The existence of \(k\)-gonal curves possessing exactly two linear systems \(g_ k^ 1\) (Q1359529)

The main theorem proved in this paper is the following. For all integers \(g\) and \(k\) satisfying the inequalities \(4\leq k\leq[(g+1)/2]\) and \(8\leq g\leq(k-1)^2\) there exists a \(k\)-gonal curve of genus \(g\) possessing exactly 2 independent linear systems \(g^1_k\) of type I and having no other linear systems \(g^1_k\). Taking into account that there is no 4-gonal curve of genus 7 having exactly two linear systems \(g_4^1\) both being of type I, this result is complete. The proof of the theorem is by means of induction on \(g\). The induction starts with a general \((k-1)\)-gonal curve of genus \(2k-1\).