diff --git a/symplectic_ball_packings_v2.txt b/symplectic_ball_packings_v2.txt
index 051ceb5..fde85de 100644
--- a/symplectic_ball_packings_v2.txt
+++ b/symplectic_ball_packings_v2.txt
@@ -254,7 +254,7 @@ tool it relies on.
          length of {f < c} equals the length of {g < c} (that is
          what re-sorting preserves); integrating these lengths
          over c (the "layer cake" identity
-         int f = int_0^infty length{f>c} dc) gives equal
+         int f = int_0^infinity length{f>c} dc) gives equal
          integrals.
     (v)  f <= h makes every window's worst value no larger, so
          the minima satisfy g_f <= g_h.  Adding a constant c to
@@ -1009,12 +1009,12 @@ captures essentially all the volume.
     F in PP^N be a hyperplane.  There is a smooth connected curve
     C, the complete intersection of N-1 quadrics, meeting F
     transversally.  With A = O(1)|_C,
-        deg A = 2^{N-1},   N_{C/PP^N} = (A^2)^{oplus (N-1)},
+        deg A = 2^{N-1},   N_{C/PP^N} = (A^2)^{⊕ (N-1)},
         g(C) = 1 + (N-3) 2^{N-2}.                              (9.4)
     Moreover, with H the hyperplane pullback and E_C the
     exceptional divisor of Bl_C PP^N, the class H - d E_C is ample
     for every rational 0 < d < 1/2, and the class A + d U is ample
-    on  PP_C( 1 + (A^2)^{oplus(N-1)} )  for the same d.
+    on  PP_C( 1 + (A^2)^{⊕(N-1)} )  for the same d.
     Proof.  Bertini's theorem, applied to the (basepoint-free on
     each smooth intermediate step, plus on F) quadratic Veronese
     system, gives N-1 quadrics whose common intersection is smooth
@@ -1024,7 +1024,7 @@ captures essentially all the volume.
     negative twists give H^0(O_C) = C, so C is connected.  The
     intersection class is (2H)^{N-1}, so deg A = 2^{N-1}.  The
     defining equations form a regular sequence, so the conormal
-    bundle is (A^{-2})^{oplus(N-1)}; dualizing gives the normal
+    bundle is (A^{-2})^{⊕(N-1)}; dualizing gives the normal
     bundle.  Adjunction K_C = K_{PP^N}|_C (x) det N_{C/PP^N}
     = A^{N-3}, giving 2g(C)-2 = (N-3)2^{N-1}, i.e. (9.4).
     Ample class H - dE_C: the quadric equations generate the ideal
@@ -1045,9 +1045,9 @@ captures essentially all the volume.
 
 9.3 The limiting base area.
     Now specialize N = n, m = n-1, and apply Lemma 8.4 to the
-    normal bundle (A^2)^{oplus m}: the exponent is b = 2 and
+    normal bundle (A^2)^{⊕ m}: the exponent is b = 2 and
     deg A = 2^{n-1} = 2^m.  Formula (8.7) gives the limiting base
-    area of the normal-bundle model Y = PP_C(1 + (A^2)^{oplus m}):
+    area of the normal-bundle model Y = PP_C(1 + (A^2)^{⊕ m}):
         H_0(p) = 2^m ( 1 - 2 sum_{j=1}^m p_j ).                 (9.5)
     Its truncation to Delta = { p >= 0 : sum_j p_j <= h }, for
     0 < h < 1/2, integrates to
@@ -1091,7 +1091,7 @@ captures essentially all the volume.
     Take any r_i' with rho_i < r_i' < r_i.  Lemma 7.4 packs
     disjoint closed balls B^{2m+2}(r_i') = B^{2n}(r_i') into
     Int(Sigma_l x V) for the normal-bundle model, with neighborhood
-    embeddings.  (This model sits inside Y = PP_C(1+(A^2)^{oplus
+    embeddings.  (This model sits inside Y = PP_C(1+(A^2)^{⊕
     m}); it is truncated at t <= h < d, so it lies in the affine
     part, away from the infinity section D_inf and away from the
     fibers over the finitely many points C cap H_inf.)
@@ -1116,7 +1116,7 @@ J_m(h) of the 1/n! total, and J_m(h) -> 1/n! as h -> 1/2.  That is
 the precise sense in which a thin normal neighborhood of a
 degree-2^{n-1} curve can hold almost the entire volume of
 projective space.  The reason: the curve's normal bundle is
-O(2)^{oplus(n-1)}, so its fiber polytope {sum p_j < d} of size
+O(2)^{⊕(n-1)}, so its fiber polytope {sum p_j < d} of size
 d up to nearly 1/2 fills almost the whole moment simplex.
 
 
@@ -1179,11 +1179,11 @@ hyperplane.
     n-1 >= 3, i.e. n >= 4) and forbidden hyperplane F.  Get a
     smooth curve C, the complete intersection of n-2 quadrics,
     meeting F transversally, with
-        deg C = 2^{n-2} = 2^q,   N_{C/D} = (O_C(2))^{oplus q},
+        deg C = 2^{n-2} = 2^q,   N_{C/D} = (O_C(2))^{⊕ q},
         g(C) = 1 + (n-4) 2^{n-3}  >= 1 .
     Deform to the normal cone of C in D with parameter d_2.  By
     Lemma 9.3 both component classes are ample when d_2 < 1/2.
-    The normal-bundle component Y_2 = PP_C(1 + (O_C(2))^{oplus q})
+    The normal-bundle component Y_2 = PP_C(1 + (O_C(2))^{⊕ q})
     has, by Lemma 8.4 with sigma = a positive curvature form of
     O_C(1) of total integral 2^q (b = 2), the affine model
         omega_X = sigma + d( sum_{j=2}^{n-1} v_j alpha_j ),