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Geometry of plane curves via toroidal resolution. (English) Zbl 0857.14014
Campillo López, Antonio (ed.) et al., Algebraic geometry and singularities. Proceedings of the 3rd international conference on algebraic geometry held at La Rábida, Spain, December 9-14, 1991. Basel: Birkhäuser. Prog. Math. 134, 95-121 (1996).
The author studies the toroidal resolution of a plane curve $$C=\{f(z,w) = 0\}$$, which is a resolution consisting of a finite composition of admissible toric blowing-ups. It is shown how to read Puiseux pairs and the intersection multiplicities among irreducible components using the data of the toroidal resolution. The results are applied to planar curves, which are obtained as an $$n$$-times iterated generic hyperplane section of a non-degenerated hypersurface, to get that:
(1) the resolution complexity of these curves are at most $$(n+1)$$;
(2) each irreducible component of the curve has at most one Puiseux pair; and
(3) no Puiseux pair implies smoothness.
For the entire collection see [Zbl 0832.00033].

##### MSC:
 14H20 Singularities of curves, local rings 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14M25 Toric varieties, Newton polyhedra, Okounkov bodies