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A variant of a base-point-free pencil trick and linear systems on a plane curve. (English) Zbl 0951.14019
Let $$X$$ be a smooth plane curve of degree $$d\geq 4$$ and $$L$$ an invertible sheaf of degree $$sd-e$$ with $$1\leq s<d-3$$ and $$0\leq e<d$$ on $$X$$. Noether’s theorem gives an upper bound of the dimension of the linear system $$|L|$$, namely $\dim|L|\leq r(s,e):=\begin{cases} {1\over 2}s (s+1)- 1\quad & \text{if } s+1\leq e<d\\ {1\over 2}(s+1)(s+2)-e-1 \quad & \text{if } 0\leq e\leq s+1. \end{cases}$ The paper considers the case when the upper bound $$r(s,e)$$ is attained. In fact, with the help of a modified base-point free pencil trick (to be found in section 1) the authors derive an explicit description of those sheafs $$L$$ with $$\dim|L|= r(s,e)$$.
##### MSC:
 14H51 Special divisors on curves (gonality, Brill-Noether theory) 14C21 Pencils, nets, webs in algebraic geometry 14H45 Special algebraic curves and curves of low genus