# zbMATH — the first resource for mathematics

Geometry of plane curves via Tschirnhausen resolution tower. (English) Zbl 0904.14014
The weight vectors of a resolution tower of toric modifications for an irreducible germ of a plane curve $$C$$ carry enough information to read off invariants such as the Puiseux pairs, multiplicities, etc. [M. Oka in: Algebraic geometry and singularities, Proc. 3rd Int. Conf. Algebraic Geometry, La Rábida 1991, Prog. Math. 134, 95-118 (1996; Zbl 0857.14014)]. However, each step of the inductive construction of a tower of toric modifications depends on a choice of the modification local coordinates. This ambiguity makes it difficult to study the equi-singularity problem of a family of germs of plane curves or to study a global curve.
The purpose of the paper under review is to make a canonical choice of the modification local coordinates $$(u_i,v_i)$$, and to obtain a canonical sequence of germs of curves $$\{C_i$$; $$i = 1,\dots,k \}$$ $$(C_k = C)$$ such that the local knot of the curve $$C_i$$ is a compound torus knot around the local knot of the curve $$C_{i-1}$$. The authors show that the local equations $$h_i (x,y)$$ of the germs $$\{C_i; i=1,\dots,k\}$$ are the Tschirnhausen approximate polynomials of the local equation $$f(x,y)$$ for $$C$$, provided that $$f(x,y)$$ is a monic polynomial in $$y$$. The paper builds on work of S. S. Abhyankar and T. Moh [J. Reine Angew. Math. 260, 47-83 and 261, 29-54 (1973; Zbl 0272.12102], giving a geometric interpretation.
The paper also gives a new method to study the equi-singularity problem. Indeed, the authors show that a family of germs of plane curves $$\{f_t(x,y)=0\}$$ with Tschirnhausen approximate polynomials $$h_i(x,y)$$, $$i=1,\dots,k-1$$, not depending on $$t$$ and satisfying an additional intersection condition is equi-singular. Furthermore they give a new proof and a generalization of a theorem of Abhyankar-Moh-Suzuki from the point of view of the equi-singularity at infinity.

##### MSC:
 14H20 Singularities of curves, local rings 14H10 Families, moduli of curves (algebraic) 32B10 Germs of analytic sets, local parametrization
Full Text: