Hi all. I'm looking for an example of a smooth projective surface $X$ and a pseudo-effective divisor $D$ on $X$ such that when I consider the Zariski decomposition $D=P+N$ there is some component $E$ of the negative part $N$ such that $(D\cdot E)\geq 0$.
Can you help me? Thank you
Gianni



Divisor intersecting non-negatively the negative part of its Zariski decomposition
EDIT: New example, hopefully this one works.


Blow up $\mathbb{P}^2$ at a point $p$, then blow up the resulting surface at a point $q$ on the exceptional divisor. The resulting surface has Picard group generated by the class $H$ of a line, the proper transform $E\_1$ of the first exceptional divisor, and the second exceptional divisor $E\_2$. We have $E\_1^2 = -2$ and $E\_2^2 = -1$, while $E\_1\cdot E\_2 = 1$. Now consider $D = E\_1 +2E\_2$. The Zariski decomposition of $D$ is $P = 0$, $N = E\_1+2E\_2$, as no effective divisor supported on $E\_1$ and $E\_2$ is nef. Then $D\cdot E\_1 = 0$. We can get strict inequality instead by taking $D = E\_1+3E\_2$, for instance. 


