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Brill--Noether loci in genus $\leq 12$

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Pith's one-line read For every genus up to 12, the complete containment pattern of Brill–Noether loci is now known, with explicit witnesses for every non-containment.

desk verdict A likely-correct complete classification of Brill–Noether containments through genus 12, with the main gap being unshown exhaustive enumerations that are load-bearing for the non-containments. read the letter →

arxiv 2507.04902 v2 pith:EFD6ULLK submitted 2025-07-07 math.AG

Richard Haburcak This is my paper · ORCID
classification math.AG MSC 14H5114J2814H60
keywords Brill–NoetherlocirelativepositionsofmodulisubvarietiesK3surfacesLazarsfeld–Mukaibundlesadmissibleassignmentsgonalitystratificationlinearseriesoncurves
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

The paper aims to determine, for each genus $g \leq 12$, exactly which Brill–Noether loci $M^r_{g,d}$ (moduli of curves carrying a linear series of degree $d$ and dimension $r$) contain which others, when the Brill–Noether number $\rho(g,r,d)$ is negative. It proves that in genus $3 \leq g \leq 6$ only trivial containments occur, and it gives complete figures for the non-trivial containments in genera 7 through 12. This matters because Brill–Noether loci are usually reducible, so the coarse structure of their inclusions is the first answer refined Brill–Noether theory can give before a component-by-component description exists. The same K3-surface machinery also yields explicit conjectural thresholds in large genus.

What carries the argument

The load-bearing object is the Lazarsfeld–Mukai bundle $E_{C,(A,V)}$ attached to a linear series on a smooth curve $C$ inside a K3 surface with Picard lattice $\mathbb{Z}[H] \oplus \mathbb{Z}[L]$; when $\rho < 0$ this bundle is non-simple. Its possible destabilizing subsheaves are organized into terminal filtrations, and the numerical data of such a filtration form an admissible assignment, a Gelfand–Tsetlin pattern of slopes together with the conditions that each quotient is nef and slope-positive. An admissible assignment gives a lower bound on $c_2(E_{C,g^s_e})$, hence a non-containment $M^r_{g,d} \not\subset M^s_{g,e}$; conversely, when a bundle realizing an assignment can be constructed, one obtains a containment. The secondary machinery consists of the gonality stratification function $\kappa(g,r,d)$ from the refined Brill–Noether theory of $k$-gonal curves, secant expected containments via the determinantal cycles $V^{r-s}_{d-e}(g^r_d)$, and classical plane-curve and Castelnuovo-curve constructions.

What would settle it

Re-run the finite enumeration of quotient non-negative, quotient slope-positive admissible assignments for Lazarsfeld–Mukai bundles on K3 surfaces with Picard lattice $\Lambda^r_{g,d}$ for each case in Propositions 3.15, 3.29, and 4.14; for instance, in genus 9 with $\Lambda^2_{9,7}$, an admissible assignment with destabilizing subsheaf other than $H-L$ or $L$ giving $c_2(E_{C,g^2_e}) \leq 6$ would disprove the claimed non-containment $M^2_{9,7} \not\subset M^2_{9,6}$.

Watch

Extended reading notes

Core claim

The central claim, Theorem A, is that the relative positions of Brill–Noether loci in genus $g \leq 6$ are governed solely by the trivial containments obtained by adding base points and subtracting non-base points, while for $7 \leq g \leq 12$ the complete set of non-trivial containments is displayed in Figures 2–5, 7, and 8. In particular, every containment among loci with $\rho < 0$ and $d \leq g-1$ is a sequence of the displayed covers, trivial containments, or Serre duality, and every non-containment is witnessed by a specific family of curves. The argument combines the refined Brill–Noether theory for curves of fixed gonality, classical bounds (Castelnuovo, Castelnuovo–Severi, Martens, Coppens), explicit constructions with nodal plane curves and Castelnuovo curves, and an analysis of unstable Lazarsfeld–Mukai bundles on K3 surfaces. On K3 surfaces, the existence of a $g^s_e$ forces an admissible assignment for a terminal filtration of the associated Lazarsfeld–Mukai bundle, and the absence of such assignments yields non-containments; when assignments exist, the bundles can sometimes be built as direct sums, yielding containments such as $M^2_{11,7} \subset M^3_{11,10}$.

