A geometric characterization of normal two-dimensional singularities of multiplicity two with \(p_ a\leq 1\)
The goal of this paper is the following theorem: A normal two-dimensional singularity (V,p) of multiplicity two over \({\mathbb{C}}\) satisfies the condition \(p_ a(V,p)\leq 1\) if and only if each normalization in Zariski's canonical resolution of (V,p) is trivial or obtained by blowing-up along (reduced) \({\mathbb{P}}^ 1\) in the singular locus. This is an analogy of the well-known characterization of the rational double point by the absolute isolatedness. Our result is proved as a corollary of four lemmas; an inequality about \(p_ a\), a formula for computation of the geometric genus due to \textit{E. Horikawa} [Invent. Math. 31, 43-85 (1975; Zbl 0317.14018)], a criterion for the condition \(p_ a\leq 1\) due to \textit{S. S.-T. Yau} [Trans. Am. Math. Soc. 257, 269-329 (1980; Zbl 0465.32008)], and an elementary computation on one-dimensional singularity with multiplicity \(\leq 5\). The effective use of the arithmetic genus \(p_ a\) of the singularity is new at all.
scientific article; zbMATH DE number 3867505
scientific article