Totally Invariant Divisors of non Trivial Endomorphisms of the Projective Space

The paper proves two claims: (i) the degree bound d < (n choose l+1)^{1/(n−l−1)} + 1 for an irreducible totally invariant divisor X ⊂ P^n with l = dim X_sing, via a Chern-class comparison on the logarithmic cotangent sheaf with an explicit computation of c_k(Ω_{P^n}(log X) ⊗ O(m)) and an iteration argument, yielding (d−1)^k < (n choose k) with k = n−l−1 (Theorem 1.2), and (ii) the cone descent statement that no cone over a non–totally invariant divisor is totally invariant (Theorem 1.3). The key ingredients are: global generation of Ω_{P^n}(log X) ⊗ O(1) away from X_sing (Proposition 2.1), a Chern-class comparison lemma (Lemma 2.2), its application on M = f^{-1}(P) for a general k-plane (Proposition 2.4 → 2.5), and the explicit Chern class formula/estimate (Proposition 2.6), followed by an asymptotic-in-m argument establishing the strict inequality; the cone result uses a projection-from-a-point construction to descend dimension while preserving total invariance (proof of Theorem 1.3) . The candidate solution reproduces the high-level strategy and the cone argument, but its core proof contains two critical flaws: (a) it asserts global generation of the untwisted Ω^1_{P^n}(log X) on a general P^{l+1} by sections “built from ∂h/∂X_i,” whereas the paper proves generation for Ω_{P^n}(log X) ⊗ O(1) by explicit sections d(X_i h)/h (the twist by O(1) is essential) ; and (b) it claims an explicit top Chern class identity c_{l+1}(Ω^1_{P^n}(log X)|_P) = (n choose l+1) − (d − 1)^{n − l − 1}, which contradicts the paper’s derived formulas and the general degree-in-d behavior of Chern classes (the paper instead computes c_k(Ω_{P^n}(log X) ⊗ O(1)) = (d − 1)^k and an asymptotic expansion for c_k(Ω_{P^n}(log X) ⊗ O(m)) with leading coefficient (n choose k) m^k) . Hence the paper’s arguments are correct, while the model’s proof, though reaching the same inequality, is not valid as written.