Abstract. We study the cones of q-ample divisors qAmp. In favourable cases, we identify a part where the closure qAmp and the nef cone have the same boundary. This is especially interesting for Fano (or almost Fano) varieties. IntroductionTotaro's landmark paper [22] has given a new impetus to the study of partially ample divisors. Let X be a smooth projective complex variety of dimension n, and let L on X be a line bundle. We recall that L is called q-ample if for every coherent sheaf F there exists an integer m 0 such that (X) in N 1 (X) (the space of R-divisors modulo numerical equivalence). We thus obtain a series of conesWhile the ample cone Amp(X) and the cone (n − 1)Amp(X) are fairly well understood, the intermediate cones qAmp(X) for 0 < q < n − 1 are still quite elusive and mysterious (see for instance [22, Section 11] for some fundamental open questions). The modest goal of this paper is to identify a part of these cones qAmp. Indeed, it turns out that in favourable cases, part of the boundary of the closed cone qAmp coincides with the boundary of the nef cone. To start with, let us restrict attention to the case that is easiest to state, that of the cone of 1-ample divisors 1Amp. Let ∂Nef(X) denote the boundary of the nef cone, and let K X ∈ N 1 X denote the class of the canonical divisor. We defineto be the part of the boundary that is visible from K X ; cf. Definition 17 for the precise definition. (We note that when K X is nef, we have ∂Nef(X) visible = ∅!)This 'K X -visible part' of the boundary turns out to be closely related to the boundary of 1Amp(X). This is detailed in the following result, where Mob(X) and Big(X) denote the cone of mobile divisors and big divisors, respectively.