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Ralf Fröberg: Curves (mostly plane) and semigroups

Time: Mon 2023-02-06 15.00 - 16.00

Location: Zoom

Video link: Meeting ID: 667 9231 0361

Participating: Ralf Fröberg, Stockholm University


A numerical semigroup S is a subset of \(\mathbb N\) such that if \(m,n\in S \text{ then } m+n\in S\), \(0\in S\), and \({\mathbb N}\setminus S\) is finite. It is easy to see that such a semigroup $S$ has a unique minimal system of generators \(\{ n_1,\ldots,n_r\}\), such that S consists of all linear combinations of the generators with nonnegative integer coefficients. If \(S=\langle n_1,\ldots,n_r\rangle\), the semigroup ring \(k[[S]]\) is then \(k[[t^{n_1},\ldots,t^{n_r}]]\). Let f(x,y) be an irreducible power series in \(k[[x,y]]\) (or analytic function around the origin) which has a singularity in $(0,0)$. For such an fone can define a numerical semigroup which determines the topological type of the singularity. (The semigroup in fact consists of all intersection numbers I(f,h), where h is prime to f.) Also for parametrized curves in higher dimensional spaces one can define a numerical semigroup, which sometimes says a lot about the curve. I will describe this connection in the first part of my talk. Then I will mention some results and open questions about numerical semigroups and semigroup rings.