ON THE VARIETY OF SPECIAL LINEAR SYSTEMS OF DEGREE g-1 ON SMOOTH ALGEBRAIC CURVES

Abstract

We classify smooth projective algebraic curves C of genus g such that the variety of special linear systems [Formula: see text] has dimension g- 7. We first prove that if [Formula: see text] has dimension g-7≥0 then C is either trigonal, tetragonal, a double covering of a curve of genus 2 or a smooth plane sextic. This result establishes the next extension of dimension theorems of H. Martens and D. Mumford on the variety of special linear systems with the fullest possible generality. We then proceed to show that, under the assumption g≥11, [Formula: see text] has dimension g- 7 if and only if C is either a trigonal curve or a double covering of a curve of genus 2.