Conic linear series and pencils of plane quartics

Gian Pietro Pirola; Lidia Stoppino; Riccardo Moschetti

REVIEW 3 major objections 3 minor 23 references

This paper constructs a non-isotrivial pencil of plane quartic curves with exactly one base point, all members irreducible and the general member smooth.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

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2026-08-03 22:27 UTC pith:36FMS2HC

load-bearing objection The conic-linear-series framework is substantial, but the advertised pencil of quartics rests on a lemma whose proof is invalid, so the main theorem is unsupported as written. the 3 major comments →

arxiv 2511.10327 v2 pith:36FMS2HC submitted 2025-11-13 math.AG

Conic linear series and pencils of plane quartics

Riccardo Moschetti , Gian Pietro Pirola , Lidia Stoppino This is my paper

classification math.AG MSC 14H5014N0514C2014H52

keywords conic linear systemsplane quarticspencils with one base pointnon-isotrivial pencilselliptic normal curveslinear limits of conescone map

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper develops a theory of conic linear series: linear systems on a smooth complex curve C ⊂ P^3 cut out by cones of fixed degree with vertex outside C, together with their limits as the vertex moves onto the curve. Using this theory, it proves that a non-isotrivial pencil of plane quartics can have all its members irreducible, a smooth general member, and a base locus reduced to a single point. The road to this result is the claim that for an elliptic normal quartic, every effective divisor of degree 16 is cut by a one-dimensional family of quartic cones; projecting one such cone together with the curve from the cone's vertex yields the desired pencil. This extends a known construction for plane cubics to degree four.

Core claim

On a smooth degree-4 genus-1 curve C ⊂ P^3, the conic map Φ from the projectivized cone bundle P_U(E) to the complete linear system P(S_4) of degree-16 divisors is dominant with one-dimensional kernel, and each effective degree-16 divisor D is claimed to be cut out by a one-dimensional family of cones whose vertices trace a curve in the open set U. Applying this to a point q with 16q ∼ 16O but 8q ≁ 8O, the paper takes one such cone, projects from its vertex along with C, and obtains a pencil of plane quartics whose base locus is exactly the projection of q; irreducibility follows from the 8q condition, smoothness of the general member from an analysis of the inverse image of 16q, and non-iso

What carries the argument

The central mechanism is the cone map ρ_d : U → G(n_d, S_d), which sends a point p to the linear system cut out on C by cones of degree d with vertex p. The paper proves that this map extends to the blow-up of P^3 along C via 'linear limits of cones': when the vertex specializes to p ∈ C along a line ℓ not tangent to C, the limiting degree-d cone is the cone from p over C together with the plane spanned by ℓ and the tangent line at p. This extension yields a morphism Φ : P_U(E) → P(S_d) whose differential has generically maximal rank, with kernel dimension 1 exactly for (d,g)=(4,1); the resulting dominance of Φ for d ≤ 4 is what forces the existence of the one-parameter families of cones cut

Load-bearing premise

The construction of the pencil rests on the claim that if the inverse image of a fixed divisor under the extended cone map projects to a positive-dimensional set for the general divisor, it must do so for every divisor — a semicontinuity assertion (Lemma 7.3) that fails for simple maps between surfaces.

What would settle it

A direct test of the load-bearing step: check whether for a specific elliptic quartic and a 16-torsion point q the inverse image (Φ'')^{-1}(16q) actually meets the open cone locus P_U(E). The proof forces this by Lemma 7.3, but that lemma has a concrete counterexample: X=P^1×P^1 with φ the first projection and μ([s:t],[u:v])=[su:tv] maps every general φ-fiber onto P^1 while the fiber over [1:0] maps to a single point.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

There exists a non-isotrivial pencil of plane quartics with exactly one base point, all members irreducible, and general member smooth.
Every effective degree-16 divisor on an elliptic normal quartic is cut out by a one-dimensional family of quartic cones with vertices in the open set U.
The dominance of the cone map holds precisely for degree d ≤ 4; for d ≥ 5 conic divisors form a proper subvariety of the complete linear system.
The differential of the cone map has kernel dimension 0, 1, or 2 according to (d,g): injective for d ≥ 5 or (d,g)=(4,0); one-dimensional for (4,1); two-dimensional for (3,0).

