Real morsifications via the trace map

Abstract

We prove that every reduced real plane curve singularity admits a real morsification. This settles a question of A'Campo and Gusein-Zade, later stated as conjectures by Leviant--Shustin and by Fomin--Pylyavskyy--Shustin--Thurston. In particular we overcome the main obstruction that remained posed by conjugate pairs of nonreal branches. Our new main ingredient is a construction that produces the divide from a nodal smoothing of two normalization disks. This is what we call the trace map. For real branches, it recovers Gusein-Zade's construction using Chebyshev polynomials. For pairs of complex conjugate branches with distinct tangents, the construction gives an explicit formula for the divide in terms of the Puiseux data. The general method consists in a delicate combination of the trace map with A'Campo's translations and contractions to produce divides and real morsifications for all reduced real plane curve singularities.