There are called 'leech tension lines' or 'leech lines'.
Their purpose is to prevent the leech from 'fluttering' or vibrating. Such vibrations will quickly 'fatigue' the fabric and stitching of the leech leading to 'premature destruction'.
They are especially important to control leech flutter between the battens (typically between the 'topmost' batten and #2 on the mainsail).
How to adjust: LOOSEN the leech line until the leech 'just begins' to vibrate, then tighten 'slightly' beyond the tension that 'stops' the flutter/vibration.
How to recognize that a leech line is too tight: The seam/fold along the leech will NOT be smooth and NOT be parallel to normal airstream flow, it may be 'hooked up' to weather (severely 'curved' leech).