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?

  • $\begingroup$ What is this statement about signature 1? What if you take two disjoint $(-1)$-curves? $\endgroup$
    – abx
    Apr 23 '19 at 5:32
  • $\begingroup$ Oh, I meant at most one positive coordinate, it is the direct implication of Hodge's theorem on index. I will fix it now. $\endgroup$ Apr 23 '19 at 6:21
  • $\begingroup$ This is known as "positive index of inertia"; the signature is the difference of the positive and negative indices. $\endgroup$ Apr 23 '19 at 14:36
  • $\begingroup$ The question is very vague. Is the surface fixed? Are the curves assumed to be irreducible? What does "a couple of curves" mean? Do you want to record the intersection indices or just geometric intersections in your "graph/matrix of incidence"? Etc. $\endgroup$ Apr 23 '19 at 14:49
  • $\begingroup$ Surface not fixed. I would like curves to be smooth, and I think geometric intersections coincide with intersection indices due to transversality assumption. $\endgroup$ Apr 23 '19 at 15:06

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.

  • $\begingroup$ Thank you for linking this - I was aware of this paper but I guess I forgot about it somehow. There is indeed a way to prove this easier for surfaces - consider n rational curves of high enough degree in general position and blow up the points you don't like. My question seems to be more about canonical class question in Kollar's paper - what are possible self-intersections for a given graph. $\endgroup$ Apr 23 '19 at 18:41
  • $\begingroup$ You're welcome! Unfortunately I don't think I can be of help with the canonical/self intersections in this case. Best of luck with it! $\endgroup$
    – cgodfrey
    Apr 23 '19 at 19:45

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