Real algebraic and pseudoholomorphic curves on the quadratic cone and smoothings of the singularity X₂₁

A fiberwise isotopy classification is completed for the smooth real algebraic and pseudoholomorphic curves of degree 8 on the quadratic cone that have a specially shaped oval crossing a given generating line of the cone at four real points. This classification is linked with an isotopy classification of smoothings of a real plane curve singularity that is the union of four smooth real local branches quadratically tangent to each other (the singularity X₂₁).