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arxiv: 2304.01851 · v4 · pith:7K5HADRWnew · submitted 2023-04-04 · 🧮 math.AG

Extensions of curves with high degree with respect to the genus

Ciro Ciliberto , Thomas Dedieu This is my paper

Pith reviewed 2026-05-24 09:04 UTC · model grok-4.3

classification 🧮 math.AG
keywords linearly normal surfacessectional genusribbon integrationGaussian mapshyperelliptic curvespluricanonical curvesuniversal extensionalgebraic curves
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The pith

Linearly normal surfaces of degree between 4g-4 and 4g+4 are classified, proving that all ribbons over pluricanonical, genus-3, and certain hyperelliptic curves are integrable and admit a universal extension.

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

The paper classifies linearly normal surfaces S in projective space whose degree d satisfies 4g-4 ≤ d ≤ 4g+4 for sectional genus g greater than 1. This classification is applied to the extension theory of curves by studying when ribbons over the curves can be integrated into surfaces, using computations of Gaussian map coranks. The result establishes that ribbons over pluricanonical curves and genus-3 curves satisfying Property N2 are always integrable, and the same holds for hyperelliptic curves precisely when the degree equals 2g+3. A sympathetic reader would care because integrability directly yields the existence of a universal extension in these cases.

Core claim

We classify linearly normal surfaces S ⊂ P^{r+1} of degree d such that 4g-4 ≤ d ≤ 4g+4, where g>1 is the sectional genus. We show that all ribbons over pluricanonical curves and genus 3 curves verifying Property N2 are integrable, and thus there exists a universal extension. For linearly normal hyperelliptic curves of degree d≥2g+3, all ribbons over C are integrable if and only if d=2g+3, in which case a universal extension exists.

What carries the argument

Integration of ribbons over curves via computation of coranks of Gaussian maps, applied after classifying surfaces having the given curve as hyperplane section.

If this is right

  • For d larger than 4g+4 the only such surfaces are cones.
  • All ribbons over the qualifying pluricanonical and genus-3 curves are integrable.
  • A universal extension exists for those curves.
  • For hyperelliptic curves integrability of all ribbons holds exactly when d equals 2g+3.
  • The corank of the relevant Gaussian maps is computed explicitly in these cases.

Where Pith is reading between the lines

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

  • The same surface classification technique might identify further degree ranges where universal extensions exist for additional curve families.
  • Verifying Property N2 for more curves would immediately extend the integrability result to those families.
  • The explicit classification could be used to parametrize the possible extensions in the moduli space of curves.
  • Similar ribbon integrability statements may hold for non-hyperelliptic curves of higher genus once analogous surface lists are obtained.

Load-bearing premise

The curves satisfy Property N2 and the prior theory of ribbon integration applies directly in the stated degree range.

What would settle it

Finding either a linearly normal surface of degree d in [4g-4,4g+4] that falls outside the listed types in the classification, or a non-integrable ribbon over a curve satisfying the hypotheses.

read the original abstract

We classify linearly normal surfaces $S \subset \mathbf{P}^{r+1}$ of degree $d$ such that $4g-4 \leq d \leq 4g+4$, where $g>1$ is the sectional genus (it is a classical result that for larger $d$ there are only cones). We apply this to the study of the extension theory of pluricanonical curves and genus $3$ curves, whenever they verify Property $N_2$, using and slightly expanding the theory of integration of ribbons of the authors and E.~Sernesi. We compute the corank of the relevant Gaussian maps, and we show that all ribbons over such curves are integrable, and thus there exists a universal extension. We carry out a similar program for linearly normal hyperelliptic curves of degree $d\geq 2g+3$. We classify surfaces having such a curve $C$ as a hyperplane section, compute the corank of the relevant Gaussian maps, and prove that all ribbons over $C$ are integrable if and only if $d=2g+3$. In the latter case we obtain the existence of a universal extension.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper classifies linearly normal surfaces S ⊂ P^{r+1} of degree d satisfying 4g−4 ≤ d ≤ 4g+4 (g>1 sectional genus), noting that only cones exist for larger d by classical results. It applies the classification to the extension theory of pluricanonical curves and genus-3 curves (under Property N2) via the authors' prior ribbon-integration framework with Sernesi, computing coranks of relevant Gaussian maps and proving all ribbons are integrable (hence a universal extension exists). An analogous program is carried out for linearly normal hyperelliptic curves of degree d≥2g+3, with integrability holding precisely when d=2g+3 (again yielding a universal extension).

