O Apéry-algoritmo para uma singularidade plana com dois ramos

Abstract

Apery showed, if (??) is a irreductible algebroid plane curve and (?? (1)) is its blowup, then the semigroups ??(??) e ??(?? (1)) associated the curves (??) and (?? (1)) respectively, they can be related. The Apery set is a special generating set of a semigroup with conductor. There is a formula to get the Apery set for ??(?? (1)) from that of ??(??) and vice versa. This does not happen in general for non plane algebroid curves. Through that result of Apery, is possible to show how one can get the semigroup from the multiplicity sequence and vice versa. The main result here is a specie of generalization of results to the case of a plane algebroid curve with two branches, this is, the results of Barucci, Fröberg e D’Anna. We will show how the semigroups ??(??) and ??(?? (1)) are strictly related in the case that (??) has two branches through of Apery set, which it is not finite, but it is a finite union of its components. Furthermore, we will characterize a multiplicity tree of a plane algebroid curve with two branches of purely numerical form