Leech line?
If this line is like dacron twine running up inside the leech, it's the leech line. Tensioning this line prevents the leech from fluttering (or "motorboating" because of the sound it makes) between the battens. This fluttering is not good, as it wears the sail fabric faster. There should be a flat nylon cleat on the leech just above the end of the boom to secure the line. Most headsails have a similar line and cleat for the same purpose.PeterH23 "Raven"