On the geometry of strongly flat semigroups and their generalizations

Abstract

Our goal is to convince the readers that the theory of complex normal surface singularities can be a powerful tool in the study of numerical semigroups, and, in the same time, a very rich source of interesting affine and numerical semigroups. More precisely, we prove that the strongly flat semigroups, which satisfy the maximality property with respect to the Diophantine Frobenius problem, are exactly the numerical semigroups associated with negative definite Seifert homology spheres via the possible 'weights' of the generic S 1 -orbit. Furthermore, we consider their generalization to the Seifert rational homology sphere case and prove an explicit (up to a Laufer computation sequence) formula for their Frobenius number. The singularities behind are the weighted homogeneous ones, whose several topological and analytical properties are exploited.

Flat semigroups and weighted homogeneous surface singularities

“…. ω n , see [LN20]. In addition, if ω i is divisible by s i for every i, then one writes b 0 − i s i ωi/si αi = 1/α, hence by setting ω i := ω i /s i the construction is finished.…”

“…For example, one can construct representable semigroups which are not symmetric, but they have a numerically Gorenstein representant: Example 6.2.2 (Non-symmetric numerically Gorenstein case [LN20]). Let Γ be the graph defined by the Seifert invariants Sf = (−2, (2, 1), (2, 1), (3, 1), (3, 1), (7, 1), (7, 1), (84, 1)).…”

“…1.2. The present article focuses on a relatively new method developed by A. Némethi and the second author in [LN20]. This method is based on a subtle connection with the theory of normal surface singularities.…”

“…section 3.3.1. Using this presentation and the arithmetics of the star-shaped minimal good dual resolution graph of the singularity, [LN20] succeeded to find a 'closed' formula for the Frobenius number. Moreover, in that work the authors proved that a special class of representable semigroups appears naturally in the classical theory of numerical semigroups: the semigroups associated with Seifert integral homology spheres are exactly the 'strongly flat semigroups' with at least 3 generators, considered by the work of Raczunas and Chrzastowski-Wachtel [RChW96] as special semigroups realizing a sharp upper bound estimate with respect to the Frobenius problem.…”

“…Section 2 presents the necessary ingredients and preliminaries regarding the topology of normal surface singularities and negative definite Seifert 3manifolds. Section 3 introduces the Frobenius problem with a view towards the 'flat' classification of [RChW96] and discusses the case of semigroups associated with weighted homogeneous surface singularities by presenting the main results of [LN20].…”