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The existence of base point free linear systems on smooth plane curves. (English) Zbl 0842.14020

One defines the Lüroth semigroup \(\text{LS} (C)\) of a smooth projective irreducible curve \(C\) defined over an algebraically closed field \(k\) as follows:
\(\text{LS} (C)= \{0\} \cup \{d\in \mathbb{N}\): there exists a base point free linear system \(g^1_d\) on \(C\}\).
In this paper we determine the Lüroth semigroup of smooth plane curves. So now let \(C\) be a smooth plane curve of degree \(d\). \(\text{LS} (C)\) is determined in case \(d\leq 14\) [cf. S. Greco and G. Raciti, Pac. J. Math. 151, No. 1, 43-56 (1991; Zbl 0691.14002)]. A very important fact is \(e\not\in \text{LS} (C)\) if \((n-1)d+ 1\leq e\leq nd- (n^2+ 1)\) for some \(n\in \mathbb{N}_0\). In the paper under review we prove that the other positive integers belong to \(\text{LS} (C)\).
Main result. \(\text{LS} (C)\) is the complement in \(\mathbb{N}\) of \(\{1; 2; \dots; d-2\} \cup \{d+ 1; d+ 2; \dots; 2d-5\} \cup\dots \cup \{(n-1) d+1; (n-1)d+2; \dots; nd-(n^2+ 1)\}\) where \(n\) is the integer part of \((d- 2)^{1/2}\).

MSC:

14H45 Special algebraic curves and curves of low genus
14C20 Divisors, linear systems, invertible sheaves
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