Nontrivial Linear Systems on Smooth Plane Curves
Abstract
Let $C$ be a smooth plane curve of degree $d$ defined over an algebraically closed 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 $mdn$ such that $g^r_n=mg^2_dE$ and $r=(m^2+3m)/2(mdn)$. In this paper, we prove the following: Theorem Let $g^r_n$ be a base point free very special nontrivial complete linear system on $C$. Write $r=(x+1)(x+2)/2b$ with $x, b$ integers satisfying $x\ge 1, 0\le b \le x$. Then $n\ge n(r):=(d3)(x+3)b$. Moreover, this inequality is best possible.
 Publication:

arXiv eprints
 Pub Date:
 January 1993
 arXiv:
 arXiv:alggeom/9301003
 Bibcode:
 1993alg.geom..1003C
 Keywords:

 Mathematics  Algebraic Geometry
 EPrint:
 15 pages, LaTeX 2.09