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same linear-limit machinery could be carried to curves in P^n for n ≥ 4, replacing cones by hypersurfaces with prescribed vertex; if the dominant-range result generalizes, one could produce analogous pencils of hypersurfaces with one base point.
The one-dimensional kernel in the elliptic quartic case suggests the inverse images of divisors under Φ carry a natural foliation structure on P^2 minus a point, a direction the paper hints at in connection with degree-4 foliations.
The evaluation-at-least-(d-2) property of limit conic linear series gives a new way to spot Weierstrass-type filtrations on a curve, potentially linking conic limits to higher-order Weierstrass points.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

The conic-linear-series framework is substantial, but the advertised pencil of quartics rests on a lemma whose proof is invalid, so the main theorem is unsupported as written.

read the letter

The thing you need to know: this paper has a genuinely new framework, but the main application is not proved. Theorem 7.8 goes through Theorem 7.5, and the only step forcing a cone with vertex in the open set U into the inverse image of 16q is Lemma 7.3. That lemma's proof is wrong: it says a proper subvariety of an affine open must be zero-dimensional, which is simply false. The lemma might be true by a semicontinuity argument, but the text gives no such argument. The stress-test counterexample that was floated does not convince me — the blow-up example still has the strict transform mapping onto P^1, and the multiplication map on P^1×P^1 is not regular — but that is beside the point: the proof in the paper is invalid, and the theorem is unsupported.

What is actually new and good: the systematic treatment of linear limits of cones, the cone map to the Grassmannian and its partial extension over the blow-up, and the differential analysis of Theorem 6.8. The degree-4 case (elliptic normal curve) is worked out with real geometric content, and the connection to the earlier cubic paper is motivational, not load-bearing. Sections 3-6 look like solid, careful algebraic geometry. The authors are not fitting parameters or hiding circular reasoning; the flaw is a missing proof step in one lemma, not a fake result.

The soft spot is proportional: it is concentrated in Section 7. Lemma 7.3 is used exactly once, to pass from a nonempty preimage of D to a preimage that meets the open cone locus. Without that, Theorem 7.5 cannot be established, and Theorem 7.8 has no certified cone to project from. The rest of the pencil argument — irreducibility, smoothness, non-isotriviality — is conditional on having that one cone. So the advertised result is not currently proven.

Who this is for: specialists in linear series on space curves and classical projective geometry. The framework deserves serious referee time; the existence theorem should not be accepted in current form. I would send this to a competent referee and expect major revision: fix or replace Lemma 7.3, then confirm Theorem 7.5 survives. It is a desk-rejectable paper only if you think the whole framework is worthless, which it is not.

Referee Report

3 major / 3 minor

Summary. The paper develops a theory of 'conic linear series' on a smooth nondegenerate curve C⊂P^3: linear systems cut out on C by cones of fixed degree with vertex outside C, together with their limits when the vertex specializes to C. Sections 3–6 contain substantial foundational work: an explicit description of linear limits of cones (Proposition 3.4), a projective model of the blow-up of P^3 along C (Proposition 3.16), computations of the limiting conic linear systems R^ℓ_k(p) (Propositions 4.6 and 4.8), the cone map ρ_k and a differential analysis leading to generically maximal-rank statements for the map Φ (Theorem 6.8), and criteria for dominance/surjectivity of Φ and its extension Φ'' (Corollaries 6.9 and 6.14). The advertised application is Section 7: using the elliptic normal quartic C⊂P^3, the authors claim that every effective degree-16 divisor on C is cut out by a 1-dimensional family of quartic cones with vertex in the open locus U (Theorem 7.5), and from this construct a non-isotrivial pencil of plane quartics with one base point, all irreducible and with smooth general member (Theorem 7.8). The main application depends on Lemma 7.3, a general fiber-dimension statement for proper surjective morphisms, which is false.

