A remark on the conjecture of Donagi-Morrison

Kenta Watanabe

REVIEW 2 major objections 4 minor 14 references

A remark on the conjecture of Donagi-Morrison

T0 review · 2 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For base point free primitive line bundles on K3 curves satisfying $g>2d-3+(r-1)^2$, a Donagi-Morrison lift always exists.

desk verdict First Donagi-Morrison lift result for non-ample big line bundles, with a proof that hinges on an unproved imported lemma. read the letter →

arxiv 2412.02256 v1 pith:Z4GFTW5E submitted 2024-12-03 math.AG

Kenta Watanabe This is my paper · ORCID

classification math.AG MSC 14J2814J6014H60

keywords K3surfacesDonagi-MorrisonconjectureLazarsfeld-MukaibundlesCliffordindexBrill-Noethertheorylineoncurvesadaptedspecial

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves a quantitative existence result for a conjecture about curves on K3 surfaces: a special line bundle on a curve should be the restriction of a line bundle on the surface. The theorem says that if the line bundle is base point free, primitive, of degree $d\ge 4$ and rank $r\ge \sqrt{d/2}$, and the curve's genus satisfies $g>2d-3+(r-1)^2$, then such a surface line bundle $N$ exists, is adapted to $|C|$, contains the original linear system, and has no larger Clifford index. The significance is that $N$ is not required to be ample, so the result covers big linear systems on arbitrary K3 surfaces. The result therefore provides a numerical threshold for Donagi-Morrison lifts and a step toward the full conjecture.

What carries the argument

The central object is the Lazarsfeld-Mukai bundle $E_{C,A}$, a rank $r+1$ vector bundle on $X$ with $c_1(E_{C,A})=\mathcal{O}_X(C)$ and $c_2(E_{C,A})=d$, obtained from the evaluation map $H^0(A)\otimes\mathcal{O}_X\to A$. The proof's load-bearing step is a decomposition $0\to M\to E_{C,A}\to F\to0$ with $M$ a saturated line subbundle, $h^0(M)\ge2$, and $F$ a torsion free globally generated rank-$r$ sheaf; the lift is $N=c_1(F^{\vee\vee})$. Adaptedness to $|L|$ is established through exact sequences and vanishing, and the Clifford comparison follows from $d=M.N+\ell(W)+c_2(F^{\vee\vee})$, the bound $\operatorname{Cliff}(F^{\vee\vee})\ge0$, and $h^0(N\otimes\mathcal{O}_C)\ge h^0(N)$.

What would settle it

Compute $E_{C,A}$ for the double cover of a smooth plane quartic described in Example 4.1 (degree 8, rank 2, genus satisfying the inequality) and check whether it has a saturated line subbundle $M$ with $h^0(M)\ge2$ and a torsion free globally generated quotient $F$ of rank 2. If any such example lacks the decomposition, Lemma 4.4 and hence the proof of Theorem 1.2 fail; if it holds in all small cases, the key step is supported.

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Extended reading notes

Core claim

The paper establishes Theorem 1.2: for a K3 surface $X$, a base point free big line bundle $L$, a smooth curve $C\in|L|$ of genus $g$, and a base point free primitive line bundle $A=g^r_d$ with $d\ge 4$ and $r\ge \sqrt{d/2}$, the inequality $g>2d-3+(r-1)^2$ implies that $|A|$ is contained in $|N\otimes\mathcal{O}_C|$ for some line bundle $N$ on $X$ adapted to $|L|$, with $\operatorname{Cliff}(N\otimes\mathcal{O}_C)\le\operatorname{Cliff}(A)$. This is a Donagi-Morrison lift in the case where $L$ is big but not necessarily ample, and Corollary 4.1 extends the conclusion to line bundles computing the special Clifford index of $C$.

Load-bearing premise

The argument stands on Lemma 4.4, imported from an earlier paper without proof, which says that the bundle $E_{C,A}$ built from $C$ and $A$ splits into a line subbundle with at least two sections and a rank-$r$ quotient generated by its global sections; if that lemma needs extra hypotheses, the main theorem has no support.

