On $\hat{I}´$-sequences and surfaces at infinity

Abstract

In most cases the semigroup at infinity $S$ of a curve $C$ with only one place at infinity is generated by a $\hat{I}´$-sequence. This sequence provides geometrical information on $C$ such as the dual graph of the resolution of the singularity of $C$ at infinity. Since different $\hat{I}´$-sequences can generate the same semigroup, it is an interesting problem to know the geometrical behaviour of curves $C$ sharing the same semigroup $S$. An analogous problem arises in a more general context when considering surfaces at infinity and their $\hat{I}´$-semigroups. We show how to construct $\hat{I}´$-sequences, and how to obtain different families that generate the same semigroup $S$, allowing us to study the geometrical content encoded by $S$.

New title, modified version with a more arithmetic approach. Comments are welcome