Non‐trivial Linear Systems on Smooth Plane Curves

Let $C$ be a smooth plane curve of degree $d$ defined over an algebraicallyclosed field $k$. A base point free complete very special linear system $g^r_n$on $C$ is trivial if there exists an integer $m\ge 0$ and an effective divisor$E$ on $C$ of degree $md-n$ such that $g^r_n=|mg^2_d-E|$ and$r=(m^2+3m)/2-(md-n)$. In this paper, we prove the following: Theorem Let$g^r_n$ be a base point free very special non-trivial complete linear system on$C$. Write $r=(x+1)(x+2)/2-b$ with $x, b$ integers satisfying $x\ge 1, 0\le b\le x$. Then $n\ge n(r):=(d-3)(x+3)-b$. Moreover, this inequality is bestpossible.