result.**
- The five changes first. The proof of Lemma 4.7 was re-derived step by step (presentation of the complete local
  ring; the kernel of B' → B; the truncated exponential sequence and its naturality; the lifting; Taylor's
  formula): correct. Proposition 4.5: the hypotheses of the lemma of Enriques–Severi–Zariski (Stacks Project,
  Tag 0FD8) hold for the invertible sheaves on the normal surface; the reduced identity component is an abelian
  variety by Kleiman's Proposition 5.3 and Theorem 5.4, the argument in the proof of his Lemma 5.1, and Tags
  047N, 047P, 0H2U: correct. Lemma 3.2(d) by induction, Lemma 2.2, the last steps of Theorem 1.5 and of
  Corollary 1.4: correct.
- All thirteen cited tags of the Stacks Project (090V, 0BXR, 0892, 031S, 01XT, 0A9W, 0FD9, 0FD8, 047N, 047P,
  0H2U, 03RP, 00NO) were opened and their statements and hypotheses compared with the use made of them: they
  fit. Every cited statement of Kleiman's article (Exercise 2.3, Theorem 2.5, Corollary 4.18.3, Lemma 5.1,
  Proposition 5.3, Theorem 5.4, Remark 5.8, Theorem 5.11, Corollaries 5.13, 5.14, Remark 5.15, Proposition 5.19)
  was read in the TeX source of arXiv:math/0504020v1: the hypotheses hold for a normal, possibly singular,
  integral projective surface and for a reduced connected projective curve over an algebraically closed field of
  any characteristic.
- Then all other proofs, line by line: correct. The proof of the statement now printed as Proposition 7.4 was
  checked in the report of run B and re-organised.
- It required eleven corrections of the text and of the package; all were made:
  1. Janssen's journal version was read: the note says so, says that the numbers of the statements agree with
     arXiv v2, and gives the pages of the journal version.
  2. Abstract (and Zenodo description): the status of the source for α(I) ≤ 2 is stated.
  3. The remark of the journal version after Question 3.1 is mentioned in Section 1.1 and proved in general
     (Remark 7.5); the sentence of its introduction on the known examples is mentioned.
  4. Remark 7.4, which quoted a result of run B without proof, is now Proposition 7.4 with its proof.
  5. Haghighi–Mosakhani: Remark 7.1 says what the review states about the field and names the configuration
     Z_{1,n}; Section 1.3 and "Scope and priority" say that the paper could not be read and that its text may
     contain more than its abstract and its review.
  6. Section 1.2: the example for isolated points is now one outside the two families (a line with two points
     not in a plane with it, of type (2,3)).
  7. Remark 4.10 and Section 1.3: what Kleiman's Remark 5.8 says about the regularity of the adjoint system is
     separated from what is recalled from memory about Mumford's paper; Remark 4.9: "general hyperplane section".
  8. Two citations made precise: Tag 0A9W gives the duality on projective space in the language of derived
     categories; that the reduction of a group scheme is a subgroup scheme is the argument in the proof of
     Kleiman's Lemma 5.1.
  9. Section 6: the multiplicity along a line agrees with the order for all powers by the proof of
     Lemma 2.1(b); in Section 1.1 the terminology "type" is cited from Janssen, who adopts it from
     Bocci–Chiantini.
  10. Section 8, the Verification paragraph and "Scope and priority" were brought to the final state.
  11. The package: this report, `reproducibility/independent_run_2/`, the README and `RERUN_LOG.txt`.
  To stay within the intended length, Figure 2, three remarks and the reporting paragraphs of Section 8 were
  shortened; no theorem, proposition, lemma or remark was removed, and the details taken out of the Verification
  paragraph (the corrections required by runs A and B, the subjects of the searches) are in this report.

## Relation to the literature, novelty and scope
- **Searches (10 October 2026).** arXiv (about 55 queries in all); OpenAlex lists of works citing Janssen's
  paper (8, and 2 for the preprint), the two records of Fashami–Haghighi–Szemberg and Haghighi–Mosakhani (0),
  Bauer–Szemberg (5); zbMATH (its lists of citing works could not be obtained); Crossref; DataCite and Zenodo;
  nine web searches in all.
