diff --git a/nef_adjoint_numerical_divisors_v2.txt b/nef_adjoint_numerical_divisors_v2.txt
index eae29f7..86df465 100644
--- a/nef_adjoint_numerical_divisors_v2.txt
+++ b/nef_adjoint_numerical_divisors_v2.txt
@@ -416,9 +416,9 @@ not be nef.
 [Minimal-model programs,
 {cite[Theorem~1.1 and Section~3.3]{MM}}]
 
-Let (X,B+mathbf M)bfM) be a projective generalized klt Q-pair over
+Let (X,B+M)M) be a projective generalized klt Q-pair over
 C, with X Q-factorial and with fixed nef rational Cartier
-b-divisor data mathbf M.bfM.  If K_X+B+M_X is pseudo-effective, there
+b-divisor data M.M.  If K_X+B+M_X is pseudo-effective, there
 is a terminating choice of its minimal-model program, with
 Q-factorial generalized klt models and a nef endpoint.  The program
 extracts no divisors and generalized log discrepancies do not decrease.
@@ -428,7 +428,7 @@ effective divisor exceptional over the final model.
 
 
 Two remarks.  First, the ordinary klt case of
-Theorem~input:models arises by taking mathbf M=0;bfM=0; then
+Theorem~input:models arises by taking M=0;M=0; then
 Theorem~input:abundance makes the endpoint a good minimal model.
 Second, neither theorem above asserts abundance for a nonzero
 generalized nef summand.  Their combination still exposes the
@@ -502,14 +502,14 @@ Q-factoriality when convenient.
 
 We start with the only new cone-theoretic input: a uniformity statement
 for negative extremal rays of generalized klt pairs.  Recall that a
-generalized klt pair (V,Delta+mathbf N)bfN) has boundary Delta and
+generalized klt pair (V,Delta+N)N) has boundary Delta and
 polarized data given by a fixed nef rational Cartier b-divisor
-mathbf NbfN determined on some projective birational model of V.
-The notation N_V denotes the trace of mathbf NbfN on V; this trace
+N determined on some projective birational model of V.
+The notation N_V denotes the trace of N on V; this trace
 need not be nef, only the polarized data on the fixing model is.
 
 
-Let (V,Delta+mathbf N)bfN) be a projective generalized klt Q-pair
+Let (V,Delta+N)N) be a projective generalized klt Q-pair
 of dimension n, with effective boundary, Q-factorial V, and
 nef rational Cartier b-divisor data.  Every
 (K_V+Delta+N_V)-negative extremal ray has an integral rational curve
@@ -536,13 +536,13 @@ Set G=K_V+Delta+N_V and fix a G-negative extremal ray generated
 by R.  Choose an ample rational divisor A on V and a rational
 eta>0 small enough that G+eta A is still negative on R, and
 such that
--(G+eta A)·gammage-tfrac12 G·gamma for every nonzero
+-(G+eta A)·gammage-1/2 G·gamma for every nonzero
 class gamma on R.
 
 We need to represent G+eta A as an ordinary klt adjoint.  Because
-the b-divisor mathbf NbfN may not descend to V, work upstairs.
+the b-divisor N may not descend to V, work upstairs.
 Choose a projective log resolution h: V'to V carrying
-mathbf NbfN as a locally free sheaf, and write
+N as a locally free sheaf, and write
 [
  K_{V'}+Delta'+N'=h^*G
 ]
@@ -568,7 +568,7 @@ rational curve generator C of the ray with
  0<-(G+eta A)· Cle2n.
 ]
 The chain
--(G+eta A)· Cge-tfrac12 G· C
+-(G+eta A)· Cge-1/2 G· C
 gives -G· Cle 4n.
 The contraction c: Vto S of the ray for G is the contraction
 of the same ray for the ordinary klt pair (V,Θ), by
@@ -626,7 +626,7 @@ multiple dU_0 rules out any positivity of U_0 along such a ray.
 
