Smooth \(k\)-gonal curves with another fixed pencil (Q2368626)

Let \(M_g\) be the moduli scheme of complex smooth complete curves of genus \(g\geq 4\). For all integers \(k, m\), with \(k\leq m\leq g/2\), let \(U(k, m; g)\) be the constructive subset of \(M_g\) parametrizing the curves, \(X\), with a base point free and complete pencils \(g_k^1\) and \(g_m^1\) such that the morphism \(X\rightarrow {\mathbb P}^1\times {\mathbb P}^1\) induced by \(g_k^1\) and \(g_m^1\) is birational. It is easily seen that \(U(k, m; g)=\emptyset\) if \(g>(m-1)(k-1)\). The author proves the following: Theorem: Let \(6\leq g\leq (m-1)(k-1)\) and \(k\geq 4\). Set \(n:=(m-1)(k-1)-g\). Then there exists \(C\in U(k, m; g)\) with \(g_k^1\) and \(g_m^1\) as linear systems such that for all nonnegative integers \(u\) and \(r\) one has \[ \dim|rg_k^1+ug_m^1|=(r+1)(u+1)+n-(k-u-1)(m-r-1)+\max\{0,(k-u-1)(m-r-1)-n. \]