Space of initial conditions for the four-dimensional Garnier system revisited

The paper constructs X_α by ten explicit blow-ups C1–C10 of P^2×P^2 (with two codimension-3 centers C7, C9), computes the anti-canonical class −K_X = 3H_q + 3H_r − Σ_{k=1,2,3,4,5,6,8,10} E_k − 2E_7 − 2E_9, and proves that deleting the six listed divisors produces a space of initial conditions for the 4D Garnier system (Theorem 2.2) . The candidate solution mirrors the same ten blow-ups and the same six divisors, then argues regularity/flatness via Hamiltonian 2-form and Ehresmann-connection methods. Two discrepancies: (i) the model claims the six removed divisors form “an anti-canonical member,” which contradicts the paper’s anti-canonical computation (the sum of the six classes equals 3H_q + H_r − E_1−…−E_6−E_8−E_10, not −K_X) ; and (ii) the model’s remark that “Bäcklund symmetries lift to pseudo-isomorphisms” omits that w_{α0} requires 21 blow-ups, as the paper explicitly adds C11–C21 to lift w_{α0} . These issues do not affect the correctness of the core SIC construction, so both are correct, but the proofs emphasize different mechanisms (paper: birational/pseudo-isomorphism resolution; model: differential/ODE regularization).