Weierstrass gaps at \(n\) points of a curve (Q2253024)

For a smooth geometrically connected curve \(C\) and \((P_1,\dots,P_n)\) distinct points on \(C\), let \(H(P_1,\dots,P_n)\) denote the subset of \(\mathbb N^n\) formed by all \((k_1,\dots,k_n)\) in \(\mathbb N^n\) such that there exists a rational function in \(K(C)\) with polar divisor \(k_1P_1+\dots+k_nP_n\). In addition let \(G(P_1,\dots,P_n)\) be the gap set of the points \((P_1,\dots,P_n)\) defined as the complement \(\mathbb N^n \setminus H(P_1,\dots,P_n)\) (see [\textit{E. Arbarello} et al., Geometry of algebraic curves. Volume I. New York etc.: Springer-Verlag (1985; Zbl 0559.14017)]; [\textit{E. Ballico} and \textit{S. J. Kim}, J. Algebra 199, No. 2, 455--471, Art. No. JA977166 (1998; Zbl 0909.14019)]). In the present paper the author studies the gap set \(G(P_1,\dots,P_n)\) as well as the gap set with \((k_1,\dots,k_n)\) in \(\mathbb Z^n\) as defined in [\textit{P. Beelen} and \textit{N. Tutaş}, J. Pure Appl. Algebra 207, No. 2, 243--260 (2006; Zbl 1105.14045)] over an algebraically closed field \(K\) with \(\mathrm{char} K =0\) or \(\mathrm{char} K \geq 2g-1\).