On the birational gonalities of smooth curves

ArticleOriginal scientific text

Title

Authors

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 {Pic}^{t} (C)

with

h^{0} (C, L) = r + 1

. Fix an integer

r \geq 3

. In this paper we prove the existence of an integer

g_{r}

such that for every integer

g \geq 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.

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.