Linear systems associated to unicuspidal rational plane curves

TL;DR

This paper investigates the properties of unicuspidal rational plane curves, establishing a link between their non-negativity and the rationality of general members in associated pencils, revealing new geometric insights.

Contribution

It proves that for unicuspidal rational plane curves, non-negativity is equivalent to the general member of a specific pencil being rational, and identifies the dicritical degree in such cases.

Findings

Non-negative curves have rational general members in their associated pencils.

All known unicuspidal rational curves are non-negative.

The pencil associated with such curves has a dicritical of degree 1.

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.