Significance. If the classification and corank computations hold, the work completes the picture of ribbon integrability and universal extensions precisely in the critical degree window relative to genus, where non-cone surfaces can appear. The explicit results for pluricanonical, genus-3, and hyperelliptic cases, together with the careful choice of range to invoke the classical cone statement outside it, constitute a solid contribution to the extension theory of curves in algebraic geometry.

minor comments (2)
  1. The abstract states that the theory of ribbon integration is 'slightly expanded'; the introduction should explicitly list the new technical ingredients (e.g., any new lemmas on corank or integrability) that go beyond the cited prior work.
  2. The standing hypotheses (linear normality, Property N2 when invoked) are mentioned in the abstract; a short dedicated paragraph early in the paper should collect all global assumptions and indicate precisely where each is used in the classification and integrability arguments.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and the recommendation to accept the manuscript. The assessment accurately captures the scope of the classification and its applications to ribbon integrability.

Circularity Check

0 steps flagged

No significant circularity; core classification and corank computations are independent

full rationale

The paper's derivation consists of classifying linearly normal surfaces S in P^{r+1} with 4g-4 ≤ d ≤ 4g+4 (a classical range where non-cones are limited) and computing coranks of Gaussian maps for pluricanonical, genus-3, and hyperelliptic curves under Property N2 and linear normality. These steps are executed directly via standard methods in algebraic geometry. The reference to the authors' prior ribbon-integration theory with Sernesi is used as an external framework for applying the results to extensions, but the classification, corank values, and integrability conclusions do not reduce to or depend circularly on that prior work. No self-definitional equations, fitted inputs renamed as predictions, uniqueness theorems imported from self-citations, or ansatzes smuggled via citation appear in the load-bearing steps. The argument remains self-contained and externally verifiable within the stated hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

No free parameters or invented entities are introduced. The paper relies on standard background assumptions of algebraic geometry.

axioms (1)
  • standard math Work over an algebraically closed field of characteristic zero
    Implicit in all statements about projective space and linear normality.

pith-pipeline@v0.9.0 · 5736 in / 1248 out tokens · 39769 ms · 2026-05-24T09:04:11.712239+00:00 · methodology

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Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    Arbarello, A

    E. Arbarello, A. Bruno, and E. Sernesi, On hyperplane sections of K 3 surfaces , Algebr.\ Geom.\ 4 (2017), no. 5, 562--596

    work page 2017

  2. [2]

    Arbarello, M

    E. Arbarello, M. Cornalba, P.\,A. Griffiths, and J. Harris, Geometry of algebraic curves. V ol. I , Grundlehren math.\ Wiss., vol. 267, Springer-Verlag, New York, 1985

    work page 1985

  3. [3]

    Beauville and J.-Y

    A. Beauville and J.-Y. M \'e rindol, Sections hyperplanes des surfaces K3 , Duke Math. J.\ 55 (1987), no. 4, 873--878

    work page 1987

  4. [4]

    Beltrametti and A.\,J

    M. Beltrametti and A.\,J. Sommese, Zero cycles and k th order embeddings of smooth projective surfaces (with an appendix by L. G\"ottsche), in: Problems in the theory of surfaces and their classification ( C ortona, 1988), pp. 33-48, Sympos.\ Math., XXXII, Academic Press, London, 1991

    work page 1988

  5. [5]

    Bertram, L

    A. Bertram, L. Ein, and R. Lazarsfeld, Surjectivity of G aussian maps for line bundles of large degree on curves , in: Algebraic geometry ( C hicago, IL , 1989), pp. 15--25, Lecture Notes in Math., vol. 1479, Springer, Berlin, 1991

    work page 1989

  6. [6]

    Besana and A

    G.\,M. Besana and A. Biancofiore, Degree eleven projective manifolds of dimension greater than or equal to three, Forum Math.\ 17 (2005), no. 5, 711--733

    work page 2005

  7. [7]

    Calabri and C

    A. Calabri and C. Ciliberto, Birational classification of curves on rational surfaces, Nagoya Math. J.\ 199 (2010), 43--93

    work page 2010

  8. [8]

    Castelnuovo, Sulle superficie algebriche le cui sezioni piane sono curve iperellittiche

    G. Castelnuovo, Sulle superficie algebriche le cui sezioni piane sono curve iperellittiche. , Rend.\ Circ.\ Mat.\ Palermo 4 (1890), 73--88

    work page

  9. [9]

    , Atti Accad.\ Sci.\ Torino 25 (1890), 695--715

    , Sulle superficie algebriche le cui sezioni sono curve di genere 3. , Atti Accad.\ Sci.\ Torino 25 (1890), 695--715

    work page

  10. [10]

    , Sui multipli di una serie lineare di gruppi di punti appartenenti ad una curva algebrica, Rend.\ Circ.\ Mat.\ Palermo 7 (1893), 89--110

    work page

  11. [11]

    Catanese and C

    F. Catanese and C. Ciliberto, Symmetric products of elliptic curves and surfaces of general type with p_g=q=1 , J. Algebraic Geom.\ 2 (1993), no. 3, 389--411

    work page 1993

  12. [12]