Editorial extensions

If this is right

For every primitive base point free $g^r_d$ with $d\ge4$, $r\ge\sqrt{d/2}$, and $g>2d-3+(r-1)^2$, there is a line bundle $N$ on $X$ adapted to $|L|$ with $|A|\subset|N\otimes\mathcal{O}_C|$ and $\operatorname{Cliff}(N\otimes\mathcal{O}_C)\le\operatorname{Cliff}(A)$.
This is the first such lift theorem that does not require $L$ to be ample, so it applies to big linear systems on K3 surfaces of arbitrary Picard number.
Corollary 4.1 gives the same conclusion for line bundles computing the special Clifford index of $C$, since those are primitive and have negative Brill-Noether number.
In the $N^2=0$ case, the proof forces $A\cong\mathcal{O}_C(r\Delta)$ for an elliptic curve $\Delta$, so the elliptic case is completely settled inside the theorem.

Reading between the lines

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

The threshold $g>2d-3+(r-1)^2$ is an artifact of the proof's estimates and Hodge-index contradictions; it is likely not sharp, so the true range of Donagi-Morrison lifts may be substantially wider.
Because Lemma 4.4 is imported without proof, a natural test is to prove the same saturation decomposition for all primitive line bundles with negative Brill-Noether number; if true, the theorem would extend to the entire negative Brill-Noether range.
The adaptedness argument suggests a general mechanism: whenever $h^1(M^\vee)$ is constant as $C'$ varies in $|L|$, the surface line bundle $N=c_1(F^{\vee\vee})$ is adapted; this could yield a standalone criterion for adaptedness.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

I'll keep this short. The paper proves a genuine extension of the Donagi-Morrison conjecture: when L is base point free and big (not assumed ample), and g > 2d - 3 + (r-1)^2, any primitive g^r_d with d≥4 and r≥sqrt(d/2) has a Donagi-Morrison lift. Previous results either assumed L ample or handled only nets. The numerical condition is a hypothesis, not an output, so there is no circularity in that sense.

The proof strategy is sound. The Lazarsfeld-Mukai bundle is decomposed, Chern classes are computed, and the two cases N^2=0 and N^2>0 are handled separately. The Clifford inequality at the end follows from c2(F^∨∨)≥2(r-1). The case analysis for the elliptic-curve situation (Lemmas 4.1–4.3) is worked out in detail and looks coherent.

The soft spot is Lemma 4.4. It asserts that E_{C,A} has a saturated line subbundle M with h0(M)≥2 whose quotient F is rank r, torsion free, and globally generated. Everything downstream depends on this: the definition N=c1(F^∨∨), the adaptedness argument, and the Clifford bound. The proof says only 'by the same way of the proof of Lemma 4.1 as in [14]', with no statement of that lemma or its hypotheses. If [14] required L ample, the theorem as stated is unsupported. This is a real gap, and the paper signals it by omitting the proof. I can't tell from the text whether the lemma carries over to the non-ample setting; the author likely knows, but a referee needs to see it.

Minor point: Example 4.1 is imported from [14] and not new, which is fine.

Net: the central idea is good and the main theorem is probably true, but the paper is not self-contained at its critical step. That is a fixable problem—state Lemma 4.4 with full hypotheses and give a proof or a precise citation—rather than a fatal flaw. I'd send this to a serious referee, asking specifically whether Lemma 4.4 holds under the hypotheses of Theorem 1.2. The rest of the argument deserves scrutiny but passes a first read.

Referee Report

2 major / 4 minor

Summary. The paper proves a quantitative sufficient condition for the existence of Donagi-Morrison lifts of line bundles on curves lying on K3 surfaces, without assuming the polarization L is ample. The main theorem (Theorem 1.2) states that if X is a K3 surface, L is base point free and big, C is a smooth curve in |L| of genus g, and A is a base point free primitive g^r_d on C with d ≥ 4, r ≥ sqrt(d/2), and g > 2d - 3 + (r-1)^2, then there is a line bundle N on X, adapted to |L|, such that |A| ⊂ |N⊗O_C| and Cliff(N⊗O_C) ≤ Cliff(A). The proof uses the Lazarsfeld-Mukai bundle E_{C,A}, a decomposition into a line subbundle and a torsion-free quotient (Lemma 4.4), Chern class identities, and case analysis for the N^2 = 0 elliptic-fiber case. A corollary is drawn for line bundles computing the special Clifford index.

