modified nef cone, with extremal rays that write ${\mathbf{R}}_+\{D\}$ for some exceptional prime $D$ of $X$. The Zariski projection $Z$ yields a projection from the pseudo-effective cone to the modified nef cone parallel to these exceptional rays, which map is concave (in some sense) and homogeneous, but not continuous up to the boundary of the pseudo-effective cone in general. The fibre $Z^{-1}(p)$ of $Z$ above a modified nef class $p$ is a countable union of simplicial cones generated by exceptional families of primes.\
When $X$ is a surface, a modified nef class is just a nef class; when $\alpha$ is the class of an effective ${\mathbf{Q}}$-divisor $D$ on a projective surface, the divisorial Zariski decomposition of $\alpha$ is just the original Zariski decomposition of $D$. More generally, we show that the divisorial decomposition of a pseudo-effective class $\alpha$ on a Kähler surface is the unique orthogonal decomposition of $\alpha$ into the sum of a modified nef class and the class of an exceptional (in some sense) effective ${\mathbf{R}}$-divisor. This fact accounts for the rationality of the Zariski decomposition on a surface, meaning that the negative part $N$ is rational when $D$ is.\
An interesting fact is that much of the 