Advertisement

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

  • Renato Vidal MartinsEmail author
  • Danielle Lara
  • Jairo Menezes Souza
Original Paper
  • 11 Downloads

Abstract

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.

Keywords

Non-Gorenstein curve Canonical model Gonality Scrolls 

Mathematics Subject Classification (1991)

Primary 14H20 14H45 14H51 

Notes

Acknowledgements

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.

References

  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)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Babbage, D.W.: A note on the quadrics through a canonical curve. J. Lond. Math. Soc. 14, 310–315 (1939)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Barucci, V., Fröberg, R.: One-dimensional almost Gorenstein rings. J. Algebra 188, 418–442 (1997)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Contiero, A., Feital, L., Martins, R.V.: Max Noether theorem for integral curves. J. Algebra 494, 111–136 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Coppens, M.: Free linear systems on integral Gorenstein curves. J. Algebra 145, 209–218 (1992)MathSciNetCrossRefzbMATHGoogle 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 MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Eisenbud, D., Harris, J.: On varieties of minimal degree. Proc. Symp. Pure Math. 46, 3–13 (1987)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)CrossRefzbMATHGoogle Scholar
  18. 18.
    Kleiman, S.L., Martins, R.V.: The canonical model of a singular curve. Geom. Dedicata 139, 139–166 (2009)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Martins, R.V.: On trigonal non-Gorenstein curves with zero Maroni invariant. J. Algebra 275, 453–470 (2004)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Miró-Roig, R.M.: The representation type of rational normal scrolls. Rend. Circ. Mat. Palermo 62, 153–164 (2012)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Rosenlicht, M.: Equivalence relations on algebraic curves. Ann. Math. 56, 169–191 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Schreyer, F.-O.: Syzygies of canonical curves and special linear series. Mathematische Annalen 275, 105–137 (1986)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Renato Vidal Martins
    • 1
    Email author
  • Danielle Lara
    • 2
  • Jairo Menezes Souza
    • 3
  1. 1.Departamento de MatemáticaInstituto de Ciências Exatas, UFMGBelo HorizonteBrazil
  2. 2.Departamento de MatemáticaUFV/CAFFlorestalBrazil
  3. 3.Unidade Acadêmica Especial de Matemática e Tecnologia - IMTec/RC/UFGCatalãoBrazil

Personalised recommendations