    Chiantini and C

    L. Chiantini and C. Ciliberto, Weakly defective varieties, Trans.\ Amer.\ Math.\ Soc.\ 354 (2002), no. 1, 151--178

    work page 2002

  13. [13]

    Ciliberto, Sul grado dei generatori dell'anello canonico di una superficie di tipo generale, Rend.\ Sem.\ Mat.\ Univ.\ Politec.\ Torino 41 (1983), no

    C. Ciliberto, Sul grado dei generatori dell'anello canonico di una superficie di tipo generale, Rend.\ Sem.\ Mat.\ Univ.\ Politec.\ Torino 41 (1983), no. 3, 83--111

    work page 1983

  14. [14]

    Ciliberto and T

    C. Ciliberto and T. Dedieu, K3 curves with index k>1 , Boll.\ Unione Mat.\ Ital.\ 15 (2022), no. 1-2, 87--115

    work page 2022

  15. [15]

    Math.\ 64 (2024), no

    , Double covers and extensions, Kyoto J. Math.\ 64 (2024), no. 1, 75--94

    work page 2024

  16. [16]

    Ciliberto , T

    C. Ciliberto , T. Dedieu , and E. Sernesi , Wahl maps and extensions of canonical curves and \(K3\) surfaces. , J. reine angew.\ Math.\ 761 (2020), 219--245

    work page 2020

  17. [17]

    Ciliberto and R

    C. Ciliberto and R. Miranda, On the G aussian map for canonical curves of low genus , Duke Math. J.\ 61 (1990), no. 2, 417--443

    work page 1990

  18. [18]

    Dedieu, Some classical formulae for curves and surfaces, preprint https://hal.archives-ouvertes.fr/hal-02914653 (2020)

    T. Dedieu, Some classical formulae for curves and surfaces, preprint https://hal.archives-ouvertes.fr/hal-02914653 (2020)

    work page 2020

  19. [19]

    Dewer, Extensions of Gorenstein weighted projective 3-spaces and characterization of the primitive curves of their surface sections, preprint arXiv:2303.01882 (2023)

    B. Dewer, Extensions of Gorenstein weighted projective 3-spaces and characterization of the primitive curves of their surface sections, preprint arXiv:2303.01882 (2023)

    work page arXiv 2023

  20. [20]

    Epema, Surfaces with canonical hyperplane sections, CWI Tract, vol

    D.\,H.\,J. Epema, Surfaces with canonical hyperplane sections, CWI Tract, vol. 1, Math.\ Centrum, Centrum Wisk.\ Inform., Amsterdam, 1984

    work page 1984

  21. [21]

    Fania and E.\,L

    M.\,L. Fania and E.\,L. Livorni, Degree nine manifolds of dimension greater than or equal to 3 , Math.\ Nachr.\ 169 (1994), 117--134

    work page 1994

  22. [22]

    , Degree ten manifolds of dimension n greater than or equal to 3 , Math.\ Nachr.\ 188 (1997), 79--108

    work page 1997

  23. [23]

    Grayson and M.\,E

    D.\,R. Grayson and M.\,E. Stillman, Macaulay2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/

    work page

  24. [24]

    Green, Koszul cohomology and the geometry of projective varieties, J

    M. Green, Koszul cohomology and the geometry of projective varieties, J. Differential Geom.\ 19 (1984), no. 1, 125--171

    work page 1984

  25. [25]

    Hartshorne, Curves with high self-intersection on algebraic surfaces, Inst.\ Hautes \' E tudes Sci.\ Publ.\ Math.\ (1969), no

    R. Hartshorne, Curves with high self-intersection on algebraic surfaces, Inst.\ Hautes \' E tudes Sci.\ Publ.\ Math.\ (1969), no. 36, 111--125

    work page 1969

  26. [26]

    52, Springer-Verlag, New York-Heidelberg, 1977

    , Algebraic geometry, Grad.\ Texts in Math., No. 52, Springer-Verlag, New York-Heidelberg, 1977

    work page 1977

  27. [27]

    Horowitz, Varieties of low -genus , Duke Math

    T. Horowitz, Varieties of low -genus , Duke Math. J.\ 50 (1983), no. 3, 667--683

    work page 1983

  28. [28]

    Ionescu, Embedded projective varieties of small invariants, in: Algebraic geometry, B ucharest 1982 ( B ucharest, 1982), pp

    P. Ionescu, Embedded projective varieties of small invariants, in: Algebraic geometry, B ucharest 1982 ( B ucharest, 1982), pp. 142--186, Lecture Notes in Math., vol. 1056, Springer, Berlin, 1984

    work page 1982

  29. [29]

