On the geometry of strongly flat semigroups and their generalizations

TL;DR

This paper explores the connection between complex surface singularities and numerical semigroups, characterizing strongly flat semigroups through topological invariants of Seifert homology spheres and providing formulas for their Frobenius numbers.

Contribution

It establishes a link between strongly flat semigroups and Seifert homology spheres, extending to rational cases and deriving explicit Frobenius number formulas.

Findings

Strongly flat semigroups correspond to weights of generic $S^1$-orbits in Seifert homology spheres.

Explicit Frobenius number formulas are derived for generalized cases.

The study leverages properties of weighted homogeneous surface singularities.

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.