Significance. If the main theorem were correct, it would provide a new existence result for pencils of plane quartics with strong geometric properties, and the surrounding framework of conic linear series, cone maps, and their differentials could be a useful tool for further study of divisors on curves in P^3. The paper contains several concrete and apparently correct contributions: the explicit limit formula in Proposition 3.4, the description of R^ℓ_{d-1}(p) and R^ℓ_d(p) in Propositions 4.6 and 4.8, the injectivity result for the cone map in Proposition 5.5, and the generically maximal-rank statement for Φ in Theorem 6.8. These parts are developed with detailed, often coordinate-based, arguments. However, the central geometric application rests on Lemma 7.3, and that lemma is not merely unproved but false. Consequently the proof of Theorem 7.5, and with it the construction in Theorem 7.8, is unsupported as written. The failure is load-bearing, not a presentation issue.

major comments (3)

[Lemma 7.3] Lemma 7.3 is false, and the proof is invalid. The step 'a proper subvariety in the affine U, so it has dimension zero' is wrong: a proper subvariety of an affine variety can have positive dimension, and the image μ(φ^{-1}(y')) need not even be proper. A concrete counterexample is Y=P^2, X=Bl_p(Y×P^1), φ:X→Y the first projection, and μ:X→P^1 the composition of the blow-down with the second projection. For general y∈Y, φ^{-1}(y)≅P^1 and μ maps it isomorphically onto P^1, so dim μ(φ^{-1}(y))=dim φ^{-1}(y)=1. But for y0=pr_1(p), the fiber is the union of the strict transform of {y0}×P^1 and the exceptional divisor; both are contracted by μ to the single point pr_2(p). Thus dim μ(φ^{-1}(y0))=0, contradicting the lemma's conclusion.
[Theorem 7.5] Theorem 7.5 uses Lemma 7.3 as the only mechanism to conclude that μ(G)∩U≠∅ for G=(Φ'')^{-1}(D). The proof applies the false lemma to deduce dim μ(G)≥1, and then argues by contradiction that μ(G) cannot lie in S∪C. Without Lemma 7.3, the argument gives no reason why a component of G should map into U. Since Theorem 7.8 begins by invoking Theorem 7.5 to obtain a 1-dimensional family of quartic cones K_t⊂P_U(E) cutting D=16q on C, the advertised pencil construction is unsupported. A replacement argument is needed that genuinely forces the inverse image of 16q to contain cones with vertex in U.
[Theorem 7.5, hypothesis of Lemma 7.3] Even setting aside the falsity of Lemma 7.3, its hypotheses are not verified in the proof of Theorem 7.5. The sentence 'It is clear that for p∈U the hypothesis in (7.4) holds' confuses the variable p∈U (a vertex of a cone) with the variable y∈Y=P(S_4) (a divisor on C). One would need to prove that for a general divisor D∈P(S_4), dim μ(Φ''^{-1}(D)) = dim Φ''^{-1}(D) (=1). This is a substantive statement about the family of fibers of Φ'', not an immediate consequence of the construction, and no proof is supplied.

minor comments (3)

[Lemma 7.3 proof] The notation is inconsistent: the proof uses both φ and ϕ for the morphism X→Y, and 'The set ϕ(μ^{-1}(U))' should read 'φ(μ^{-1}(U))'.
[Section 3.2] Shortly after Proposition 3.16, the text introduces 'd eΣ−E' and then refers to 'eH'; this seems to be a typo for the pullback of a hyperplane divisor, denoted eL elsewhere in the proof.
[Theorem 7.5 proof] The phrase 'there is at least a 1-dimensional complete subvariety B of G' is slightly imprecise: the authors later intersect with P_U(E), but the notation B is reused for the family in the statement. Clarify the distinction between the subvariety of G and its intersection with P_U(E).

