On the birational gonalities of smooth curves

E. Ballico

Annales UMCS, Mathematica (2014)

  • Volume: 68, Issue: 1, page 11-20
  • ISSN: 2083-7402

Abstract

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Let C be a smooth curve of genus g. For each positive integer r the birational r-gonality sr(C) of C is the minimal integer t such that there is L ∈ Pict(C) with h0(C,L) = r + 1. Fix an integer r ≥ 3. In this paper we prove the existence of an integer gr such that for every integer g ≥ gr there is a smooth curve C of genus g with sr+1(C)/(r + 1) > sr(C)/r, i.e. in the sequence of all birational gonalities of C at least one of the slope inequalities fails

How to cite

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E. Ballico. "On the birational gonalities of smooth curves." Annales UMCS, Mathematica 68.1 (2014): 11-20. <http://eudml.org/doc/267082>.

@article{E2014,
abstract = {Let C be a smooth curve of genus g. For each positive integer r the birational r-gonality sr(C) of C is the minimal integer t such that there is L ∈ Pict(C) with h0(C,L) = r + 1. Fix an integer r ≥ 3. In this paper we prove the existence of an integer gr such that for every integer g ≥ gr there is a smooth curve C of genus g with sr+1(C)/(r + 1) > sr(C)/r, i.e. in the sequence of all birational gonalities of C at least one of the slope inequalities fails},
author = {E. Ballico},
journal = {Annales UMCS, Mathematica},
keywords = {Birational gonality sequence; smooth curve; nodal curve; Hirzebruch surface; birational gonality sequence},
language = {eng},
number = {1},
pages = {11-20},
title = {On the birational gonalities of smooth curves},
url = {http://eudml.org/doc/267082},
volume = {68},
year = {2014},
}

TY - JOUR
AU - E. Ballico
TI - On the birational gonalities of smooth curves
JO - Annales UMCS, Mathematica
PY - 2014
VL - 68
IS - 1
SP - 11
EP - 20
AB - Let C be a smooth curve of genus g. For each positive integer r the birational r-gonality sr(C) of C is the minimal integer t such that there is L ∈ Pict(C) with h0(C,L) = r + 1. Fix an integer r ≥ 3. In this paper we prove the existence of an integer gr such that for every integer g ≥ gr there is a smooth curve C of genus g with sr+1(C)/(r + 1) > sr(C)/r, i.e. in the sequence of all birational gonalities of C at least one of the slope inequalities fails
LA - eng
KW - Birational gonality sequence; smooth curve; nodal curve; Hirzebruch surface; birational gonality sequence
UR - http://eudml.org/doc/267082
ER -

References

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  1. [1] Coppens, M., Martens, G., Linear series on 4-gonal curves, Math. Nachr. 213, no. 1 (2000), 35-55. Zbl0972.14021
  2. [2] Eisenbud, D., Harris, J., On varieties of minimal degree (a centennial account), Algebraic Geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 3-13, Proc. Sympos. Pure Math., 46, Part 1, Amer. Math. Soc., Providence, RI, 1987. 
  3. [3] Harris, J., Eisenbud, D., Curves in projective space, S´eminaire de Math´ematiques Sup´erieures, 85, Presses de l’Universit´e de Montr´eal, Montr´eal, Que., 1982. 
  4. [4] Hatshorne, R., Algebraic Geometry, Springer-Verlag, Berlin, 1977. 
  5. [5] Laface, A., On linear systems of curves on rational scrolls, Geom. Dedicata 90, no. 1 (2002), 127-144; generalized version in arXiv:math/0205271v2. Zbl1058.14011
  6. [6] Lange, H., Martens, G., On the gonality sequence of an algebraic curve, Manuscripta Math. 137 (2012), 457-473.[WoS] Zbl1238.14022

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