Weierstrass Gap Sets for Quadruples of Points on Compact Riemann Surfaces

Let M be a compact Riemann surface of genus g, and let P1,…,P4 be distinct points on M. We study the Weierstrass gap set G(P1,…,P4) and prove the conjecture of Ballico and Kim on the upper bound of #G(P1,…,P4) affirmatively in case M is d-gonal curve of genus g ≥ 5 with d = 2 or d ≥ 5.