On the birational gonalities of smooth curves

E. Ballico

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2014)

  • Volume: 68, Issue: 1
  • ISSN: 0365-1029

Abstract

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Let C be a smooth curve of genus g . For each positive integer r the birational r -gonality s r ( C ) of C is the minimal integer t such that there is L ∈ Pic t ( C ) with h 0 ( C , L ) = r + 1 . Fix an integer r ≥ 3 . In this paper we prove the existence of an integer g r such that for every integer g ≥ g r there is a smooth curve C of genus g with s r + 1 ( C ) / ( r + 1 ) > s r ( 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 Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 68.1 (2014): null. <http://eudml.org/doc/289735>.

@article{E2014,
abstract = {Let $C$ be a smooth curve of genus $g$. For each positive integer $r$ the birational $r$-gonality $s_r(C)$ of $C$ is the minimal integer $t$ such that there is $L\in \mbox\{Pic\}^t(C)$ with $h^0(C,L) =r+1$. Fix an integer $r\ge 3$. In this paper we prove the existence of an integer $g_r$ such that for every integer $g\ge g_r$ there is a smooth curve $C$ of genus $g$ with $s_\{r+1\}(C)/(r+1) > s_r(C)/r$, i.e. in the sequence of all birational gonalities of $C$ at least one of the slope inequalities fails.},
author = {E. Ballico},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {},
language = {eng},
number = {1},
pages = {null},
title = {On the birational gonalities of smooth curves},
url = {http://eudml.org/doc/289735},
volume = {68},
year = {2014},
}

TY - JOUR
AU - E. Ballico
TI - On the birational gonalities of smooth curves
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2014
VL - 68
IS - 1
SP - null
AB - Let $C$ be a smooth curve of genus $g$. For each positive integer $r$ the birational $r$-gonality $s_r(C)$ of $C$ is the minimal integer $t$ such that there is $L\in \mbox{Pic}^t(C)$ with $h^0(C,L) =r+1$. Fix an integer $r\ge 3$. In this paper we prove the existence of an integer $g_r$ such that for every integer $g\ge g_r$ there is a smooth curve $C$ of genus $g$ with $s_{r+1}(C)/(r+1) > s_r(C)/r$, i.e. in the sequence of all birational gonalities of $C$ at least one of the slope inequalities fails.
LA - eng
KW -
UR - http://eudml.org/doc/289735
ER -

References

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  1. Coppens, M., Martens, G., Linear series on 4-gonal curves, Math. Nachr. 213, no. 1 (2000), 35–55. 
  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, Séminaire de Mathématiques Supérieures, 85, Presses de l’Université de Montréal, Montréal, 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. 
  6. Lange, H., Martens, G., On the gonality sequence of an algebraic curve, Manuscripta Math. 137 (2012), 457–473. 

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