Load-bearing premise

The exhaustive enumerations of admissible assignments in the non-containment proofs are assumed complete; if a single admissible assignment was missed, one of the claimed non-containments could fail.

Editorial extensions

If this is right

  • For $g \leq 12$, the inclusion partial order among Brill–Noether loci with $\rho < 0$ is finite and explicit: any containment question reduces to reading the appropriate figure together with trivial containments and Serre duality.
  • Families realizing the non-containments (smooth plane sextics, bielliptic covers, trigonal curves, Castelnuovo curves, nodal plane curves) certify that the omissions in the figures are genuine.
  • The realized K3-expected containments, such as $M^2_{11,7} \subset M^3_{11,10}$ and $M^2_{12,7} \subset M^3_{12,10}$, provide explicit decompositions of Lazarsfeld–Mukai bundles as sums of line bundles and smaller Lazarsfeld–Mukai bundles.
  • Conjecture A gives a concrete numerical threshold, $e \geq d - 2r + s + \frac{g-d+r+1}{2} + \frac{(s-2)(r-1)-1}{s-1}$, predicting exactly when $M^r_{g,d} \subset M^s_{g,e}$ in large genus, and Conjecture B predicts that no containments into $M^2_{g,e}$ occur outside the secant range for $r \geq 3$.

Reading between the lines

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

  • The same admissible-assignment calculus should be run in genus 13–15; even a partial run could show where the low-genus pattern breaks and how quickly the Conjecture A threshold becomes accurate.
  • The equalities in low Clifford index, e.g. $M^{e-1}_{g,2e}$ all equal for $g \geq 11, e \geq 4$ and $M^4_{g,11}=M^4_{g,10}=M^3_{g,8}$ for $g \geq 13$, suggest a general Clifford-coalescence principle: loci of a fixed small Clifford index eventually coincide, a principle that could be tested by computing $\kappa$ in higher genus.
  • The explicit constructions through linear systems on nodal plane curves indicate that many K3-expected containments can be realized without K3 surfaces; this raises the possibility of a purely classical proof of Conjecture A's containment direction.
  • If the hand-checked enumerations are replaced by a certified computer enumeration, the non-containment half of Theorem A would be independently verifiable and the method could be extended mechanically to bounded genus.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper studies the relative positions (containments and non-containments) of Brill–Noether loci M^r_{g,d} with ρ(g,r,d)<0 in low genus. The main result, Theorem A, asserts that for g≤6 all containments are trivial, while for 7≤g≤12 the complete poset of non-trivial containments is described by the Hasse-like diagrams in Figures 2–5, 7–8. The proof combines classical results (Castelnuovo–Severi, Max Noether, Mori complete intersections), the refined Brill–Noether theory for fixed gonality, chains of elliptic curves, explicit nodal-plane-curve constructions, and an analysis of admissible assignments of terminal filtrations of Lazarsfeld–Mukai bundles on K3 surfaces. The paper also proposes Conjectures A and B on expected containments in large genus and raises questions on irreducibility of Brill–Noether loci.

Significance. If Theorem A is correct, the paper gives the first complete classification of non-trivial containments among Brill–Noether loci in genera 7 through 12, a finite list that should be a benchmark for refined Brill–Noether theory. A notable strength is that many containments are proved by explicit geometric constructions rather than by a black-box computation, and non-containments are often witnessed by concrete curves (smooth plane curves, bielliptic covers, Castelnuovo curves, K3 sections). The K3-admissible-assignment formalism is a useful heuristic and is clearly connected to the Donagi–Morrison circle of ideas. However, the classification is not independently verifiable as written: several load-bearing non-containment proofs rest on assertions of exhaustive finite enumerations for which no tables, code, or detailed case analysis are supplied. The result is plausible and the methods are appropriate, but the missing evidence for these enumerations is a correctness risk that must be addressed before the classification can be accepted.

