The nef cone of divisors is an important invariant of a scheme as it records its birational geometry. For some time it has been known that the nef cone of \overline{M}_{g,n}, the moduli space of stable n-pointed curves of genus g, is contained in a polyhedral cone and the F-conjecture asserts that this is in fact the nef cone. Moreover, it is known that if the conjecture holds on \overline{M}_{0,m} for m=g+n then it holds for \overline{M}_{g,n}. In this talk I will describe a polyhedral subcone of the nef cone of \overline{M}_{0,m}. These cones are exactly the same for m at most 6 and we have reason to believe they are always the same. (This is joint work with Diane Maclagan.)