 Proof. 
 Apply Theorem~input:models to the ordinary klt pair (X,B)
-(i.e. with mathbf M=0)bfM=0) and pseudo-effective adjoint T.  The
+(i.e. with M=0)M=0) and pseudo-effective adjoint T.  The
 output is a terminating T-MMP ending on a Q-factorial klt pair
 (Y_0,B_0) with nef ordinary adjoint U_0=K_{Y_0}+B_0.
 Theorem~input:abundance makes U_0 semiample.  The
@@ -638,10 +638,10 @@ program has finitely many steps, there is a rational s>0 such that
 every step is still negative for the transform of T+uM for every
 uin(0,s], and such that the strict sign on the flipped sides is
 preserved.  Hence these steps are generalized klt steps for the pairs
-(X,B+umathbf M),bfM), so (Y_0,B_0+umathbf M)bfM) is a generalized klt
+(X,B+uM),M), so (Y_0,B_0+uM)M) is a generalized klt
 pair for uin(0,s].  In particular its trace
-(Y_0,B_0+smathbf M)bfM) is a generalized klt pair with polarized data
-smathbf MbfM contained in the fixed b-divisor.  Pushforward under a
+(Y_0,B_0+sM)M) is a generalized klt pair with polarized data
+sbfM contained in the fixed b-divisor.  Pushforward under a
 birational contraction to a Q-factorial target preserves
 pseudo-effectivity: on a common resolution, an upstairs divisor
 numerically trivial implies it is the pullback of its pushforward,
@@ -682,7 +682,7 @@ the extremal contraction, so it descends to the base; pulling back to
 the flipped model gives the transform of dU_i, again Cartier and
 free.  Negativity for the generalized adjoint with coefficient s
 persists, so the generalized klt condition on the pair (B_i+smathbf
-M)bfM) continues to hold.  The induction closes on the finite program.
+M)M) continues to hold.  The induction closes on the finite program.
 
 Let Y be its output.  Because each step is U_0-trivial,
 U=U_0 is pulled back unchanged, hence semiample with the same
@@ -770,7 +770,7 @@ S.  Fix an ample Cartier divisor H on Y and shrink S so that
 h is flat over the chosen open.  For every positive integer ell
 clearing the denominators of Q, the locus
 [
- {sin S: H^0(F_s,OO_{F_s}((ell Q-H)|_{F_s}))neq0}
+ {sin S: H^0(F_s,OO_{F_s}((ell Q-H)|_{F_s}))ne0}
 ]
 is closed in S, by upper semicontinuity of cohomology in flat
 families.  Bigness on a very general fibre puts the base point of
@@ -916,7 +916,7 @@ widetilde Y.  Effectivity of F and nefness of P_0 give
 
 For fge2, put s_i=P_0^{f-i}· H^i for 0le ile f.
 Both divisors are nef and big, so each s_i is positive.  The
-Khovanskii--Teissier inequalities
+Khovanskii--Teissier ineualities
 cite[Corollary~1.6.3(i) and Example~1.6.4]{LazarsfeldPAGI}, namely
 s_i^2ge s_{i-1}s_{i+1}, show that the ratios s_i/s_{i-1} are
 decreasing; hence
@@ -1008,13 +1008,13 @@ smooth projective resolution, because the subvariety itself may be
 singular.
 
 A linear system H H^0(Z,OO_Z(kR)) has a
-base ideal mathfrak bfrakb defined as the ideal generated by its
+base ideal b defined as the ideal generated by its
 sections in OO_Z; its zero locus is the base locus.  A base
 component V is isolated at the generic point of V when the
-localized ideal mathfrak bfrakbOO_{Z,eta_V} is primary for the
-maximal ideal mathfrak mfrakm of OO_{Z,eta_V}.  In this case there
+localized ideal bOO_{Z,eta_V} is primary for the
+maximal ideal m of OO_{Z,eta_V}.  In this case there
 is a well-defined multiplicity h with
-mathfrak bfrakbOO_{Z,eta_V}subsetmathfrak m^h,frakm^h, measuring how
+bOO_{Z,eta_V}subsetm^h,m^h, measuring how
 strongly H vanishes along V at its generic point.
 