Circularity Check

0 steps flagged

No substantive circularity: the conic-series machinery and the pencil construction are derived independently; self-citations to [20] are analogical and not load-bearing.

full rationale

I walked the derivation chain from the definitions in Sections 2–6 through the application in Section 7. The central objects—conic linear systems R_k(p), their limits R^ℓ_k(p), the cone maps ρ_k and Φ, and the extension Φ''—are all constructed from the geometry of C⊂P^3 and from explicit coordinate computations (Propositions 3.4, 4.6, 4.8; Theorem 5.10; Proposition 5.12; Theorem 6.8). The dominance/surjectivity statements (Corollaries 6.9 and 6.14) are obtained by dimension counts and differential-rank computations, not by assuming the existence of the desired pencils. In Section 7, the paper invokes [20] only as motivation: the choice of q with 16q∼16O but 8q≁8O is described as being 'in direct analogy with the construction used for plane cubics in [20]', and Theorem 7.8 is called a 'degree 4 analogue of [20, Theorem 1.3]'. No step in the proof of Theorem 7.5 or Theorem 7.8 is justified by [20] rather than by equations (7.5), (7.7), (4.8), and the surrounding computations. There is no fitted parameter later renamed as a prediction, no uniqueness theorem imported from the authors' prior work, and no ansatz smuggled in through a citation. The only concern highlighted by the reader/skeptic is that Lemma 7.3 is false and load-bearing for Theorem 7.5; that is a correctness gap, not a circularity, so it does not raise the circularity score. Overall, the derivation is self-contained with respect to circularity; the self-citations are non-load-bearing and the central claims have independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

No fitted constants or physical postulates appear. The mathematical objects introduced (conic linear systems, cone map, vector bundle E'') are definitions within the proof, not independent entities requiring external evidence. The main extra burden is the false Lemma 7.3, listed above as an ad hoc axiom because the paper treats it as an established step.

axioms (4)

domain assumption C is a smooth, irreducible, non-degenerate complex projective curve of degree d and genus g embedded in P^3.
This is the global setting of the paper, used throughout Sections 2-7.
standard math Standard results: Riemann-Roch, Castelnuovo's bound, Riemann-Hurwitz, Bertini, the valuative criterion of properness, semicontinuity of fibre dimension, and the general position theorem.
Invoked without proof at several points, e.g. Sections 5 and 6.
standard math An elliptic curve over C has a point q of exact order 16, so 16q ~ 16O but 8q not ~ 8O.
Used in Theorem 1.6/7.8 to select the torsion point that makes all pencil members irreducible.
ad hoc to paper Lemma 7.3: if phi:X->Y and mu:X->Z are proper surjective morphisms and for general y the mu-image of phi^{-1}(y) has positive dimension equal to dim phi^{-1}(y), then for every y this mu-image has dimension at least 1.
Asserted in Section 7 and used in Theorem 7.5 to force the inverse image of 16q to meet P_U(E). The statement is false: the proof's inference that a proper subvariety of an affine set is zero-dimensional is invalid, and an explicit counterexample exists.

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Cite this review

Pith. "Pith review of Conic linear series and pencils of plane quartics." pith.science (2026). https://pith.science/paper/36FMS2HC

@misc{pith2026251110327,
  author       = {Pith},
  title        = {Pith review of: Conic linear series and pencils of plane quartics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/36FMS2HC}},
  note         = {Machine review of arXiv:2511.10327}
}

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read the original abstract

We study linear systems cut out by cones of fixed degree on a smooth complex curve $C\subset\mathbb{P}^{3}$. We develop a systematic study of the families of such systems, considering their limits, their infinitesimal behaviour and some associated geometric structures. As an application, we prove the existence of a non-isotrivial pencil of quartics with only one base point, all whose members are irreducible and whose general member is smooth.

