The Lüroth semigroup of plane algebraic curves

S. Greco, G. Raciti
1991 Pacific Journal of Mathematics  
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 Hubert functions. It turns out that for d < 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
more » ... nds are also given for the conductor of the Luroth semigroup, depending on d . Introduction. The Luroth semigroup (LS) of a smooth irreducible complete algebraic curve C is the additive semigroup Sc containing all the degrees of the rational functions of C (equivalently: the degrees of the linear series on C without base points). As such it was introduced by Heinzer and Moh [HM], but the problems related to it are as old as the theory of curves: indeed the knowledge of the degrees of the rational functions on C is a remarkable step towards the knowledge of the geometry of C. A systematic account on Sc is available only in a few cases, namely curves with general moduli, hyperelliptic curves, and plane curves up to degree 9 (see 1.3 below). The main purpose of this paper is to study Sc for a plane curve C of degree d > 4 (the situation being trivial for d < 3). After some preliminaries collected in §1, we prove, in §2, that if a curve carries a vety ample linear series of degree m and dimension r, then all the integers n such that m-r+l<n<m belong to its LS (Corollary 2.2). This fact follows from a theorem of Bertini (proved in arbitrary characteristic by Laksov [L]), and implies that, if C is as above, then Sc contains all integers n such that ad -a(a + 3)/2 + 1 < n < ad for a e N and 1 < a < d -1 (Corollary 2.6). In §3 we show that no integer n with (a -\)d < n < ad -a 2 {a € N) can belong to Sc (Theorem 3.1). To prove this we use some results of Davis [D] to study the Hubert function of a zerodimensional subscheme of P 2 contained in C in this way we can 43 44
doi:10.2140/pjm.1991.151.43 fatcat:6sxl2ycpnvf4zl2rrpnzzvndoy