# On the Apéry algorithm for a plane singularity

## Dettagli dell'evento

### Quando

dalle 16:30 alle 18:00

### Dove

### Persona di riferimento

Abstract:

A classical tool to get numerical invariants of a curve singularity is the study of its value

semigroup. In case of a one branch singularity this semigroup is a numerical semigroup

(i.e. a submonoid of N with finite complement in it); in case the singularity has h branches,

this semigroup is a subsemigroup of Nh, belonging to the class of the so-called ”good semi-

groups”. Despite their name, the combinatoric of good semigroups is quite problematic;

moreover, for h ≥ 2, it is an open problem to understand which good semigroups can be

realized as value semigroups.

In case of a plane singularity with one branch, an old result of Ap ́ery shows that there

is a particularly strict connection between the value semigroups of the singularity and

of its blowup; this connection is obtained using a particular set of generators of the

semigroup, named ”Ap ́ery set”. In fact, using that result, it is possible to show very

easily, that the equisingularity classes given by the multiplicity sequence and by the

value semigroup coincide. In particular, this method allows to reconstruct the numerical

semigroup associated to a plane branch singularity starting from the multiplicity sequence.

When the singularity has more than one branch, in order to generalize the Ap ́ery result,

two main problems arise: firstly, the Ap ́ery set becomes an infinite set; secondly, in the

process of blowing-up it is necessary to deal with semilocal rings, that cannot be presented

as quotients of a power series ring in two variables, as it happens in the local case. These

problems where partially solved twenty years ago in the two branch case, but the general

case is still open.

In my talk, after describing some key definitions and results on value semigroups and

good semigroups of a curve singularity with h branches, I will explain explaining the

Ap ́ery process for a plane branch and then, I will present some recent results obtained in

a joint project with F. Delgado de la Mata, L. Guerrieri, N. Maugeri and V. Micale, that

are a significant progress toward a complete solution of this problem.

Dipartimento di Matematica e Informatica - University of Catania

V.le A. Doria 6, 95125

Catania

Italy

E-mail address: marco.danna@unict.it