Birational properties of tangent to the identity germs without non-degenerate singular directions

The paper proves, for a generic germ in the family (1), that every strongly adapted regular modification dominating π0 yields only degenerate characteristic directions, via a complete combinatorial analysis of invariant classes (simple corners, degenerate spikes, spinning corners, half corners) and their stability under both point and curve blow-ups. The candidate solution correctly computes the first blow-up in one chart, derives the local normal form x' = x + α x^3, u' = u + A x^2, v' = v + B x^2 with α(u,v)=uv(u−v), A(u,v)=1−v^2−u^2v(u−v), B(u,v)=v(u−v)(1−uv), and gives a valid degeneracy test at non-singular and certain singular points. However, it (i) only treats one blow-up chart and two singular points, ignoring the other characteristic directions and charts; (ii) does not handle curve blow-ups essential for strongly adapted modifications; (iii) does not start from or reconcile with the resolved model π0 with its nine isolated canonical singularities; and (iv) overclaims that the statement holds for the entire family, whereas the paper’s theorem is stated for a generic set of parameters. Hence the model’s proof is incomplete and overstates the result, while the paper’s result and proof are sound.