# On the birational gonalities of smooth curves

Annales UMCS, Mathematica (2014)

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

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topE. 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

top- [1] Coppens, M., Martens, G., Linear series on 4-gonal curves, Math. Nachr. 213, no. 1 (2000), 35-55. Zbl0972.14021
- [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] 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] Hatshorne, R., Algebraic Geometry, Springer-Verlag, Berlin, 1977.
- [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] 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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