On $\delta$-sequences and surfaces at infinity

In most cases the semigroup at infinity $S$ of a curve $C$ with only one place at infinity is generated by a $\delta$-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 $\delta$-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 $\delta$-semigroups. We show how to construct $\delta$-sequences, and how to obtain different families that generate the same semigroup $S$, allowing us to study the geometrical content encoded by $S$.