On the birational gonalities of smooth curves

E. Ballico

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.

Full Text:

PDF

References


Coppens, M., Martens, G., Linear series on 4-gonal curves, Math. Nachr. 213, no. 1 (2000), 35–55.

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.

Harris, J., Eisenbud, D., Curves in projective space, Séminaire de Mathématiques Supérieures, 85, Presses de l’Université de Montréal, Montréal, Que., 1982.

Hatshorne, R., Algebraic Geometry, Springer-Verlag, Berlin, 1977.

Laface, A., On linear systems of curves on rational scrolls, Geom. Dedicata 90, no. 1 (2002), 127–144; generalized version in arXiv:math/0205271v2.

Lange, H., Martens, G., On the gonality sequence of an algebraic curve, Manuscripta Math. 137 (2012), 457–473.




DOI: http://dx.doi.org/10.17951/a.2014.68.1.11
Data publikacji: 2015-05-23 16:29:36
Data złożenia artykułu: 2015-05-04 21:16:17

Refbacks

  • There are currently no refbacks.


Copyright (c) 2015 Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica