Foliations with Isolated Singularities on Hirzebruch Surfaces

The paper’s main theorem (Theorem 5.2) proves that if a foliation [s] on Sδ has isolated singularities and L = O(−d1,−d2) satisfies d2 ≥ 1 and d1 ≥ 1 (δ = 0), d1 ≥ 0 (δ = 1), or d1 ≥ 2 (δ ≥ 2), then any other section s′ with the same singular scheme Z must be of the form s′ = Φ(s) for a global endomorphism Φ of TSδ; moreover the three explicit formulas for Ω′ in the cases δ = 0, 1, ≥ 2 are precisely as stated, yielding uniqueness for δ = 1 . The proof proceeds via the Koszul resolution and a long exact sequence that makes the surjectivity of the map Φ ↦ Φ(s) hinge on the vanishing H1(Sδ, ΘSδ ⊗ E) = 0, which the authors establish (Lemma 5.1) under the cited degree hypotheses . The structure of H0(End TSδ) in Theorem 4.1 matches the model’s description: diagonal scalars for δ = 1, two independent scalars for δ = 0, and an additional off-diagonal family parameterized by H0(P1, O(δ − 2)) for δ ≥ 2; the induced transformations of affine 1-forms also agree with the three displayed cases . By contrast, the model’s Step 1–2 constructs, without additional hypotheses, a “canonical” automorphism Φ on U = Sδ \ Z using ωs and claims it extends across Z by reflexivity to give s′ = Φ(s) globally. This relies on a non-existent canonical splitting of the short exact sequence 0 → s(L∗) → E|U → det(E) ⊗ L → 0 and thus does not establish a well-defined O-linear automorphism on U. The paper correctly avoids this and uses cohomology to guarantee surjectivity, which is essential; without those vanishing conditions, s′ need not be of the form Φ(s). Hence the model’s existence-and-uniqueness claim is overstated, while its computation of H0(End TSδ) and the δ-dependent formulas for Ω′ agree with the paper .