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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.

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