Dynamically Stabilising Birational Surface Maps: Two Methods

The paper states and proves two results: (i) blowing up a minimal destabilising orbit strictly decreases the number of α-exceptional components on the smooth graph (ep(αm+1) = ep(αm) − 1), hence Method A terminates in finitely many steps; this is established via Lemma 5 and the ensuing proof of Theorem 1 (ep(αm)) strictly decreases to 0 . (ii) For Method B (iterated graphs), the lift to the smooth graph is untangled (Corollary 8) and Proposition 10 constructs an injection ι that maps any upstairs destabilising triple of length n to a downstairs one of length n+1, so either ep(fm) drops or all orbits shorten by one; iterating shows eventual stability (Theorem 2) . By contrast, the model’s core monotonicity claims are reversed: it asserts that after each step the minimal orbit length strictly increases for both methods, whereas the paper shows the opposite effect for Method B (orbits shorten under the lift) and, in the analysis of Method A, the key invariant is ep(α), not orbit length, with Lemma 5 exhibiting a length-shortening phenomenon in the stepwise construction . The model also misidentifies the indeterminacy set for the lifted map on the smooth graph, writing I(f1) = α(Eα) for f1: Σf ⇢ Σf; the correct relation is that f1 = α−1 ∘ β has indeterminacy β−1(α(Eα)), and in any case f̂ is untangled (E(f̂) ∩ I(f̂) = ∅) by Corollary 8 . Finally, the model’s fallback to Diller–Favre for Method A does not by itself show that the specific “blow up minimal orbit” algorithm follows the stabilising sequence; the paper provides a direct, elementary proof that it does via the invariant ep(α) .