modified nef cone and the exceptional divisors; our divisorial Zariski decomposition is orthogonal, and is thus a rational decomposition, which fact accounts for the usual existence statement of a Zariski decomposition on a projective surface, which is thus extended to the hyper-Kähler case. Finally, we explain how the divisorial Zariski decomposition of (the first Chern class of) a big line bundle on a projective manifold can be characterized in terms of the asymptotics of the linear series $|kL|$ as $k\to\infty$.\
2000 Mathematics Subject Classification: 32J25

Introduction
============

It is known since the pioneering work of O.Zariski \[Zar62\] that the study of the linear series $|kL|$ where $L$ is a line bundle on a projective surface can be reduced to the case where $L$ is numerically effective (nef). The more precise result obtained by Zariski is that any effective ${\mathbf{Q}}$-divisor $D$ on a projective surface $X$ can be uniquely decomposed into a sum $D=P+N$ where $P$ is a nef ${\mathbf{Q}}$-divisor (the positive part), $N=\sum a_j D_j$ is an effective ${\mathbf{Q}}$-divisor (the negative part) such that the Gram matrix $(D_i\cdot D_j)$ is negative definite, and $P$ is orthogonal to $N$ with respect to the intersection form. Zariski shows that the natural inclusion 