  - Subjects of the arXiv queries of the first stage and of run B (35): the title of Janssen's paper; fattening
    in titles; fattening with ACM or Cohen–Macaulay; pseudostar and pseudo-star; initial degree with symbolic
    powers and lines; Bocci and Chiantini; points fattening; initial sequence; the authors Haghighi, Mosakhani,
    Fashami, Janssen; symbolic powers with lines and ACM; symbolic square with lines or curves; star
    configurations with initial degree; Fermat-like arrangements; surfaces singular along lines; recent
    listings.
  - Subjects of the arXiv queries of run 3 (18): "fattening of lines"; title fattening; pseudostar, pseudo-star;
    symbolic square with lines or curves; initial degree with symbolic and lines or curves; Janssen with
    symbolic; arithmetically Cohen–Macaulay with lines and symbolic; Waldschmidt constant with lines, linear
    subspaces or flats; symbolic with configurations of lines; adjoint surfaces (two forms); adjoint with
    conductor; conductor with normalization and hypersurface; singular along or singular locus with "degree
    d-2"; star configuration with symbolic and lines or codimension two; symbolic with lines and Cohen–Macaulay;
    initial degree or initial sequence with symbolic, newest first; the authors Mosakhani, Fashami, Janssen. Run
    3 also queried OpenAlex (11 requests), zbMATH (6), Crossref (15, of which 12 for the DOIs of the
    bibliography), DataCite (3) and Zenodo (4), and made one web search.
  - No publication was found that answers Janssen's question in general or states Theorem 1.3 for α(I) ≥ 3,
    Theorem 1.5, or Theorem 1.6 in this generality. No record for the corpus identifier exists in DataCite or
    Zenodo.
  - Of the citing works only titles and available abstracts were examined (apart from Fashami–Haghighi–Szemberg,
    which was read). In particular the paper of Haghighi and Mosakhani, which cites Janssen's paper, could not
    be read; its text may contain more than its abstract and its review say. Three repository pages from the web
    searches refused automated access and were not examined.
  - The classical literature on adjoint surfaces cannot be searched in this way; for Theorem 1.6 the negative
    result is weak.
- **What was read.** The Oberwolfach report (publisher's PDF); Janssen's paper in the journal version (publisher's
  PDF provided by the author of the note; run 3) and in arXiv v2 (run A also v1); Fashami–Haghighi–Szemberg in
  arXiv v1; the zbMATH reviews Zbl 1473.14102, 1327.14171, 1437.13014; of Geramita–Harbourne–Migliore the
  definitions and Proposition 2.9 in arXiv v1 (run 3; the Remark 2.13 cited by Janssen is not in that version);
  of Kleiman's article (arXiv:math/0504020) the statements used, Lemma 5.1 and Proposition 5.19 with proofs,
  Remarks 5.8 and 5.15 (run A also the proof of Theorem 5.4); the statements of the Stacks Project tags cited;
  of Bauer–Szemberg the abstract (run B: introduction and Section 6 of arXiv v2).
- **Not read.** Haghighi–Mosakhani (abstract and review only; closed access, not obtained); the journal versions
  of Fashami–Haghighi–Szemberg and of Geramita–Harbourne–Migliore; Bocci–Chiantini; Mumford's "Pathologies
  III"; Zariski's "Algebraic Surfaces"; Hartshorne's book (two standard theorems are quoted from memory, with
  Stacks tags as second references); Derksen–Sidman.
- **Caveats.** The deduction of the case α(I) ≤ 2 rests on a theorem known only from a review. The proofs of the
  cited statements of Kleiman's article and of the Stacks Project were not checked. Standard facts quoted from
  memory are listed in the note under "Scope and priority". A search that finds nothing is not a proof of
  novelty.
- **Scope.** The note answers the first question of the abstract completely and the third under the stated
  reading. No priority is 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/