Generic forms

We determine the Hilbert series of some classes of ideals generated by generic forms of degree two and three, and investigate the difference to the Hilbert series of ideals generated by powers of linear generic forms of the corresponding degrees.Comment: 2 page

Arf characters of an algebroid curve

Two algebroid branches are said to be equivalent if they have the same multiplicity sequence. It is known that two algebroid branches $R$ and $T$ are equivalent if and only if their Arf closures, $R'$ and $T'$ have the same value semigroup, which is an Arf numerical semigroup and can be expressed in terms of a finite set of information, a set of characters of the branch. We extend the above equivalence to algebroid curves with $d>1$ branches. An equivalence class is described, in this more general context, by an Arf semigroup, that is not a numerical semigroup, but is a subsemigroup of $\mathbb N^d$. We express this semigroup in terms of a finite set of information, a set of characters of the curve, and apply this result to determine other curves equivalent to a given one.Comment: 17 page

Pragmatic 2011

This is a presentation of articles started in Pragmatic 2011

On differential operators of numerical semigroup rings

If S=〈d1,...,dν〉 is a numerical semigroup, we call the ring C[S]=C[td1,...,tdν] the semigroup ring of S. We study the ring of differential operators on C[S], and its associated graded in the filtration induced by the order of the differential operators. We find that these are easy to describe if S is a so-called Arf semigroup. If I is an ideal in C[S] that is generated by monomials, we also give some results on Der(I,I) (the set of derivations which map I into I). © 2012 Elsevier B.V

Normal Hilbert functions of one-dimensional local rings

Let (R, m) be a one-dimensional reduced (Noetherian) local ring with finite integral closure ((R) over bar, M-1,..., M-t). We assume further that R/m similar or equal to (R) over bar /M-i for each i and that Card(R/m) greater than or equal to t. We study for such a ring R the associated graded ring and the Hilbert series, with respect to the normal filtration of an m-primary ideal I, R superset of or equal to (I) over bar superset of or equal to (I-2) over bar superset of or equal to (...). We make use of the value semigroup of R and in particular of some results of (7)

On free resolutions of some semigroup rings

For some numerical semigroup rings of small embedding dimension, namely those of embedding dimension 3, and symmetric or pseudosymmetric of embedding dimension 4, presentations has been determined in the literature. We extend these results by giving the whole graded minimal free resolutions explicitly. Then we use these resolutions to determine some invariants of the semigroups and certain interesting relations among them. Finally, we determine semigroups of small embedding dimensions which have strongly indispensable resolutions. (C) 2013 Elsevier B.V. All rights reserved