The Lüroth semigroup of plane algebraic curves
Let \(C\) be a smooth complete irreducible algebraic curve over an algebraically closed field. The Lüroth semigroup of \(C\) is the additive semigroup \(S_ C:=\{n\in {\mathbb{N}}|\) there is a rational function of degree n on \(C\}\) [see \textit{W. Heinzer} and \textit{T. T. Moh}, J. Algebra 77, 62-73 (1982; Zbl 0533.14010)]. If \(C\) is a plane curve of degree \( d\geq 4\) and let a be an integer with \(1\leq a\leq d-1\), then the following are proved: (1) If \(n\) is an integer such that \(ad\geq n\geq ad+1-a(a+3)/2\), then \(n\in S_ C.\) (2) There is at least one integer \(n\in S_ C\) such that \(ad-a(a+3)/2\geq n\geq ad-a_ 2\). In particular \(3d-9\in S_ C.\) (3) If n is an integer such that \(ad-a^ 2+1\geq n\geq (a-1)d+1\), then \(n\not\in S_ C.\) From the above one can get lower and upper bounds for the conductor of \(S_ C\) and one can compute exactly \(S_ C\) for all plane curves of degree \(d\leq 14\), thus extending results in the paper cited above (first unsettled case: is \(46\in S_ C\) for all plane curves of degree 15 ?). The proofs of (1) and (2) consist in constructing base point free linear series by using linear systems of plane curves of degree a with a suitable number of base points on \(C\). The proof of (3) is achieved by a careful study of the Hilbert function of the closed zero-dimensional subschemes of \(C\) having degree n.
constructing base point free linear series
scientific article; zbMATH DE number 65397