On the effective cone of the moduli space of pointed rational curves

For a smooth projective variety, Kleiman’s criterion for ample divisors states that the closed ample cone (i.e., the nef cone) is dual to the closed cone of effective curves. Since the work of Mori, it has been clear that extremal rays of the cone of effective curves play a special role in birational geometry. These correspond to certain distinguished supporting hyperplanes of the nef cone which are negative with respect to the canonical class. Contractions of extremal rays are the fundamental operations of the minimal model program. Fujita [F] has initiated a dual theory, with the (closed) cone of effective divisors playing the central role. It is natural then to consider the dual cone and its generators. Those which are negative with respect to the canonical class are called coextremal rays, and have been studied by Batyrev [Ba]. They are expected to play a fundamental role in Fujita’s program of classifying fiber-space structures on polarized varieties. There are relatively few varieties for which the extremal and coextremal rays are fully understood. Recently, moduli spaces of pointed rational curves M 0,n have attracted considerable attention, especially in connection with mathematical physics and enumerative geometry. Keel and McKernan first considered the ‘Fulton conjecture’: The cone of effective curves of M 0,n is generated by one-dimensional boundary strata. This is proved for n ≤ 7 [KeMc]. The analogous statement for divisors, namely, that the effective cone of M 0,n is generated by boundary divisors, is known to be false ([Ke] and [Ve]). The basic idea is to consider the map