On gonality, scrolls, and canonical models of non-Gorenstein curves


Let C be an integral and projective curve; and let \(C'\) be its canonical model. We study the relation between the gonality of C and the dimension of a rational normal scroll S where \(C'\) can lie on. We are mainly interested in the case where C is singular, or even non-Gorenstein, in which case \(C'\not \cong C\). We first analyze some properties of an inclusion \(C'\subset S\) when it is induced by a pencil on C. Afterwards, in an opposite direction, we assume \(C'\) lies on a certain scroll, and check some properties C may satisfy, such as gonality and the kind of its singularities. At the end, we prove that a rational monomial curve C has gonality d if and only if \(C'\) lies on a \((d-1)\)-fold scroll.

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  1. 1.

    Andreotti, A., Mayer, A.L.: On period relations for abelian integrals on algebraic curves. Annali della Scuolla Normale Superiore di Piza 21(2), 189–238 (1967)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Babbage, D.W.: A note on the quadrics through a canonical curve. J. Lond. Math. Soc. 14, 310–315 (1939)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Bresinsky, H.: Monomial space curves in \(\mathbb{A}^3\) as set-theoretic complete intersections. Proc. Am. Math. Soc. 75, 23–24 (1979)

    Google Scholar 

  4. 4.

    Barucci, V., Fröberg, R.: One-dimensional almost Gorenstein rings. J. Algebra 188, 418–442 (1997)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Brundu, M., Sacchiero, G.: Stratification of the moduli space of four-gonal curves. Proc. Edinb. Math. Soc. 57(03), 631–686 (2014)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Casnati, G., Ekedahl, T.: Covers of algebraic varieties. I. A general structure theorem, covers of degree 3,4 and Enriques surfaces. J. Algebraic Geom. 5(3), 439–460 (1996)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Contiero, A., Stoehr, K.-O.: Upper bounds for the dimension of moduli spaces of curves with symmetric Weierstrass semigroups. J. Lond. Math. Soc. 88, 580–598 (2013)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Contiero, A., Feital, L., Martins, R.V.: Max Noether theorem for integral curves. J. Algebra 494, 111–136 (2018)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Coppens, M.: Free linear systems on integral Gorenstein curves. J. Algebra 145, 209–218 (1992)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Cotterill, E., Feital, L., Martins, R. V.: Dimension counts for singular rational curves via semigroups. arXiv:1511.08515v2

  11. 11.

    Cotterill, E., Feital, L., Martins, R.V.: Singular rational curves with points of nearly-maximal weight. J. Pure Appl. Algebra 222, 3448–3469 (2018). https://doi.org/10.1016/j.jpaa.2017.12.017

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Eisenbud, D., Harris, J., Koh, J., Stillmann, M.: Determinantal equations for curves of high degree. Am. J. Math. 110, 513–539 (1988)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Eisenbud, D., Harris, J.: On varieties of minimal degree. Proc. Symp. Pure Math. 46, 3–13 (1987)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Enriques, F.: Sulle curve canoniche di genera \(p\) cello spazio a \(p-1\) dimensioni. Rend. Accad. Sci. Ist. Bologna 23, 80–82 (1919)

    Google Scholar 

  15. 15.

    Herzog, J.: Generators and relations of abelian semigroups and semigroup rings. Manuscr. Math. 3, 175–193 (1970)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Hotchkiss, J., Ullery, B.: The gonality of complete intersection curves. arXiv:1706.08169

  17. 17.

    Jäger, J.: Längeberechnungen und Kanonische Ideale in Eindimensionalen Ringen. Arch. Math. 29, 504–512 (1977)

    Article  Google Scholar 

  18. 18.

    Kleiman, S.L., Martins, R.V.: The canonical model of a singular curve. Geom. Dedicata 139, 139–166 (2009)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Lara, D., Marchesi, S., Martins, R.V.: Curves with canonical models on scrolls. Int. J. Math. 27(5), 1650045-1-30 (2016)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Martins, R.V.: On trigonal non-Gorenstein curves with zero Maroni invariant. J. Algebra 275, 453–470 (2004)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Matsuoka, T.: On the degree of singularity of one-dimensional analytically irreducible noetherian rings. J. Math. Kyoto Univ. 11, 485–491 (1971)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Miró-Roig, R.M.: The representation type of rational normal scrolls. Rend. Circ. Mat. Palermo 62, 153–164 (2012)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Reid, M.: Chapters on algebraic surfaces. 6 Feb 1996. Lectures of a summer programm Park City, UT. arXiv:alg-geom/9602006v1 (1993)

  24. 24.

    Rosa, R., Stöhr, K.-O.: Trigonal Gorenstein curves. J. Pure Appl. Algebra 174, 187–205 (2002)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Rosenlicht, M.: Equivalence relations on algebraic curves. Ann. Math. 56, 169–191 (1952)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Schreyer, F.-O.: Syzygies of canonical curves and special linear series. Mathematische Annalen 275, 105–137 (1986)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Stöhr, K.-O.: On the poles of regular differentials of singular curves. Boletim da Sociedade Brasileira de Matemática 24, 105–135 (1993)

    MathSciNet  Article  Google Scholar 

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We specially thank the Referee for many suggestions and very discerning remarks, which made us restructure considerably some parts of the original version of the present article. The first named author is partially supported by CNPq Grant Number 306914/2015-8.

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Correspondence to Renato Vidal Martins.

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Martins, R.V., Lara, D. & Souza, J.M. On gonality, scrolls, and canonical models of non-Gorenstein curves. Geom Dedicata 203, 111–133 (2019). https://doi.org/10.1007/s10711-019-00428-2

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  • Non-Gorenstein curve
  • Canonical model
  • Gonality
  • Scrolls

Mathematics Subject Classification (1991)

  • Primary 14H20
  • 14H45
  • 14H51