Complex factors and singularities

Characterize the complex irreducible factors and non-real singularities of the finite free curve C_w defined by h_w(x,y,t)=\mathcal F_n(x,yα+tu,yβ+tv) directly in terms of the input velocity w=u⊕v, for fixed simple real root vectors α and β.

Background

The paper determines the real singularities of finite free curves: every real singularity is an ordinary totally real multiple point, all real singularities lie over at most one real projective parameter, and the real collision loci admit an explicit linear arrangement.

The complex geometry remains less understood. A single cycle in the real normalization covering forces absolute irreducibility, but multiple real components can still belong to one complex irreducible curve. The question asks for a direct description of both factorization and non-real singular behavior from the input velocities.