The end curve theorem for normal complex surface singularities

Abstract

Abstract. We prove the "End Curve Theorem," which states that a normal surface singularity (X, o) with rational homology sphere link is a splice quotient singularity if and only if it has an end curve function for each leaf of a good resolution tree.An "end curve function" is an analytic function (X, o) → (C, 0) whose zero set intersects in the knot given by a meridian curve of the exceptional curve corresponding to the given leaf.A "splice quotient singularity" (X, o) is described by giving an explicit set of equations describing its universal abelian cover as a complete intersection in C t , where t is the number of leaves in the resolution graph for (X, o), together with an explicit description of the covering transformation group.Among the immediate consequences of the End Curve Theorem are the previously known results: (X, o) is a splice quotient if it is weighted homogeneous (Neumann 1981), or rational or minimally elliptic (Okuma 2005).Keywords. Surface singularity, splice quotient singularity, rational homology sphere, complete intersection singularity, abelian cover, numerical semigroup, monomial curve, linking pairing We consider normal surface singularities whose links are rational homology spheres (QHS for short). The QHS condition is equivalent to the condition that the resolution graph of a minimal good resolution be a rational tree, i.e., is a tree and all exceptional curves are genus zero.Among singularities with QHS links, splice quotient singularities are a broad generalization of weighted homogeneous singularities. We recall their definition briefly here and in more detail in Section 1. Full details can be found in [20].Recall first that the topology of a normal complex surface singularity is determined by and determines the minimal resolution graph . Let t be the number of leaves of . For i = 1, . . . , t, we associate the coordinate function x i of C t to the i-th leaf. This leads to a natural action of the "discriminant group" D = H 1 ( ) by diagonal matrices on C t (see Section 1).

Generalized Monodromy Conjecture in dimension two

“…Another case is when Γ is minimal and it represents a minimally elliptic (automatically Gorenstein) singularity. These facts follow from the 'End Curve Theorem' [23,24].…”

On splice quotients of the form {zn = f (x,y )}

“…Remark 3.5 End curve functions for the splice quotient ( X g , n , 0) (as defined by Neumann and Wahl in 14) corresponding to the leaves of the splice diagram that provide nontrivial elements of the discriminant group can be determined from equation (3.4). Namely, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\big \lbrace \widetilde{M_s}(x,y)-\zeta ^j z^{n^{\prime }} : 1\le j\le h\big \rbrace$\end{document} are end curve functions.…”

“…The resulting quotient singularities ( X , 0) are called splice quotients . This work has led to a recent interest in splice quotients and universal abelian covers in general (see 5, 7, 14–16, 19).…”

“…However, there are nice classes of singularities for which all analytic types are splice quotients: weighted homogeneous singularities, as shown by Neumann in 8, and rational and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb Q{\rm HS}$\end{document}‐link minimally elliptic singularities, as shown by Okuma in 15. More recently, the “End Curve Theorem” of Neumann and Wahl 14 gives an analytic characterization of splice quotients. This theorem says that a singularity is a splice quotient if and only if there exist functions in the local analytic ring that have certain properties.…”

The multiplicity of abelian covers of splice quotient singularities

“…It is very natural to expect that some analytic invariants of splice quotients can be computed from their weighted dual graphs. In fact, since the end curve theorem (a characterization of splice quotients) was given by Neumann–Wahl , we have been able to compute the geometric genus (), the dimension of cohomology groups of certain invertible sheaves (, , , ), and the multiplicity () of splice quotients from their weighted dual graphs. However, we should remark that the embedding dimension, which is one of the most fundamental analytic invariants, is not topological even for quasihomogeneous singularities.…”