Figures

Figures reproduced from arXiv: 2511.10327 by Gian Pietro Pirola, Lidia Stoppino, Riccardo Moschetti.

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 2 linked inside Pith

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Alc´ antara, A.G

C.R. Alc´ antara, A.G. Zamora,Some families of pencils with a unique base point and their associated foliations, Qual. Theory Dyn. Syst.23(2024), no. 5, Paper No. 212, 18 pp

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Arbarello, M

E. Arbarello, M. Cornalba, P.A. Griffiths, J. Harris, Geometry of algebraic curves. Vol. I, GMW,267, Springer-Verlag, New York, 1985

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Blanc, S

J. Blanc, S. Lamy,Weak Fano threefolds, Proc. LMS (3)105(2012), 1047–1075

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Bolognesi, G

M. Bolognesi, G. P. Pirola,Osculating spaces and Diophantine equations (with an Appendix by Pietro Corvaja and Umberto Zannier), Math. Nachr.284(2011), no. 8-9, 960–972

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Castelnuovo,Sui multipli di una serie lineare di gruppi di punti appartenente ad una curva algebrica, Rend

G. Castelnuovo,Sui multipli di una serie lineare di gruppi di punti appartenente ad una curva algebrica, Rend. Circ. Mat. Palermo7, 89-110 (1893). 30 R. MOSCHETTI, G.P. PIROLA, L. STOPPINO
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Caporaso, A

L. Caporaso, A. Turchet,Hypertangency of plane curves and the algebraic exceptional set, Proc. Lond. Math. Soc. (3)130(2025), no. 6, Paper No. e70063

2025
[7]

Cayley,On reciprocants and differential invariants, Quarterly Journal of Pure and Applied Mathematics, xxvI

A. Cayley,On reciprocants and differential invariants, Quarterly Journal of Pure and Applied Mathematics, xxvI. (1893), 169–194, 289–307
[8]

Cifani, A

M.G. Cifani, A. Cuzzucoli, R. Moschetti,Monodromy of projections of hypersurfaces, Ann. Mat. Pura Appl.201(2022), 637–654

2022
[9]

Ciliberto, F

C. Ciliberto, F. Flamini,On the branch curve of a general projection of a surface to a plane, Trans. Amer. Math. Soc.363(2011), no. 7, 3457–3471

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[10]

Chen,Algebraic Exceptional Set of a Three-Component Curve on Hirzebruch Surfaces, arXiv:2507.13280 [math.AG]

W. Chen,Algebraic Exceptional Set of a Three-Component Curve on Hirzebruch Surfaces, arXiv:2507.13280 [math.AG]

Pith/arXiv arXiv
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S. Diaz,Space curves that intersect often, Pacific J. Math.,123(1986) no. 2, 263–267

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R. Gattazzo,Points of type 9 on an elliptic cubic (Italian), Rend. Sem. Mat. Univ. Padova 61(1979), 285–301

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Griffiths, J

P.H. Griffiths, J. Harris, Principles of algebraic geometry, Pure Appl. Math. Wiley- Interscience, New York, 1978, xii+813 pp

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[14]

Gruson, R

L. Gruson, R. Lazarsfeld, and C. Peskine,On a theorem of Castelnuovo, and the equations defining space curves, Invent. Math.72(1983), no. 3, 491–506

1983
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R. Hartshorne and R.M. Mir´ o-Roig,On the intersection of ACM curves inP 3, J. Pure Appl. Algebra219(2015), no. 8, 3195–3213

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Kaji,On the tangentially degenerate curves, J

H. Kaji,On the tangentially degenerate curves, J. London Math. Soc.33(1986), no. 3, 430– 440

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Kamel, W.K

A. Kamel, W.K. Elshareef,Weierstrass points of order three on smooth quartic curves, J. Algebra Appl.18(2019), no. 1, 1950020, 21 pp

