Singularities of plane rational curves via projections

We consider the parameterization ${mayhbf{f}}=(f_0:,f_1:f_2)$ of a plane rational curve $C$ of degree $n$, and we study the singularities of $C$ via such parameterization. We use the projection from the rational normal curve $C_nsubsetmathbb{P}^n$ to $C$ and its interplay with the secant varieties to $C_n$. IN particular, we define via $mathbf{f}$ certain 0-dimensioal schemes $X_ksubset mathbb{P}^k$, $2leq k leq (n-1)$, which encode all information on the singularities of multiplicity $geq k$ of $C$ (e.g. using $X_2$ we can give a criterion to determine whether $C$ is a cuspidal curve or has only ordinary singularities). We give a series of algorithms which allow one to obtain information about the singularities from such schemes.