LOWER BOUNDS FOR GENERA OF FIBER PRODUCTS

The paper’s Theorem 1.1 is stated and proved rigorously using a lifting lemma to the normalization and an orbifold–Euler characteristic inequality, followed by Hurwitz’s automorphism bound, yielding g(E) ≥ (g(R)−1)(deg V−1) + 1 + deg P/84 under the reduced, deg V=deg W>1, and g(N_W)>1 hypotheses. The model’s proof sketch reaches the same numerical bound but makes a crucial inequality error when combining Riemann–Hurwitz identities: it drops a nonpositive term −(deg V−1)R_P, incorrectly concluding R_V + (2g(R)−2) ≥ deg P·(R_W + 2g(C)−2). It also asserts R_W ≥ Σ_p(m_p−1), which is false in general. The claim that the reduced hypothesis is unnecessary is not justified in the model’s argument, whereas the paper relies essentially on reducedness (via Lemma 3.3).