On the relation between Betti numbers of an Arf semigroup and its blowup

TL;DR

This paper establishes a precise mathematical relation between the Betti numbers of an Arf semigroup and its blowup, revealing how their algebraic invariants are interconnected when they share the same multiplicity.

Contribution

It proves a specific formula linking the Betti numbers of an Arf semigroup and its blowup, advancing understanding of their algebraic structure.

Findings

Betti numbers of S' relate to those of S via a shift in degree

The relation holds when S and S' have the same multiplicity n

Provides a new tool for studying Arf semigroups and their invariants

Abstract

In this note we prove the relation between Betti numbers of an Arf semigroup $S$ and its blowup $S'$ in the case when they have the same multiplicity $n$. The relation is then $\beta_{i,s}(S')=\beta_{i,{s+(i+1)n}}(S)$.