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

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.

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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.

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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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Keywords

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

Mathematics Subject Classification (1991)

  • Primary 14H20
  • 14H45
  • 14H51