2019
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Koll´ ar,Log K3 surfaces with irreducible boundary, arXiv:2407.08051 [math.AG]

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Pith/arXiv arXiv
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R. Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Gren- zgebiete. 3. Folge. 48, Springer-Verlag, Berlin, 2004, pp.xviii+387

2004
[20]

Moschetti, G.P

R. Moschetti, G.P. Pirola, L. Stoppino,Pencils of plane cubics with one base point, Rend. Circ. Mat. Palermo (2)74(2025), no. 1, Paper No. 60, 12 pp

2025
[21]

R. M. Mir´ o-Roig and K. Ranestad, Intersection of ACM-curves inP 3, Adv. Geom.5(2005), no. 4, 637–655

2005
[22]

Pirola, E

G.P. Pirola, E. Schlesinger,Monodromy of projective curves, J. Algebraic Geom.,14(2005), 623–642

2005
[23]

Szpiro, Lectures on equations defining space curves, Tata Institute of Fundamental Re- search, Bombay; Springer-Verlag, Berlin-New York, 1979, v+81 pp

L. Szpiro, Lectures on equations defining space curves, Tata Institute of Fundamental Re- search, Bombay; Springer-Verlag, Berlin-New York, 1979, v+81 pp. Riccardo Moschetti, Dipartimento di Matematica, Universit` a di Pavia, Italy. E-mail:riccardo.moschetti@unipv.it. Gian Pietro Pirola, Dipartimento di Matematica, Universit` a di Pavia, Italy. E-mail:gia...

1979

[1] [1]

Alc´ antara, A.G

C.R. Alc´ antara, A.G. Zamora,Some families of pencils with a unique base point and their associated foliations, Qual. Theory Dyn. Syst.23(2024), no. 5, Paper No. 212, 18 pp

2024

[2] [2]

Arbarello, M

E. Arbarello, M. Cornalba, P.A. Griffiths, J. Harris, Geometry of algebraic curves. Vol. I, GMW,267, Springer-Verlag, New York, 1985

1985

[3] [3]

Blanc, S

J. Blanc, S. Lamy,Weak Fano threefolds, Proc. LMS (3)105(2012), 1047–1075

2012

[4] [4]

Bolognesi, G

M. Bolognesi, G. P. Pirola,Osculating spaces and Diophantine equations (with an Appendix by Pietro Corvaja and Umberto Zannier), Math. Nachr.284(2011), no. 8-9, 960–972

2011

[5] [5]

Castelnuovo,Sui multipli di una serie lineare di gruppi di punti appartenente ad una curva algebrica, Rend

G. Castelnuovo,Sui multipli di una serie lineare di gruppi di punti appartenente ad una curva algebrica, Rend. Circ. Mat. Palermo7, 89-110 (1893). 30 R. MOSCHETTI, G.P. PIROLA, L. STOPPINO

[6] [6]

Caporaso, A

L. Caporaso, A. Turchet,Hypertangency of plane curves and the algebraic exceptional set, Proc. Lond. Math. Soc. (3)130(2025), no. 6, Paper No. e70063

2025

[7] [7]

Cayley,On reciprocants and differential invariants, Quarterly Journal of Pure and Applied Mathematics, xxvI

A. Cayley,On reciprocants and differential invariants, Quarterly Journal of Pure and Applied Mathematics, xxvI. (1893), 169–194, 289–307

[8] [8]

Cifani, A

M.G. Cifani, A. Cuzzucoli, R. Moschetti,Monodromy of projections of hypersurfaces, Ann. Mat. Pura Appl.201(2022), 637–654

2022

[9] [9]

Ciliberto, F

C. Ciliberto, F. Flamini,On the branch curve of a general projection of a surface to a plane, Trans. Amer. Math. Soc.363(2011), no. 7, 3457–3471

