Maximal contact is unchanged by tuning: MC_s(W_s(I)) = MC_m(I) #
Step 2.2 of [Kol07, 104] (the amplified Step 2 of the proof of Theorem 103) replaces I by W(I),
so that I may be assumed MC-invariant, and does "not pick a new hypersurface of maximal contact"
but uses only the birational transforms H_{r(s)} of the old ones — the Warning of [Kol07, 104]:
after a blow-up one should "stick with the birational transforms of the old ones". The old
hypersurface stays one of maximal contact for the tuned ideal because tuning does not change the
maximal-contact ideal: [Kol07, Proposition 99 (4)], MC_s(W_s(I)) = W_1(I) = MC_m(I) for
m = max-ord I ≥ 1 and s ≥ 1. This is RegularCoords.MC_W of Hironaka.Local at the stalk of
every closed point, lifted to the sheaf exactly as
Hironaka.Resolution.Algebraic.Tuning.Corollary101 lifts Proposition 99 (5) (isMCInvariant_W):
the stalks of MC and of W are their counterparts in regular coordinates (stalkIdeal_MC,
stalkIdeal_W), and an equality of ideal sheaves on a Jacobson scheme is checked at the closed
points (le_of_forall_stalkIdeal_le_of_isClosed).
[Kol07, Proposition 99 (4)]: for m = max-ord I ≥ 1 and s ≥ 1, MC_s(W_s(I)) = MC_m(I);
RegularCoords.MC_W at every closed point.
At the tuning parameter s = s(m) (tuningParam): the tuned ideal W_{s(m)}(I) has the
maximal-contact ideal of I.
"We do not pick a new hypersurface of maximal contact" (Step 2.2 of [Kol07, 104], and its
Warning): a hypersurface of maximal contact for (I, m) is one for the tuned (W_{s(m)}(I), s(m)).