Sextic curves with simple singularities. (English) Zbl 0866.14014
In this paper the author studies the configuration of simple singularities in a reduced sixtic curve in \(\mathbb{P}^2\). Recall that singularities are described by their dual graph in an embedded resolution of singularities and simple singularities are in correspondence with simple Dynkin diagrams, of type \(A_i\) \((i\geq 1)\), \(D_j\) \((j\geq 4)\) and \(E_k\) \((k=6,7,8)\). The main theorem in this article is:
There exists a reduced sixtic curve in \(\mathbb{P}^2\) with only simple singularities given by a finite Dynkin graph \(G\) if and only if \(G\) is a subgraph of a graph listed by the author.
For the proof the author uses full version of Nikulin’s embedding theorem of even lattices.
There exists a reduced sixtic curve in \(\mathbb{P}^2\) with only simple singularities given by a finite Dynkin graph \(G\) if and only if \(G\) is a subgraph of a graph listed by the author.
For the proof the author uses full version of Nikulin’s embedding theorem of even lattices.
Reviewer: M.Morales (Saint-Martin-d’Heres)
MSC:
| 14H20 | Singularities of curves, local rings |
| 14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |
| 14Q05 | Computational aspects of algebraic curves |
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