Significance. If correct, the result is a genuine advance: it extends the Donagi-Morrison lift theorem to polarizations that are only base point free and big, under an explicit numerical inequality, and it provides a new sufficient condition that is checkable. The proof strategy is elegant, particularly the use of the Chern class identity c2(E) = M.N + ℓ(W) + c2(F∨∨) together with nonnegativity of Cliff(F∨∨) to obtain the Clifford inequality. The paper is short and mostly self-contained, with clearly stated hypotheses and a concrete corollary for the special Clifford index. However, the central structural lemma (Lemma 4.4) is imported from the author's previous work [14] without proof or statement, and at least one numerical argument in Lemma 4.1 is under-justified; these gaps must be addressed before the theorem can be considered fully established.

major comments (2)

[Section 4, Lemma 4.4] Lemma 4.4 is the central structural input: it asserts that E_{C,A} has a saturated line subbundle M with h0(M) ≥ 2 whose quotient F is torsion free of rank r and globally generated. The proof of Theorem 1.2 depends on this decomposition at every stage: equation (3), the definition N = c1(F∨∨), the base-point-freeness and adaptedness of N, and the final bound c2(F∨∨) ≥ 2(r−1) all rely on it. However, the lemma is neither stated nor proved in the present paper; the text says only that it follows 'by the same way of the proof of Lemma 4.1 as in [14]'. The statement and hypotheses of [14, Lemma 4.1] are not reproduced, and there is no verification that its assumptions are satisfied in the setting of Theorem 1.2, where L is only base point free and big (not necessarily ample). Please provide a complete proof of Lemma 4.4, or state [14, Lemma 4.1] explicitly and check its hypotheses in detail for the present non-ample situation.
[Section 4, Lemma 4.1] In the proof of Lemma 4.1, after the displayed lower bounds for (C−r∆−kΓ1)^2 and C·(C−r∆−kΓ1), the conclusion h0(L⊗O_X(−r∆−kΓ1)) > 2 is stated to follow from the Riemann-Roch theorem because C is nef. This step is not immediate: the divisor H := C − r∆ − kΓ1 is not shown to be nef, and in fact H·Γ1 ≤ −2 can occur (for k = 0 when C·Γ1 ≤ r−2), so h1(H) need not vanish. The Riemann-Roch formula alone gives h0(H) − h1(H) + h0(H∨) = 2 + H^2/2, which does not yield h0(H) > 2 unless h1(H) is controlled. Since Lemma 4.1 feeds directly into Proposition 4.2 and hence into the N^2 = 0 case of Theorem 1.2, this gap is load-bearing. Please supply a missing argument (for example, a suitable vanishing theorem or a subtraction of (−2)-curves) to justify the claimed lower bound on h0.

minor comments (4)

[Introduction, p.3] There is a typo: 's ee [1], [2]' should read 'see [1], [2]'.
[Section 4, proof of Lemma 4.2] The deduction 'by Proposition 2.4, we have D^2 = 0' is not explained; Proposition 2.4 states 2-connectedness of members of |L| and does not directly imply D^2 = 0. Please add a short justification (e.g., using the hyperbolic lattice structure of NS(X) and the fact that D is a movable, hence nef, divisor orthogonal to an isotropic class).
[Section 4, proof of Theorem 1.2] Equation (5) is quoted from [7, (0.3)]. For self-containedness, please derive it from the exact sequences (3) and (4) together with the Riemann-Roch theorem and the additivity of Chern classes.
[Remark 1.1] The wording is confusing: condition (i) of Conjecture 1.1 is the containment condition, whereas the adaptedness conditions are defined separately in the introduction. Please rephrase to clarify that when h1(L⊗N∨) = 0, the restriction map H0(N) → H0(N⊗O_C) is surjective, so that |N⊗O_C| is the restriction of |N|.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the proof of Theorem 1.2 does not assume its own conclusion, and the only self-citation (Lemma 4.4 from [14]) is a proof gap rather than a circular step.