 The first input is a numerical version of the
@@ -1035,7 +1035,7 @@ normalization, then
 ]
 
 
-Only generic log canonicity is required: the inequality lives at the
+Only generic log canonicity is required: the ineuality lives at the
 generic point of V.  The proof in [cite] passes through the dlt
 modification of cite[Theorem~2.10]{FH}, whose truncated boundary
 leaves an effective surplus supported over the non-lc locus.  On a
@@ -1056,11 +1056,11 @@ Let Z be a projective Q-factorial klt variety of dimension d,
 let R be a nef rational Cartier divisor, and let k>0 be an integer
 such that kR is Cartier.  Let
 0neH H^0(Z,OO_Z(kR)) be a linear subspace with
-proper base locus and base ideal mathfrak b.frakb.  Suppose that
+proper base locus and base ideal b.b.  Suppose that
 V is an integral base-locus component of dimension f<d, isolated
 at its generic point eta_V, and that eta_Vin Z_{sm}.
-Let mathfrak mfrakm be the maximal ideal of OO_{Z,eta_V}.  If
-mathfrak bfrakbOO_{Z,eta_V}subsetmathfrak m^hfrakm^h for an integer h>0,
+Let m be the maximal ideal of OO_{Z,eta_V}.  If
+bOO_{Z,eta_V}subsetm^hfrakm^h for an integer h>0,
 there are a rational 0<cle(d-f)k/h and an effective rational
 divisor Θsim_Q cR such that (Z,Θ) is log canonical
 at eta_V and has a log canonical place centered on V.  The
@@ -1080,7 +1080,7 @@ The degree estimate is the local multiplicity bound h^{d-f}
 combined with successive general cuts by the given linear series.
 The boundary is obtained by averaging general members with total
 coefficient equal to the generic log canonical threshold of
-mathfrak b;frakb; thus it is an actual divisor to which
+b;b; thus it is an actual divisor to which
 Lemma~j:subadjunction applies.
 
 The third input produces the moving base components in the first
@@ -1095,7 +1095,7 @@ tau_0<...<tau_d with common gap delta>0.  For xin
 Z_{sm}, put
 [
  H_i(x)=H^0(Z,OO_Z(kR)
-              ⊗ mathfrak m_x^frakm_x^{lceil ktau_irceil}).
+              ⊗ m_x^m_x^{lceil ktau_irceil}).
 ]
 Suppose that for very general x all these spaces are nonzero, and
 that the base locus of H_0(x) has a positive-dimensional
@@ -1105,7 +1105,7 @@ are: an integral parameter variety T with a dominant marked-point
 map x: Tto Z_{sm}; an index iin{0,...,d-1}
 fixed throughout; and a family of integral subvarieties V_t
 containing x(t) and common to the base loci of mathcal
-H_i(x(t))calH_i(x(t)) and H_{i+1}(x(t)).  The incidence
+H_i(x(t))H_i(x(t)) and H_{i+1}(x(t)).  The incidence
 [
  Gamma={(t,z):zin V_t}subset Ttimes Z
 ]
@@ -1178,7 +1178,7 @@ a finite list once the branch divisors do not sweep:
 
 Let Y be a normal integral projective variety, let
 V T be a smooth projective family with integral fibres
-over an integral parameter variety, and let gcolonV Y
+over an integral parameter variety, and let g:V Y
 be a morphism.  Suppose the general fibre maps
 g_t: V_tto Y are dominant and generically finite of degree
 at most a fixed integer e.
@@ -1186,7 +1186,7 @@ at most a fixed integer e.
 After a finite parameter extension and shrinking, spread the
 geometric generic components of the zero divisor of the relative
 Jacobian over Y_{sm} to divisors
-R_1,...,calR_1,...,R_N in V.  Retain this
+R_1,...,R_1,...,R_N in V.  Retain this
 notation after the base change.  Let I consist of those j for
 which g_t((R_j)_t)bar is a divisor in Y for
 general t.  Assume
@@ -1262,7 +1262,7 @@ X has positive integral L-degree.
 Fix an ample Cartier divisor A on X and choose a rational
 ε>0 with
 
- (14ε)^n<tfrac14,  80ε<tfrac34.
+ (14ε)^n<1/4,  80ε<3/4.
 
 Let t>0 tend to zero through positive rationals.  Since L is nef
 and not big, L^n=0; we choose positive integers r=r(t) by
@@ -1365,7 +1365,7 @@ then force
 [
  1le(k/h_0)^nP^n < 1,
 ]
-contradiction: the strict inequality follows because
+contradiction: the strict ineuality follows because
 k/h_0le 1/tau_0 = 2/(3rho), so
 (k/h_0)^nP^nle(2/3)^n(P^n/rho^n)=(2/3)^n<1, since nge2.
 Hence the lowest base locus has a positive-dimensional component
@@ -1522,7 +1522,7 @@ general point, a curve C with R· C/mult_zC smaller than the
 lowest level tau_0.  Every section in the lowest kernel vanishes
 identically on C: otherwise its intersection with C would be at
 least its vanishing order times mult_zC, contradicting the degree
-inequality R· C<tau_0mult_zC and the order bound ktau_0.
+ineuality R· C<tau_0mult_zC and the order bound ktau_0.
 Thus the lowest base locus has a positive-dimensional component
 through the marked point.
 