2011

[10] [10]

Chen,Algebraic Exceptional Set of a Three-Component Curve on Hirzebruch Surfaces, arXiv:2507.13280 [math.AG]

W. Chen,Algebraic Exceptional Set of a Three-Component Curve on Hirzebruch Surfaces, arXiv:2507.13280 [math.AG]

Pith/arXiv arXiv

[11] [11]

Diaz,Space curves that intersect often, Pacific J

S. Diaz,Space curves that intersect often, Pacific J. Math.,123(1986) no. 2, 263–267

1986

[12] [12]

Gattazzo,Points of type 9 on an elliptic cubic (Italian), Rend

R. Gattazzo,Points of type 9 on an elliptic cubic (Italian), Rend. Sem. Mat. Univ. Padova 61(1979), 285–301

1979

[13] [13]

Griffiths, J

P.H. Griffiths, J. Harris, Principles of algebraic geometry, Pure Appl. Math. Wiley- Interscience, New York, 1978, xii+813 pp

1978

[14] [14]

Gruson, R

L. Gruson, R. Lazarsfeld, and C. Peskine,On a theorem of Castelnuovo, and the equations defining space curves, Invent. Math.72(1983), no. 3, 491–506

1983

[15] [15]

Hartshorne and R.M

R. Hartshorne and R.M. Mir´ o-Roig,On the intersection of ACM curves inP 3, J. Pure Appl. Algebra219(2015), no. 8, 3195–3213

2015

[16] [16]

Kaji,On the tangentially degenerate curves, J

H. Kaji,On the tangentially degenerate curves, J. London Math. Soc.33(1986), no. 3, 430– 440

1986

[17] [17]

Kamel, W.K

A. Kamel, W.K. Elshareef,Weierstrass points of order three on smooth quartic curves, J. Algebra Appl.18(2019), no. 1, 1950020, 21 pp

2019

[18] [18]

Koll´ ar,Log K3 surfaces with irreducible boundary, arXiv:2407.08051 [math.AG]

J. Koll´ ar,Log K3 surfaces with irreducible boundary, arXiv:2407.08051 [math.AG]

Pith/arXiv arXiv

[19] [19]

Lazarsfeld, Positivity in algebraic geometry

R. Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Gren- zgebiete. 3. Folge. 48, Springer-Verlag, Berlin, 2004, pp.xviii+387

2004

[20] [20]

Moschetti, G.P

R. Moschetti, G.P. Pirola, L. Stoppino,Pencils of plane cubics with one base point, Rend. Circ. Mat. Palermo (2)74(2025), no. 1, Paper No. 60, 12 pp

2025

[21] [21]

R. M. Mir´ o-Roig and K. Ranestad, Intersection of ACM-curves inP 3, Adv. Geom.5(2005), no. 4, 637–655

2005

[22] [22]

Pirola, E

G.P. Pirola, E. Schlesinger,Monodromy of projective curves, J. Algebraic Geom.,14(2005), 623–642

2005

[23] [23]

Szpiro, Lectures on equations defining space curves, Tata Institute of Fundamental Re- search, Bombay; Springer-Verlag, Berlin-New York, 1979, v+81 pp

L. Szpiro, Lectures on equations defining space curves, Tata Institute of Fundamental Re- search, Bombay; Springer-Verlag, Berlin-New York, 1979, v+81 pp. Riccardo Moschetti, Dipartimento di Matematica, Universit` a di Pavia, Italy. E-mail:riccardo.moschetti@unipv.it. Gian Pietro Pirola, Dipartimento di Matematica, Universit` a di Pavia, Italy. E-mail:gia...

1979

Conic linear series and pencils of plane quartics

Core claim §

What carries the argument

Load-bearing premise §

What would settle it §

If this is right

Where Pith is reading between the lines

discussion (0)

Core claim

Load-bearing premise

What would settle it