The Columbia plot specifies the order of the QCD thermal transition as a function of the quark masses. Since massless quarks cannot be simulated directly, the nature of the phase transition in the limit of vanishing -quark masses has remained elusive, with different discretisations showing different orders of the transition in the small mass regime. We propose a modified analysis in the parameter space of degenerate quark masses, variable number of flavours and lattice spacing. Using unimproved staggered fermions with and , we map out regions of first-order transitions and crossover transitions, separated by a critical surface, in the bare parameter space of the lattice theory. This constrains the possibilities for an eventual continuum approach.