diff --git a/projective_hodge_iitaka_v2.txt b/projective_hodge_iitaka_v2.txt
index 4f4d0ce..e923e83 100644
--- a/projective_hodge_iitaka_v2.txt
+++ b/projective_hodge_iitaka_v2.txt
@@ -60,7 +60,7 @@ Three big inputs drive the proof:
       Hashizume) together with Fujino's weak positivity.
 
 The genuinely new content is (A): proving that the Hodge line M is
--positive- in the precise sense needed.  Everything else assembles (A)
+positive in the precise sense needed.  Everything else assembles (A)
 with (B) and (C).  The appendices (Sections 10 and 11) give two other
 proofs of (A); the main text proves (A) in Sections 3-7.
 
@@ -377,11 +377,12 @@ Proof outline of Theorem 3.1.
 
   (b) Section 5: prove nu(J) = rank of ell, identify the boundary line
       with a graded piece attached to the monodromy logarithm, and
-      prove the local rank principle (Lemma 5.1).
+      prove the local rank principle (Lemma 5.4).
 
   (c) Section 6: use (b) to force nu(J|_E) < nu(J) on every boundary
-      component E with nonzero monodromy (Lemma 6.3), and on every
-      divisor fixed by a nontrivial finite inertia element (Lemma 6.4).
+      component E with nonzero monodromy, and on every divisor fixed
+      by a nontrivial finite inertia element. Both drops are proved
+      inside Theorem 6.1.
 
   (d) Section 7: build the actual period quotient, apply the volume
       lemma (Lemma 7.1) to subtract the boundary from an adjoint
@@ -1122,11 +1123,11 @@ Now base change to etabar.  X~_etabar and Y_etabar are smooth, the
 former birational to F.  Restrictions of canonical divisors agree up to
 linear equivalence, so
 
-      K_{X~_etabar}  <=Q  g_etabar*(D|_{Y_etabar}) + R^+|...,
+      K_{X~_etabar}  <=Q  g_etabar*(D|_{Y_etabar}) + (R^+)|_X~_etabar,
 
 where <=Q means Q-linearly equivalent up to an effective divisor.
 Proper direct image commutes with this flat base change, so
-(g_etabar)_* O(mR^+|_{...}) = O_{Y_etabar} for divisible m, and the
+(g_etabar)_* O(m (R^+)|_X~_etabar) = O_{Y_etabar} for divisible m, and the
 projection formula gives
 h^0(X~_etabar, mK) <= h^0(Y_etabar, mD|).  Hence
 d = kappa(F) <= kappa(Y_etabar, D|) <= dim Y_etabar = d; so D on the