Linear systems associated to unicuspidal rational plane curves

2012/02/28 by Daigle, Daniel, Hernández, Alejandro Melle
#14C20 #14J26 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1202.6146

Abstract

A curve C in the projective plane is called non-negative if the self-intersection number of C after the minimal resolution of singularities of C is non-negative. Given a unicuspidal rational plane curve C with singular point P, we study the unique pencil Lambda_C on the projective plane satisfying C is in Lambda_C and P is its unique base point. We show that the general member of Lambda_C is a rational curve if and only if the curve C is non-negative. We also show that in such a case then Lambda_C has a dicritical of degree 1. Note that all currently known unicuspidal rational curves C in the projective plane are non-negative.

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