@@ -1588,10 +1588,10 @@ cases arise.
 Case 1: F is generically finite over a proper
 subvariety.:   Suppose Fto W_0 is generically finite with
 dim W_0<n.  Let P_Y=R_{sl}|_{F^*}, and let L_Y be the
-pullback of L from X.  The first inequality in
+pullback of L from X.  The first ineuality in
 (j:slice-bounds) gives a uniform positive volume bound; since
 F meets neither axis generically, K_{Z_{sl}}|_Fle0, so
-the second inequality gives a uniform canonical ratio
+the second ineuality gives a uniform canonical ratio
 K_{F^*}P_Y^{f_{sl}-1}le cP_Y^{f_{sl}}.  Moreover
 P_Y-rL_Y is nef, since
 P_Y-rL_Y=(P|_{W_0}-rL|_{W_0})+qxi_{sl}|_{F^*} is the sum
@@ -1608,7 +1608,7 @@ coincide with the full PP^1-bundle over W_0.  Write
 w=dim W_0=f_{sl}-1.
 
 If w=0, then F is a projective-line fibre and
-R_{sl}· F=q.  The first inequality in
+R_{sl}· F=q.  The first ineuality in
 (j:slice-bounds), together with (j:bundle-volumes) and
 (j:slice-gap), gives
 [
@@ -1638,7 +1638,7 @@ right, the expansion used for (j:bundle-volumes) gives
  le C'_n qP^w· W_0,
 
 using rge q.  The leading coefficient in the lower bound is the
-term with exactly one xi, as before.  The first inequality in
+term with exactly one xi, as before.  The first ineuality in
 (j:slice-bounds) then bounds 0<P^w· W_0 uniformly from
 above, and (j:saturated-volume)'s leading coefficient bounds it
 from below.  Equations (j:saturated-canonical)--(j:saturated-volume)
@@ -1789,7 +1789,7 @@ On a resolution J^*, Lemma~j:subadjunction, applied to
  
 
 The pair is lc at that generic point because X is normal and hence
-smooth there.  The middle inequality is the Hodge index inequality
+smooth there.  The middle ineuality is the Hodge index ineuality
 on a resolution, obtained by ample approximation of the pulled-back
 testing class; it does not require J to be nef.  The final constant
 is uniform by the volume lower bound and the bound for d_J.
@@ -1941,7 +1941,7 @@ nonzero after reduction to positive characteristic.  A small
 polarization bounds the rank on the diagonal, forcing high vanishing
 at each diagonal point.  A fixed movable curve on a point blowup
 bounds the vanishing of every section of each determinant factor, and
-gives the opposite inequality.
+gives the opposite ineuality.
 
 ## The finite-data statement
 
@@ -1964,7 +1964,7 @@ numbers r>1, ε>0.  Suppose
  H^nle(2ε)^n,
  K_WH^{n-1}le 2 H^n/r,
  (2D+qS)H^{n-1}le 4H^n,
- (14ε)^n<tfrac14, 80ε<tfrac34.
+ (14ε)^n<1/4, 80ε<3/4.
  
 
 Assume that the following fixed data exist.
@@ -1989,19 +1989,19 @@ z_* in their complement, together with finitely many sections of
 k_0M for some integer k_0>0 with k_0D Cartier, generate jets
 through order (2n+2)k_0 at z_*.
  A point xin W, its blowup
-pi_xcolonwidehatwidehat WWto W with exceptional divisor J, and a
+pi_x:WWtohat W with exceptional divisor J, and a
 curve class
 [
  gamma=f_*(A_1... A_{n-1}),
 ]
-where f: Vtowidehatwidehat WW is one fixed birational morphism,
+where f: VtoWWhat is one fixed birational morphism,
 V is smooth integral projective, and the A_i are ample integral
 Cartier divisors, such that J·gamma>0 and
 
  pi_x^*(D+K_W/2+lambda S)·gamma
  le 40ε, J·gammaquad(0lelambdale q).
 
