On curves on sandwiched surface singularities

TL;DR

This paper investigates curves on sandwiched surface singularities obtained via blow-ups, providing criteria for principality, formulas for multiplicity and intersection numbers, and determining the semigroup of branches.

Contribution

It introduces new criteria and formulas for analyzing curves on sandwiched surface singularities, including the semigroup of branches, advancing understanding of their structure.

Findings

Criteria for principality of curves on sandwiched singularities

Formulas for multiplicity, intersection numbers, and singularity order

Effective determination of the semigroup of branches

Abstract

Fixed a point O on a non-singular surface S and a complete mO-primary ideal I in its local ring, the curves on the surface X obtained by blowing-up I are studied in terms of the base points of I. Criteria for the principality of these curves are obtained. New formulas for their multiplicity, intersection numbers and order of singularity at the singularities of X are given. The semigroup of branches going through a sandwiched singularity is effectively determined, too.