full rationale

The proof of Theorem 1.2 starts from the stated hypotheses and derives the existence of N from the structural exact sequence provided by Lemma 4.4, not from the existence of a Donagi-Morrison lift. The numerical inequality g > 2d - 3 + (r - 1)^2 is used as an input in Lemma 4.1 and Proposition 4.2, not fitted to force the conclusion. Equation (5) is cited from Green-Lazarsfeld [7], and the final Clifford inequality follows from c2(F^∨∨) >= 2(r - 1) via Proposition 3.3; neither step assumes the target lift. No equation in the paper reduces to the theorem's conclusion. The only circularity-burden item is Lemma 4.4, which is loaded from [14] without proof and is essential to the construction, but it is a proof gap rather than a reduction by construction or a self-citation of the theorem itself.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof builds on standard K3 surface geometry and on the published theory of Lazarsfeld-Mukai bundles. The only non-routine input is Lemma 4.4, taken from the same authors' earlier paper [14], whose statement and proof are not included. No free parameters or invented entities are introduced.

assumptions (6)

standard math Riemann-Roch, Serre duality, and the Hodge index theorem on K3 surfaces.
Used throughout Section 4 to compute h0, h1, and to obtain contradictions from positivity of intersection numbers; these are standard background in the field.
standard math Saint-Donat classification of base point free linear systems (Propositions 2.2, 2.3, 2.4 in [13]).
Used to identify O_X(r∆), to ensure base point freeness of N, and to get 2-connectivity of members of |L|.
standard math Knutsen-Lopez criterion for h1(L) on K3 surfaces (Proposition 2.1 in [8]).
Used to produce (-2)-divisors Γ with small intersection with C-r∆, creating the cases handled by Lemma 4.2 and Proposition 4.2.
domain assumption Properties of Lazarsfeld-Mukai and generalized LM bundles (Propositions 3.1, 3.2, 3.3 in [11]).
Supplies global generation of E_{C,A}, the rank-one decomposition when c1^2=0, and nonnegativity of Cliff(E) for c1^2>0.
domain assumption Chern class formula c2(E_{C,A}) = M.N + ℓ(W) + c2(F∨∨) from [7, (0.3)].
This equality is the bridge between the degree d of A and the Clifford index of N⊗O_C; it is quoted without derivation.
domain assumption Lemma 4.4, imported from [14, Lemma 4.1] without proof.
The paper states that a saturated line subbundle M and a globally generated rank-r torsion free quotient F exist 'by the same way' as Lemma 4.1 in [14]; this is the main unverified assumption of the proof.

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Pith. "Pith review of A remark on the conjecture of Donagi-Morrison." pith.science (2026). https://pith.science/paper/Z4GFTW5E

@misc{pith2026241202256,
  author       = {Pith},
  title        = {Pith review of: A remark on the conjecture of Donagi-Morrison},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z4GFTW5E}},
  note         = {Machine review of arXiv:2412.02256}
}

abstract

Let $X$ be a K3 surface, let $C$ be a smooth curve of genus $g$ on $X$, and let $A$ be a base point free and primitive line bundle $g_d^r$ on $C$ with $d\geq4$ and $r\geq\sqrt{\frac{d}{2}}$. In this paper, we prove that if $g>2d-3+(r-1)^2$, then there exists a line bundle $N$ on $X$ which is adapted to $|C|$ such that $|A|$ is contained in the linear system $|N\otimes\mathcal{O}_C|$, and ${\rm{Cliff}}(N\otimes\mathcal{O}_C)\leq {\rm{Cliff}}(A)$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 13 canonical work pages

[14]

Watanabe, K., Komeda, J., Lifts of line bundles on curves on K3 surfaces, Abh. Math. Semin. Univ. Hambg. 94 (2024), 95-106. 15

work page 2024
[1]

