Geometry of Puiseux expansions
Tom 93 / 2008
Streszczenie
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 $\mathrm{Expan}$.
