Singularities of plane rational curves via projections

2016/09/07 by Bernardi, Alessandra, Gimigliano, Alessandro, Idà, Monica
#Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1609.01877

Abstract

We consider the parameterization f=(f₀,f₁,f₂) of a plane rational curve C of degree n, and we want to study the singularities of C via such parameterization. We do this by using the projection from the rational normal curve C_n⊂ ℙⁿ to C and its interplay with the secant varieties to C_n. In particular, we define via f 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 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 to get info about the singularities from such schemes.

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