Auel, A., Brill–Noether special cubic fourfolds of discriminant 14 , London Mathematical Society Lecture Note Series, Cambridge University Press, vol.1 (2022), p. 29–53

work page 2022
[2]

Auel, A., Haburcak, R., Maximal Brill-Noether loci via K3 sur- faces, arXiv:2206.04610 [math.AG] (2023). 14

arXiv 2023
[3]

Donagi, R., Morrison, D, Linear systems on K3 sections , J. Diﬀ. Geom. 29 (1989), 49-64

work page 1989
[4]

Haburcak, R., Curves on Brill-Noether special K3 surfaces , Math. Nach. (2024), https://doi.org/10.1002/mana.202300403

work page doi:10.1002/mana.202300403 2024
[5]

Algebra 611 (2022), 630–650

Hoﬀ, M., Staglian´ o, G., Explicit constructions of K3 surfaces and unirational Noether–Lefschetz divisors , J. Algebra 611 (2022), 630–650

work page 2022
[6]

Huybrechts, D., Lectures on K3 Surfaces , Cambridge University Press, Cambridge (2015)

work page 2015
[7]

Green, M., Lazarsfeld, R., Special divisors on curves on K3 sur- faces, Inv. Math. 89 (1987), 357-370

work page 1987

Show all 14 references

[8]

Knutsen, A.L., Lopez, A.F., A sharp vanishing theorem for line bundles on K3 or Enriques surfaces , Proc. Am. Math. Soc. 135(11), (2007), 3495–3498

2007
[9]

Lazarsfeld, R., Brill-Noether-Petri without degenerations, J. Diﬀer. Geom. 23 (1986), no.3, 299-307

1986
[10]

London Math

Lelli-Chiesa, M., Stability of rank-3 Lazarsfeld–Mukai bundles on K3 surfaces , Proc. London Math. Soc. 107(2), (2013), 451–479

2013
[11]

Lelli-Chiesa, M., Generalized Lazarsfeld-Mukai bundles and a con- jecture of Donagi and Morrison , Adv. Math. 268 (2015), 529-563

2015
[12]

H., Pencils and nets on curves arising from rank 1 sheaves on K3 surfaces , Math

Rasmussen, N. H., Pencils and nets on curves arising from rank 1 sheaves on K3 surfaces , Math. Scand. 122(2), (2018), 197–212

2018
[13]

Saint-Donat, B., Projective models of K3 surfaces , Am. J. Math. 96 (1974), 602-639

1974

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [14]

Watanabe, K., Komeda, J., Lifts of line bundles on curves on K3 surfaces, Abh. Math. Semin. Univ. Hambg. 94 (2024), 95-106. 15

work page 2024

[2] [1]

Auel, A., Brill–Noether special cubic fourfolds of discriminant 14 , London Mathematical Society Lecture Note Series, Cambridge University Press, vol.1 (2022), p. 29–53

work page 2022

[3] [2]

Auel, A., Haburcak, R., Maximal Brill-Noether loci via K3 sur- faces, arXiv:2206.04610 [math.AG] (2023). 14

arXiv 2023

[4] [3]

Donagi, R., Morrison, D, Linear systems on K3 sections , J. Diﬀ. Geom. 29 (1989), 49-64

work page 1989

[5] [4]

Haburcak, R., Curves on Brill-Noether special K3 surfaces , Math. Nach. (2024), https://doi.org/10.1002/mana.202300403

work page doi:10.1002/mana.202300403 2024

[6] [5]

Algebra 611 (2022), 630–650

Hoﬀ, M., Staglian´ o, G., Explicit constructions of K3 surfaces and unirational Noether–Lefschetz divisors , J. Algebra 611 (2022), 630–650

work page 2022

[7] [6]

Huybrechts, D., Lectures on K3 Surfaces , Cambridge University Press, Cambridge (2015)

work page 2015

[8] [7]

Green, M., Lazarsfeld, R., Special divisors on curves on K3 sur- faces, Inv. Math. 89 (1987), 357-370

work page 1987