Lee McEwan | Searcharxiv

arXiv2023

The generic value set of a two-generator semigroup

We give a simple recursion for the minimal generators $G=\{g_i\}$ of the generic value set $Λ_{gen}$ of a plane curve germ $C$ with a two-generator semigroup $Γ\langle p, m \rangle$. The main result provides for explicit calculation of all $g_i$ and shows that they are in fact minimal generators. Explicit formulas for the cardinality of the minimal generators $|G|$ and for the conductor $c(Λ_{gen})$ of $Λ_{gen}$ are given. The recursion can be used to compute the generic Tjurina number $τ_{gen}$ in this case, a method we compare to that given by Alberich-Carramiñana, Almirón, Blanco, and Melle-Hernández (arXiv:1904.02652). The main result is based on the non-explicit algorithm and ideas of Delorme.

math.AG↗

arXiv2019

On the Milnor fiber boundary of a quasi-ordinary surface

We give a recursive formula, expressed in terms of the characteristic tuples, for the Betti numbers of the boundary of the Milnor fiber of an irreducible quasi-ordinary surface. The singular locus of the surface consists of two components, and for each component we introduce a sequence of increasingly simpler surfaces. Our recursion depends on a detailed comparison of these two sequences. We indicate how we expect pieces of these associated surfaces to glue together to reconstruct the Milnor fiber and its boundary.

math.AG↗