-It suffices to check the two endpoints in this affine inequality.
+It suffices to check the two endpoints in this affine ineuality.
 
 These data cannot coexist.
 
@@ -2119,7 +2119,7 @@ Let F: Wto W' be the relative Frobenius to the base-field
 twist.  It is finite flat because W is smooth.  Write S' for the
 twist of S, so F^*S'=pS, and define
 [
- E=F_*calE=F_*OO_W(B_p),
+ E=F_*E=F_*OO_W(B_p),
   s=rkE=p^n,
   R=s^2.
 ]
@@ -2130,7 +2130,7 @@ evaluation map on W'times W':
  ⊕_{substack{a+b=qa,bge0}}
  U_a ⊗  U_b ⊗ 
  OO(-aS'_1-bS'_2)
- → E E.calE.
+ → E E.E.
 
 
 
@@ -2198,7 +2198,7 @@ On the diagonal of W'times W', every source twist of
 (fr:two-slot-evaluation) becomes -qS'.  Finite base change
 and the projection formula identify the twisted target as
 [
- (E ⊗ E)(qS')calE)(qS')~= h_*L_p.
+ (E ⊗ E)(qS')E)(qS')~= h_*L_p.
 ]
 The product sections defining the columns restrict to global
 sections of this one bundle L_p.  Their values at a
@@ -2322,7 +2322,7 @@ list.  Combining (fr:intersection-bounds),
 (fr:flag-degrees), and Lemma~fr:truncated-slope,
 therefore gives, uniformly for 0le jle u,
 
- mu_{max}(G_j|_C)calG_j|_C)
+ mu_{max}(G_j|_C)G_j|_C)
  &le p(4+2/r)Λ H^n+O(1)notag
  &le M_p:=7pΛ H^n+c_0,
 
@@ -2352,8 +2352,8 @@ b_1,...,b_{k-1} and put
 ]
 For each b_kge0, the divisor restriction sequence is
 [
- 0→F_k(-(b_k+1)h_k)calF_k(-(b_k+1)h_k)
- →F_k(-b_kh_k)calF_k(-b_kh_k)
+ 0→F_k(-(b_k+1)h_k)F_k(-(b_k+1)h_k)
+ →F_k(-b_kh_k)F_k(-b_kh_k)
  →F_k(-b_kh_k)|_calF_k(-b_kh_k)|_{W_{n-k}}
  →0.
 ]
@@ -2362,12 +2362,12 @@ member of the flag bounds the section space by
 
  h^0(W,G_j)
  sum_{b_1,...,b_{n-1}ge0}
- h^0(C,G_j|_C(-calG_j|_C(-sum_i b_ih_i|_C)).
+ h^0(C,G_j|_C(-G_j|_C(-sum_i b_ih_i|_C)).
 
 At each stage, the remainder is zero after sufficiently negative
 ample twisting.  Its stopping index need not be uniform in j,p or
 in the preceding twists: enlarging the nonnegative finite sums to
-the displayed lattice sum preserves the inequality.
+the displayed lattice sum preserves the ineuality.
 
 A summand in (fr:flag-lattice-sum) vanishes when
 sum_i b_i(h_i· C)>M_p, because then its maximum slope is
@@ -2392,7 +2392,7 @@ the number of graded pieces.  Finally,
 [
  7^nH^nle(14ε)^n<1/4
 ]
-by (fr:intersection-bounds), proving the strict inequality for
+by (fr:intersection-bounds), proving the strict ineuality for
 all large p.
 
 
@@ -2403,7 +2403,7 @@ Choose R generically independent columns of
 (fr:two-slot-evaluation) from bases of its source vector spaces.
 Their determinant is a nonzero section
 [
- 0nesigmain H^0(W'times W',Q_1 Q_2)calQ_2)
+ 0nesigmain H^0(W'times W',Q_1 Q_2)Q_2)
 ]
 of an actual external-product line bundle.  To specify its factors,
 let m_i be the sum of the weights of the R chosen columns in slot
@@ -2411,7 +2411,7 @@ i.  The target determinant is
 (detE)^s, and the chosen source
 determinant is OO(-m_1S')boxtimesOO(-m_2S').  Thus
 [
- Q_i=( E)^calE)^{ ⊗  s}
+ Q_i=( E)^E)^{ ⊗  s}
               ⊗ OO_{W'}(m_iS'),
   lambda_i=m_i/R.
 ]
@@ -2433,10 +2433,10 @@ Chern class e_j(p-1)K_W.  Summing gives
 [
  sum_jc_1(A_j(Ω_W^1))=frac{s(p-1)}2K_W.
 ]
-Thus c_1(F^*E)=sB_p+s(p-1)K_W/2,calE)=sB_p+s(p-1)K_W/2, and Frobenius pullback of
+Thus c_1(F^*E)=sB_p+s(p-1)K_W/2,E)=sB_p+s(p-1)K_W/2, and Frobenius pullback of
 twisted divisor classes multiplies by p, as required.
 
