On surface singularities of multiplicity three

Let $P$ be a normal singularity of multiplicity $d = 2$ or $3$ of a complex surface $X$. It is well-known that $X$ is locally an irreducible finite cover $\pi : X \to Y$ of degree $d$ over a smooth surface $Y$, and the singularity $(X, P)$ can be resolved by the canonical resolution $X_k \to X_{k-1} \to \dots \to X_0 = X$, which is the pullback of the embedded resolution of the corresponding singularity $p = \pi (P)$ of the branch locus. Let $F$ be the maximal ideal cycle of this resolution. We will prove that $F$ has a unique decomposition $F = Z_1 + \dots + Z_d$ with $Z_1 \geq Z_2 \geq \dots \geq Z_d \geq 0$, where $Z_i$ is a fundamental cycle or zero. We show that $w = p_a (Z_1) + \dots + p_a (Z_d)$ is an invariant of $(X, P)$ that can also be computed from the multiplicity of the branch locus at $p$. $(X, P)$ is a rational singularity iff all of the singular points in the canonical resolution satisfies $w \leq d - 1$. In order to get the minimal resolution from the canonical one, we need to blow down some exceptional curves, the number of blowing-downs is exactly that of fundamental cycles $Z$ in the canonical resolution satisfying $p_a (Z) = 0$ and $Z^2 = -1$.

Keywords

Jung’s resolution, canonical resolution, fundamental cycle, surface singularity, triple cover

2010 Mathematics Subject Classification

Primary 14E15. Secondary 14B05, 32S45.

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Published 9 October 2014