We consider the parameterization f=(f0:f1:f2)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 Cn⊂Pn to C and its interplay with the secant varieties to Cn. In particular, we define via f certain 0-dimensional schemes Xk⊂Pk, 2≤k≤(n−1), which encode all information on the singularities of multiplicity ≥k of C (e.g. using X2 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.
Singularities of plane rational curves via projections
Gimigliano, Alessandro;Idà, Monica
2018
Abstract
We consider the parameterization f=(f0:f1:f2)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 Cn⊂Pn to C and its interplay with the secant varieties to Cn. In particular, we define via f certain 0-dimensional schemes Xk⊂Pk, 2≤k≤(n−1), which encode all information on the singularities of multiplicity ≥k of C (e.g. using X2 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.File | Dimensione | Formato | |
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