The monomial order-reduction run on the combinatorial state: worked examples #

Examples of the run step3 of [Kol07, Theorem 107, Step 3 of the proof] (order reduction for a monomial ideal supported on a simple normal crossing divisor with ordered components) on the combinatorial state MonomialState, decided by decide (never native_decide) through the fuelled copies step3Fuel/maxMultRun and the agreement theorem step3_map_eq_of_step3Fuel.

[Kol07, Example 112]: two curves meeting at a point, a₁ = a₂ = m + 1. The maximal-multiplicity procedure ("look for the highest multiplicity locus and blow it up … does not work, not even for surfaces", [Kol07, Theorem 107, Step 3 of the proof]) blows up the point, then E² ∩ E³, E³ ∩ E⁴, …, and the exceptional exponents are m + p_i with p_i the Fibonacci numbers (3, 4, 6, 9 for m = 1): unbounded. Kollár's Step 3 blows up E¹, then E² (trivial blow-ups: the components die and the exceptional components carry 1), and is done with max-ord = 2 for m ≥ 3; for m = 2 phase 2 blows up the remaining double point once; for m = 1 phase 1 takes two steps per curve: 4, 3, 2, 2, 2 blow-ups for m = 1, 2, 3, 5, 10.
A three-phase instance: n = 3, m = 8, a = (4, 9, 3, 6), maximal faces {0} and {1, 2, 3}; phase 1 blows up {1}, phase 2 {2, 3}, phase 3 {3, 4, 5}, final max-ord = 7 < 8; the measure vectors ((m_s, n_s))_{s = 1, 2, 3} are ((9,1), (15,1), (18,1)), ((6,1), (9,1), (10,1)), ((6,1), (7,2), (8,1)), ((6,1), (7,2), (7,2)): no pair ever increases.
The split example: E¹ = V(x(x-1)), E² = V(y(y-1)), I = (xy), m = 1, exponents 1, 0, 1, 0 on the four lines; Step 3 blows up V(x) then V(y) on X and on U = D((x-1)(y-1)) alike.
The nerve rule: three lines in 𝔸² blown up at a double point; the coordinate planes of 𝔸³ blown up at the origin and then along the transform of a line.