major comments (4)
  1. [§3.4, Proposition 3.15] The non-containment proofs for genus 9 depend on the assertion that certain lists of admissible assignments are complete. For instance, Proposition 3.15(i) states that for Λ^2_{9,6} the only possible admissible assignments of a g^1_e are those with c1(E1) ∈ {2H−4L, H−L, 2L, −H+4L}, and Proposition 3.15(ii) states that for Λ^2_{9,7} all admissible assignments of a g^2_e are of type 1⊂3 and have destabilizing sub-line bundle H−L or L. The text gives no derivation, no table of the checked cases, and no code. Since by §2.2 every terminal filtration yields an admissible assignment, an overlooked admissible assignment could directly produce a g^s_e below the asserted c2 threshold and invalidate the claimed non-containment. This is load-bearing for Theorem A in genus 9.
  2. [§3.5, Proposition 3.29] The same issue occurs in the genus 10 classification. Proposition 3.29 states that for Λ^2_{10,7} the only admissible assignments for g^1_k and g^2_e are as listed, and for Λ^3_{10,9} that the destabilizing classes for g^2_e are exactly {2H−3L, H−L, L, −H+3L}, that the only types for g^3_e are 1⊂4, 2⊂4, and 1⊂2⊂4, and that there are no admissible assignments of type 1⊂2⊂4. These exhaustive statements are asserted with phrases such as “checking all possibilities” and “one can check,” without displaying the enumeration or the inequalities from Equations (4) and (5) that eliminate the remaining cases. An omitted admissible assignment in any of these lists would falsify one of the non-containments (i)–(iv) and hence the genus 10 part of Theorem A.
  3. [§3.5, Proposition 3.27] The non-containment M^2_{10,7} ⊄ M^3_{10,9} is proved by degenerating to a chain of ten elliptic curves and asserting: “There is no admissible filling with the entries 1,…,10 of a 4×4 square with the same torsion conditions.” This is a finite combinatorial check, but no table of attempted fillings, no explicit obstruction, and no reproducible algorithm is provided. The statement is essential to the proof of the proposition, and the reader cannot verify it from the text. Please provide the complete case analysis or a short script with documented output.
  4. [§2.3–§2.4, Lemma 2.10 and the enumeration method] The paper would benefit from a precise description of how the “check all possibilities” enumerations are performed. Lemma 2.10 gives finite ranges for the coefficients (x,y) of c1(E/E_i), so an exhaustive search is in principle possible, but the manuscript does not state the algorithm: which terminal filtration types are considered in each genus, how the Gelfand–Tsetlin inequalities together with Equations (4) and (5) are used to prune the search, and how the 'quotient non-negative' and 'quotient slope-positive' conditions of Remark 2.6 are applied. Supplying this information, together with the resulting tables for each Λ^r_{g,d} used in Propositions 3.4(iii), 3.8, 3.15, and 3.29, would make the central classification checkable and would remove the main correctness risk.
minor comments (5)
  1. [§3.2, Proposition 3.4(iii)] The statement appears as “M^2_{1,6} ⊈ M^1_{7,3}”; the subscript “1,6” should presumably be “7,6”. Please correct this typo.
  2. [§4.2, Proposition 4.16] The proof says that a basepoint on the g^2_7 sends C into M^2_{12,6} ⊂ M^3_{12,9} “by Proposition 4.14,” but Proposition 4.14 does not state M^2_{12,6} ⊂ M^3_{12,9}. The intended argument is likely via Remark 4.13 together with κ(12,3,9)=3; please fix the reference.
  3. [§4.2, Proposition 4.16] The phrase “4 dimensional linear system of conics through 2 of the nodes” is ambiguous: the vector space of conics through two fixed points has dimension 4, but the resulting projective linear system has dimension 3, which is what produces the g^3_{10}. Please clarify whether “dimension” refers to the projective system or the vector-space dimension.
  4. [Figures 2–5, 7–8] The diagrams are difficult to read in the current rendering: many labels and arrows overlap, and in places the arrows are difficult to distinguish from the 'trivial containment' arrows that are intentionally omitted. A larger vector graphic or a separate list of the cover relations would improve usability.
  5. [Throughout] There are several typographical errors, e.g. “satsify” in Section 1.1 and “Corolalry” in the proof of Proposition 4.14; also the notation for Brill–Noether loci sometimes omits spaces, as in “M2 1,6.” A careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: K3-expected containments are used only as heuristics, and every asserted containment/non-containment in genus ≤ 12 is justified by explicit constructions or external, parameter-free theorems.

