The Lüroth semigroup of plane algebraic curves

S. Greco and G. Raciti

Vol. 151 (1991), No. 1, 43–56

DOI: 10.2140/pjm.1991.151.43

Abstract

A “gap” for a smooth irreducible complete algebraic curve C is a non-negative integer n such that no rational function on C has degree n. The non-gaps form the so called “Lüroth semigroup” of C.

We give methods to find gaps and non-gaps when C is a plane curve of degree d, based on properties of linear series and Hilbert functions. It turns out that for d ≤ 14 the Lüroth semigroup depends only on d; and for larger d we point out where two curves might have different gaps. Bounds are also given for the conductor of the Lüroth semigroup, depending on d.

Mathematical Subject Classification 2000

Primary: 14H05

Secondary: 20M99

Milestones

Received: 28 February 1990

Published: 1 November 1991