THE LUROTH SEMIGROUP OF PLANE ALGEBRAIC-CURVES

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 "Luroth 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 less-than-or-equal-to 14 the Luroth 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 Luroth semigroup, depending on d.