-For 0lelambdale q, the fixed movable inequality (fr:movable-input)
+For 0lelambdale q, the fixed movable ineuality (fr:movable-input)
 gives
 [
  (D+K_W/2+lambda S)·gammale40ε J·gamma,
@@ -2535,7 +2535,7 @@ and the nefness of P-rL give
  
  (2D+qS)H_0^{n-1}le(2+q/r)H_0^n.
 ]
-At H_0, the inequalities
+At H_0, the ineualities
 K_WH_0^{n-1}<2H_0^n/r and (2D+qS)H_0^{n-1}<4H_0^n are strict.
 Thus, by continuity of intersection products, a sufficiently small
 rational ample perturbation H=H_0+eta A_W, with A_W an ample
@@ -2581,7 +2581,7 @@ divisible k_0, Serre vanishing gives
 ]
 Since z_* is smooth, the sections on the thickened exceptional
 divisor k_0aG identify with the jet quotient at z_* modulo
-mathfrak m_frakm_{z_*}^{k_0a}, in a local frame for k_0M.  The
+m_{z_*}^{k_0a}, in a local frame for k_0M.  The
 restriction sequence therefore shows that sections of k_0M
 generate jets through order k_0a-1, and hence through order
 (2n+2)k_0.  Increase the divisibility of k_0 to make k_0P
@@ -2598,7 +2598,7 @@ required in Hypothesis~(ii).
 The movable curve class.:   Choose a very general point
 xin W outside the exceptional locus, with image in the locus where
 Lemma~lemma:scalar-bound holds.  Let
-pi_xcolonwidehatwidehat WWto W be the blowup at x and J its
+pi_x:WWtohat W be the blowup at x and J its
 exceptional divisor.  The rational divisor
 [
  D^+=2D+E
@@ -2629,7 +2629,7 @@ movable curves [cite], the strict separation
 in~(fr:application-not-pe) is detected by a class
 [
  gamma=f_*(A_1... A_{n-1}),
-  f: V→widehatwidehat W,W,
+  f: V→What,W,
 ]
 where V is smooth projective, f is birational, and the A_i are
 ample integral Cartier divisors.  To obtain this form, first choose a
@@ -2647,7 +2647,7 @@ For every 0lelambdale q,
  D^+-(D+K_W/2+lambda S)sim_R w^*(P-lambda L)+E/2
 ]
 is pseudo-effective: P-lambda L=(P-rL)+(r-lambda)L is nef, and
-E is effective.  Pulling back to widehatwidehat WW and pairing with
+E is effective.  Pulling back to WWhat and pairing with
 gamma therefore gives
 [
  pi_x^*(D+K_W/2+lambda S)·gamma
@@ -2788,7 +2788,7 @@ Hence
  p^*(sM)ge q^*(sN)
 ]
 as actual divisors.  If N is big, so is q^*N, and the
-inequality makes p^*(sM) big, hence sM (and M) big.
+ineuality makes p^*(sM) big, hence sM (and M) big.
 
 In the case kappa(T_*)>0, Proposition~prop:full-positive
 applied to the pair (X,(1+alpha)B) with M gives T_*+cM big
@@ -2887,7 +2887,7 @@ A positive power of the line bundle on the right is globally
 generated, because A_W is semiample.  Since p_*OO_W=OO_X for
 a birational morphism from a smooth variety, the projection formula
 identifies the sections with those of the same power of mathcal
-A_X.calA_X.  These sections generate downstairs.  Hence A_X is
+A_X.A_X.  These sections generate downstairs.  Hence A_X is
 globally generated after a sufficiently high multiple, so the
 divisor representing A_X, divided by r, is a semiample
 Q-divisor on X numerically equivalent to D.