Geometry of Puiseux expansions
Volume 93 / 2008
Abstract
We consider the space 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}.