full rationale

The derivation chain in this paper is not circular. The K3-expected containments, defined in Definition 2.12, are explicitly labeled as a heuristic: Section 2.4 states that they are 'a useful heuristic in determining the relative positions of Brill–Noether loci,' and the actual containments are proven independently. For example, the containment M^2_{11,7} ⊂ M^3_{11,10} is first seen as K3-expected in Remark 4.10, but Proposition 4.11 gives an explicit construction using plane septic curves with four nodes; similarly, M^2_{12,7} ⊂ M^3_{12,10} is proven in Proposition 4.16 using conics through nodes of a plane curve, not by the admissible-assignment heuristic of Example 4.15. Non-containments are derived from the necessity of admissible assignments for a terminal filtration of a Lazarsfeld–Mukai bundle: if a g^s_e existed, its LM bundle would have a terminal filtration and hence an admissible assignment producing a c2-bound; the paper then rules out the relevant e by showing all admissible assignments give larger bounds. This does not assume the conclusion. The uses of the author's own prior work are not load-bearing in a circular way. The κ(g,r,d) formula from [7] is used for many non-containments via Proposition 1.10, but [7] derives it from the external refined Brill–Noether theory of Jensen–Ranganathan (Theorem 1.9), and the formula is parameter-free with assumptions not containing Theorem A. The irreducibility and chain-of-elliptic-curves degeneration of M^2_{10,7} cited from [30] in Proposition 3.27 is an independent theorem about the structure of that locus, not a disguised form of the non-containment M^2_{10,7} ⊈ M^3_{10,9} being proved. The cited results [5], [6], and [47] supply definitions, flexible-decomposition rigidity, or generalized LM-bundle facts, but the low-genus classifications are established in this paper by direct constructions and by admissible-assignment computations performed here. The reader's flagged risk—that some enumerations of admissible assignments in Propositions 3.15, 3.29, and elsewhere are asserted without displayed casework—is a completeness/correctness concern about the enumeration, not circularity: the method does not assume the non-containment it proves, and a missed assignment would falsify a claim rather than tautologically confirm it. Accordingly, the paper's central claim is self-contained against external benchmarks and no step reduces by construction to its own inputs.

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

The classification rests on standard Brill–Noether theory, on the refined theory for fixed gonality, on the existence of K3 surfaces with prescribed Picard lattices, and on the paper's own method of admissible assignments. The main unverified input is the exhaustive finite enumeration of admissible assignments; no code or tables are given. There are no fitted parameters or invented physical entities. The conjectural part is clearly separated from the theorems.

assumptions (4)
  • standard math Brill–Noether–Petri theorem
    Gives the threshold ρ ≥ 0 for existence of g^r_d on general curves; fundamental background used throughout.
  • domain assumption Existence of K3 surfaces with Pic(Λ^r_{g,d}) when ∆ < 0
    Used to construct smooth curves in M^r_{g,d} on K3 surfaces; relies on surjectivity of the period map and Noether–Lefschetz theory as cited in §1.5.
  • domain assumption Theorem 1.9 (Pflueger, Jensen–Ranganathan)
    Computes dimensions of Brill–Noether loci for general k-gonal curves; used to derive κ(g,r,d) and many non-containments.
  • ad hoc to paper Completeness of admissible-assignment enumerations in §§3.4, 3.5, 4.3
    Propositions 3.15, 3.29, and 4.14 assert that a finite list of admissible assignments is exhaustive without showing the enumeration; this is load-bearing for the exactness of the non-containment results.

