Singularities of plane rational curves via projections. (May 2018) - Bernardi, Alessandra; Gimigliano, Alessandro; Idà, Monica
Abstract: We consider the parameterization 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 n ⊂ P n to C and its interplay with the secant varieties to C n . In particular, we define viaf certain 0-dimensional schemes X k ⊂ P k, 2 ≤ k ≤ ( n − 1 ), which encode all information on the singularities of multiplicity ≥ 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.