Possible configurations of rational curves on a rational surface Consider a set of smooth rational curves on a rational surface, say, with normal crossings between curves. Is anything known on what combinatorics of configurations are possible?
Say, what obstructions are known to finding the rational curves with fixed graph of incidence and fixed self-intersections numbers?
I understand that its matrix of incidence should be embeddable into $\mathbb{Z}^{n+1}$ with standard form of signature $(1, n)$, (in particular, have s̶i̶g̶n̶a̶t̶u̶r̶e̶ ̶1̶  index at most 1). Are there any other obstructions?
 A: This question is addressed in Kollár's paper
Simple normal crossing varieties with prescribed dual
complexes. For a variety $X$
with a (reduced, effective) simple normal crossing divisor $\Delta \subset
X$, write $\Delta = \bigcup_{i} D_{i}$ where $D_{i} \subset D$ are the
components. The dual complex $D(X, \Delta)$ is the simplicial complex
with 


*

*0-cells corresponding to the components $D_{i}$

*1-cells corresponding to the non-empty intersections
$D_{i} \cap D_{j}$

*2-cells corresponding to the non-empty intersections $D_{i} \cap
D_{j} \cap D_{k}$
and so on. (Part of) theorem 1 of that paper shows that given any
simplicial complex $\mathcal{C}$ of dimension $n-1$ there is a smooth,
projective, rational variety $X$ and a simple normal crossing divisor
$\Delta \subset X$ with rational components so that $D(X, \Delta)
\simeq C$ (isomorphism, not homotopy equivalence!).
When $X$ is a surface, the triple intersections are automatically
empty, so $D(X, \Delta)$ is a graph, the dual graph if you
will. So in this case the theorem says any 1-dimensional simplicial
complex can be realized as the dual graph of a simple normal crossing
divisor with rational components on a smooth rational surface. 
Remark: the theorem I cited holds in all dimensions; it's
quite possible that there's an easier way to prove the result for
surfaces. 
