Construction of log canonical singularity

Question

I know there's classification about normal log canonical surface singularity in the sense of configuration of exceptional curves. There is one type of log canonical singularity(not klt) whose resolution is a string with two additional rational curves attached to each end. I wonder how to construct a singularity with the exceptional curves I described above? According to some reference, it's uniformized by cusp log canonical singularity whose exceptional curves are a circle of rational curves. Thanks for the help or any detailed reference.

Answer 1

For a given weighted graph whose associated intersection matrix is negative definite, it is relatively easy to construct a normal surface singularity whose resolution graph is that one. Whether it is log canonical or not is determined by the graph.

To construct a concrete singularity with a given resolution graph one could do the following. First get a collection of curves in the right configuration. Let me use the example you asked about, but I think the method can be tweaked to get just about any log canonical singularity.

Start with two transversal conics in $\mathbb P^2$, $C_1,C_2$ and their tangent lines, $L_1,L_2$ in one of the intersection points. Blow up the four intersection points and let $E_0$ be the exceptional curve lying over the point $C_1\cap C_2\cap L_1\cap L_2$. For simplicity I will use the same notation for the strict transform of every curve on any blow ups appearing here.

So after blowing up the four intersection points we have (the strict transforms of) the curves $C_1,C_2,L_1,L_2,E_0$ such that $C_1$ and $L_1$ intersect $E_0$ in the same point pairwise transversally and the same for $C_2$ and $L_2$, but $C_1$ does not intersect $C_2$ or $L_2$ and symmetrically for $C_1$. Now $E_0$ is a $(-1)$-curve and the other four curves each has self-intersection $0$. So, now if we blow up the points $C_1\cap L_1\cap E_0$ and $C_2\cap L_2\cap E_0$, with exceptional curves $E_1$ and $E_2$, then we get a chain of three curves $E_1,E_0,E_2$ where the outside ones are $(-1)$-curves and the middle one is a $(-3)$-curve, and there are two $(-2)$-curves hanging off $E_1$ and $E_2$ respectively. To get a longer chain you can blow up the intersection points $E_1\cap E_0$ and/or $E_2\cap E_0$ and iterate it.

Now this gives you already the incidence graph you are looking for. To adjust the intersection numbers, you can blow up a general point on any curve to lower their self-intersection and you can blow down attached $(-1)$-curves intersecting only one curve to increase the self-intersection of that curve.

The first thing is easy. For the second one you need to find such a curve. In this example you can take the proper transform of a line through $C_1\cap C_2\cap L_1\cap L_2\subseteq \mathbb P^2$ and one of the other original intersection points of the conics. That will become a $(-1)$-curve hanging off $E_0$. Similarly, conics going through $C_1\cap L_2$ and $C_2\cap L1$ and two additional original intersection points of the conics will have the same property. So this gives you a bunch of $(-1)$-curves to blow down in case you need it. To give an air-tight argument at this point one would need to show that you will always have enough $(-1)$-curves hanging that you can blow down. In order to do this the major question is what if you ran out of the types describes above. After all there are only a finite number of them. In that case you can take another conic, transversal to all of the previous ones, but going through the quadruple intersection point that was blown-up first. Now blowing up the intersection points of this one with the other conics will lower the self-intersection of $C_1$ and $C_2$, but there will be more $(-1)$-curves available to blow down in case these get too negative. Of course, this is not a complete proof that this can be done in every case, but it seems that the fact that your intersection matrix has to be negative definite should help.

So, this way you get a collection of curves with the prescribed weighted graph (i.e., the desired incidence relations and self-intersections). Then by Artin's criterion this union of curves is contractible and the result is the desired singularity.

This gives you at least one example of what you wanted. I think that a variant of this method should allow you to construct most log canonical surface singularities that are rational. If you cannot get one this way you can try to start with a different surface and for a non-rational singularity you might need to use some elliptic curves (or a cycle of rational ones).