Mori cone: extremal ray intersections On an algebraic surface, much can be said about the Mori cone, or cone of curves. In this question, I will be particularly interested in intersection properties.
Several sweeping statements can be made about the self-intersections of the extremal rays. For example, they must all have non-positive self-intersection.
My question is: what can be said about intersections between distinct extremal rays?
Trivially, these intersections must be non-negative. But I suspect there is much more one can say. For example, here are some obvious questions.


*

*Are the distinct intersections bound, for example to be less than or equal to 1?

*Can an extremal ray have non-zero intersection with arbitrarily many other extremal rays?
One can think of many other questions. For an answer, I would be very happy to simply be pointed to a relevant reference.
 A: I don't know a specific reference for thid sort of question, so let me just address your two example questions. (Everything I say is discussed in the books of  Kollár--Mori, Lazarsfeld, Debarre, etc.)
Let $X$ be the blowup of $\mathbf P^2$ in 9 very general points. Then the Mori cone of $X$ is well-known: its extremal rays are


*

*the class $-K$ of the anticanonical divisor (i.e. the proper transform of the unique cubic through the 9 points)

*the 9 exceptional divisors $E_1,\ldots,E_9$

*the images of the $E_i$ under the action of the Cremona group: it doesn't matter what these look like, only that there are infintely many of them.
We can write any class on $X$ in the form
$$ aH + \sum_{i=1}^9 b_i E_i $$
for some $a, \, b_i \in \mathbf Z$, and intersections with the exceptionals are given by $( aH + \sum_{i=1}^9 b_i E_i) \cdot E_j = -b_j$. (For classes of the third kind above other than the $E_i$ themselves, the coefficients $b_i$ are therefore nonpositive. 
Since there are infinitely many classes of the third kind above, the corresponding coefficients $(b_1,\ldots,b_9)$ cannot be bounded, and so the answer to 

Are the distinct intersections bound, for example to be less than or equal to 1?

is no. 
On the other hand if we blow up 8 or fewer points in $\mathbf P^2$, we get a del Pezzo surface whose Mori cone has finitely many extremal rays. That means that of the infinite set of classes above, there are only finitely many in which one of the coefficients $b_i$ equals zero. So back on $X$ a given exceptional $E_i$ has nonzero intersection with all but finitely many extremal rays. Therefore the answer to 

Can an extremal ray have non-zero intersection with arbitrarily many other extremal rays?

is yes. 
Finally let me mention an obvious example in which one has positive answers to your questions: if $X$ is a nonsingular toric surface, then any extremal ray of the Mori cone is spanned by a torus-invariant curve. Two such curves $C_1$ and $C_2$ intersect as follows:


*

*if the corresponding rays span a cone of the fan of $X$, then $C_1 \cdot C_2=1$;

*otherwise $C_1 \cdot C_2=0$.  


So intersections are bounded by 1, and each ray has nonzero intersection with at most 2 other rays. 