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Pith. "Pith review of Brill--Noether loci in genus $\leq 12$." pith.science (2026). https://pith.science/paper/EFD6ULLK

@misc{pith2026250704902,
  author       = {Pith},
  title        = {Pith review of: Brill--Noether loci in genus $\leq 12$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EFD6ULLK}},
  note         = {Machine review of arXiv:2507.04902}
}
abstract

A refined Brill--Noether theory seeks to determine which linear series are admitted by a ``general'' curve in a particular Brill--Noether locus. However, as Brill--Noether loci are not irreducible in general, a coarse answer is given by the relative positions of Brill--Noether loci. Via an analysis of unstable Lazarsfeld--Mukai bundles on K3 surfaces, we distinguish Brill--Noether loci and provide expectations for the relative positions of Brill--Noether loci in general. Together with classical results, the refined Brill--Noether theory for curves of fixed gonality and on Hirzebruch surfaces, and explicit constructions, we identify the relative positions of all Brill--Noether loci in genus $g\leq 12$.

Figures

Figures reproduced from arXiv: 2507.04902 by the authors.

Figure 1
Figure 1. Lines ℓ, ℓ+, and ℓ− for (g, r, d) = (14, 3, 13). This defines a triangle bounded by the lines ℓ+, ℓ− and ℓ for allowed (x, y), with vertices (0, 0), Q+ := ℓ ∩ ℓ− = [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Non-trivial containments of Brill–Noether loci in genus 7. We begin by noting containments, and then show all the required non-containments. Proposition 3.3. We have the following containments (i) M2 7,5 = M1 7,2 , (ii) M2 7,6 ⊂ M1 7,4 , and (iii) M1 7,3 ⊂ M2 7,6 . Proof. Statement (i) follows from Lemma 1.3, (ii) follows from Lemma 1.5, and (iii) follows directly from Theorem 1.9. It remains to prove the non-contai… view at source ↗
Figure 3
Figure 3. Non-trivial containments of Brill–Noether loci in genus 8. We begin by noting the containments resulting from Lemma 1.3 and the refined Brill–Noether theory for curves of fixed gonality, see Theorem 1.9. Proposition 3.6. We have the containments (i) M1 8,2 = M2 8,5 = M3 8,7 , (ii) M1 8,3 ⊆ M2 8,6 , and (iii) M1 8,4 ⊆ M2 8,7 . It remains to show the non-containments. We begin with the noncontainments which are a dire… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Non-trivial containments of Brill–Noether loci in genus 9. We begin by noting the containments resulting from Lemma 1.3 and the refined Brill–Noether theory for curves of fixed gonality, see Theorem 1.9. Proposition 3.10. We have the following containments (i) M1 9,2 =…
Figure 5
Figure 5. Figure 5: Non-trivial containments of Brill–Noether loci in genus 10. Proof. Let C ∈ M3 10,8 . We see from Theorem 1.1 that the g 3 8 cannot be both basepoint free and birationally very ample. If it is not basepoint free, then C ∈ M3 10,7 = M1 10,2 ⊂ M1 10,4 , as claimed. We may…
Figure 6
Figure 6. Figure 6: Admissible filling giving a g 2 7 . In particular, the curves E2, and E9 have a 2-torsion condition on the nodes, and E3, E5, and E7 have a 4-torsion condition on the nodes. There is no admissible filling with the entries 1, . . . , 10 of a 4 × 4 square with the same t…
Figure 7
Figure 7. Figure 7: Non-trivial containments of Brill–Noether loci in genus 11 [PITH_FULL_IMAGE:figures/full_fig_p026_7.png]
Figure 7
Figure 7. Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p027_7.png]
Figure 8
Figure 8. Figure 8: Non-trivial containments of Brill–Noether loci in genus 12. Remark 4.13. It is straightforward to show that the curves in M2 12,6 are either trigonal or hyper￾elliptic. We give a few results using Castelnuovo curves, following [11, Corollary 5.2, Example 5.3], which do…

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