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

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


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.


Non-Gorenstein curve Canonical model Gonality Scrolls 

Mathematics Subject Classification (1991)

Primary 14H20 14H45 14H51 



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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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

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