Geometry of Puiseux expansions

Maciej Borodzik¹ ; Henryk /Zo/l/adek¹

¹Institute of Mathematics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Annales Polonici Mathematici, Tome 93 (2008) no. 3, pp. 263-280

Résumé

We consider the space $\mathrm{Curv}$ of complex affine lines $% t\mapsto(x,y)=(\phi(t),\psi(t))$ with monic polynomials $\phi$, $\psi$ of fixed degrees and a map $\mathrm{Expan}$ from $\mathrm{Curv}$ to a complex affine space $\mathrm{Puis}$ with $\dim\mathrm{Curv}=\dim\mathrm{Puis}$, which is defined by initial Puiseux coefficients of the Puiseux expansion of the curve at infinity. We present some unexpected relations between geometrical properties of the curves $(\phi,\psi)$ and singularities of the map $\mathrm{% Expan}$. For example, the curve $(\phi,\psi)$ has a cuspidal singularity iff it is a critical point of $\mathrm{Expan}$. We calculate the geometric degree of $\mathrm{Expan}$ in the cases $\gcd(\deg\phi,\deg\psi)\le 2$ and describe the non-properness set of $\mathrm{Expan}$.

Détail

DOI : 10.4064/ap93-3-7

Keywords: consider space mathrm curv complex affine lines mapsto phi psi monic polynomials phi psi fixed degrees map mathrm expan mathrm curv complex affine space mathrm puis dim mathrm curv dim mathrm puis which defined initial puiseux coefficients puiseux expansion curve infinity present unexpected relations between geometrical properties curves phi psi singularities map mathrm expan example curve phi psi has cuspidal singularity critical point mathrm expan calculate geometric degree mathrm expan cases gcd deg phi deg psi describe non properness set mathrm expan