On V-Weierstrass sets and gaps

Let X be a nonsingular projective curve of genus g⩾1 defined over a field F which is assumed to be the full field of constants of F(X), and let V be a linear system of divisors on X. We define the set G(V;P1,…,Pn) of V-Weierstrass gaps at the rational points P1,…,Pn of X as being formed by the n-tuples α=(α1,…,αn) such that V(α)⊊V(α−ej) for some j∈{1,…,n} (here and ej=(δji)1⩽i⩽n∈Nn). We call Nn∖G(V;P1,…,Pn) the V-Weierstrass set associated to V and P1,…Pn. The main result of this paper proves a sharp lower bound for #G(V;P1,…,Pn).