    Knutsen, Global sections of twisted normal bundles of K3 surfaces and their hyperplane sections , Atti Accad.\ Naz.\ Lincei Rend.\ Lincei Mat.\ Appl.\ 31 (2020), no

    A.\,L. Knutsen, Global sections of twisted normal bundles of K3 surfaces and their hyperplane sections , Atti Accad.\ Naz.\ Lincei Rend.\ Lincei Mat.\ Appl.\ 31 (2020), no. 1, 57--79

    work page 2020

  30. [30]

    Knutsen and A.\,F

    A.\,L. Knutsen and A.\,F. Lopez, Surjectivity of Gaussian maps for curves on Enriques surfaces, Adv.\ Geom.\ 7 (2007), no. 2, 215--247

    work page 2007

  31. [31]

    Lanteri and E.\,L

    A. Lanteri and E.\,L. Livorni, Complex surfaces polarized by an ample and spanned line bundle of genus three, Geom.\ Dedicata 31 (1989), no. 3, 267--289

    work page 1989

  32. [32]

    Lopez, On the Extendability of Projective Varieties: A Survey (with an appendix by T.\ Dedieu), in: The Art of Doing Algebraic geometry, pp

    A.\,F. Lopez, On the Extendability of Projective Varieties: A Survey (with an appendix by T.\ Dedieu), in: The Art of Doing Algebraic geometry, pp. 261-291, Trends Math., Birkh\"auser/Springer, Cham, 2023

    work page 2023

  33. [33]

    Mella and E

    M. Mella and E. Polastri, Equivalent birational embeddings. II : Divisors , Math. Z.\ 270 (2012), no. 3-4, 1141--1161

    work page 2012

  34. [34]

    Miranda, Triple Covers in Algebraic Geometry, Amer

    R. Miranda, Triple Covers in Algebraic Geometry, Amer. J.\ Math.\ 107 (1985), no. 5, 1123--1158

    work page 1985

  35. [35]

    Pareschi, Gaussian maps and multiplication maps on certain projective varieties, Compos.\ Math.\ 98 (1995), no

    G. Pareschi, Gaussian maps and multiplication maps on certain projective varieties, Compos.\ Math.\ 98 (1995), no. 3, 219--268

    work page 1995

  36. [36]

    Peeva, Graded syzygies, Algebr.\ Appl., vol

    I. Peeva, Graded syzygies, Algebr.\ Appl., vol. 14, Springer-Verlag London, Ltd., London, 2011

    work page 2011

  37. [37]

    Reider, Vector bundles of rank 2 and linear systems on algebraic surfaces , Ann.\ of Math

    I. Reider, Vector bundles of rank 2 and linear systems on algebraic surfaces , Ann.\ of Math. (2) 127 (1988), no. 2, 309--316

    work page 1988

  38. [38]

    Schreyer, A standard basis approach to syzygies of canonical curves, J

    F.-O. Schreyer, A standard basis approach to syzygies of canonical curves, J. reine angew.\ Math.\ 421 (1991), 83--123

    work page 1991

  39. [39]

    Segre, Ricerche sulle rigate ellittiche di qualunque ordine , Atti Accad.\ Sci.\ Torino 21 (1885--86), 628--651

    C. Segre, Ricerche sulle rigate ellittiche di qualunque ordine , Atti Accad.\ Sci.\ Torino 21 (1885--86), 628--651

    work page

  40. [40]

    , Recherches g\'en\'erales sur les courbes et les surfaces r\'egl\'ees alg\'ebriques. II. , Math.\ Ann.\ 34 (1889), 1--25

    work page

  41. [41]

    Serrano, The adjunction mappings and hyperelliptic divisors on a surface, J

    F. Serrano, The adjunction mappings and hyperelliptic divisors on a surface, J. reine angew.\ Math.\ 381 (1987), 90--109

    work page 1987

  42. [42]

    Sommese and A

    A.\,J. Sommese and A. Van de Ven, On the adjunction mapping, Math.\ Ann.\ 278 (1987), no. 1-4, 593--603

    work page 1987

  43. [43]

    Voisin, Courbes t\'etragonales et cohomologie de K oszul , J

    C. Voisin, Courbes t\'etragonales et cohomologie de K oszul , J. reine angew.\ Math.\ 387 (1988), 111--121

    work page 1988

  44. [44]

    3-4, 249--272

    , Sur l'application de W ahl des courbes satisfaisant la condition de B rill- N oether- P etri , Acta Math.\ 168 (1992), no. 3-4, 249--272

    work page 1992

  45. [45]

    Wahl, On cohomology of the square of an ideal sheaf, J

    J. Wahl, On cohomology of the square of an ideal sheaf, J. Algebraic Geom.\ 6 (1997), no. 3, 481--511

    work page 1997