Stable cohomology of discriminant complements for an algebraic curve

Let [Formula: see text] be a degree [Formula: see text] line bundle on a smooth projective complex algebraic curve [Formula: see text] of genus [Formula: see text]. Let [Formula: see text] (respectively, [Formula: see text]) denote the space of algebraic (respectively, [Formula: see text]) sections of [Formula: see text] with isolated regular zeroes. We show that (1) The inclusion [Formula: see text] induces an isomorphism between [Formula: see text] and [Formula: see text] for [Formula: see text]. (2) [Formula: see text] is aspherical. (3) [Formula: see text] is a group closely related to the surface braid group [Formula: see text].