A characterization of double coverings of curves

The gonality sequence ({(d_{r})_{r}}) of a curve X of genus g which doubly covers a curve of genus h satisfies ({d_{r} = 2(r + h)}) for all ({r = h, h + 1, ldots, g - 3h}) provided that ({g gg h}). In this paper we explore if this striking feature of ({(d_{r})_{r}}) actually characterizes such a covering.