diff --git a/nef_adjoint_numerical_divisors_v2.txt b/nef_adjoint_numerical_divisors_v2.txt
index 1ae32d0..eae29f7 100644
--- a/nef_adjoint_numerical_divisors_v2.txt
+++ b/nef_adjoint_numerical_divisors_v2.txt
@@ -46,7 +46,7 @@ maketitle
 # Introduction
 
 
-A divisor D on a projective variety is nef if Dcdot Cge 0
+A divisor D on a projective variety is nef if D· Cge 0
 for every integral curve C, and semiample if some positive
 multiple mD is globally generated, so that mD defines a morphism to
 projective space.  Nefness is numerical and semiampleness is geometric;
@@ -132,12 +132,12 @@ Shape of the proof.:   Two simultaneous inductions on the
 dimension n=dim X organise the argument.
 [(P_n)] Suppose T=K_X+B is pseudo-effective, M is a nef
 Q-Cartier divisor on a projective klt Q-pair (X,B) of dimension
-n, and there is a rational a>0 such that (T+aM)cdot C>0 for
+n, and there is a rational a>0 such that (T+aM)· C>0 for
 every integral curve C through a very general point of X.  Then
 T+cM is big for every rational c>0.
 [(Z_n)] Under the same hypotheses, without the
 very-general-curve condition, there is a projective birational model
-wcolon Wto X from a smooth projective W such that the positive
+w: Wto X from a smooth projective W such that the positive
 part P_sigma(w^*(T+M)) of the divisorial Zariski decomposition is a
 rational divisor numerically equivalent to a semiample Q-divisor on
 W.
@@ -220,7 +220,7 @@ Extension to klt Calabi--Yau pairs.:   Once
 Theorem~thm:key-ingredient is proved for terminal X with trivial
 boundary, Section~sec:completion extends it in two steps.  First,
 a klt X with K_Xequiv 0 admits a finite quasi-'etale index-one
-cover picolon X^{ind}to X with K_{X^{ind}}sim 0
+cover pi: X^{ind}to X with K_{X^{ind}}sim 0
 and canonical singularities; a crepant terminalization then gives a
 terminal Q-factorial widetilde Xto X^{ind}.  Full nef
 dimension and bigness both survive the finite quasi-'etale cover, so
@@ -288,7 +288,7 @@ each appearance.
 
 A Q-pair (X,B) consists of a normal projective variety X,
 an effective rational boundary B, and a Q-Cartier divisor K_X+B.
-Log discrepancies are defined on a log resolution hcolon Yto X,
+Log discrepancies are defined on a log resolution h: Yto X,
 via the equality K_Y+B_Y=h^*(K_X+B): a prime divisor Esubset Y
 with coefficient b_E in B_Y carries log discrepancy 1-b_E.
 The pair is klt if all log discrepancies are positive and
@@ -300,7 +300,7 @@ The symbols sim_Q and equiv denote rational linear and numerical
 equivalence.  A divisor D is pseudo-effective if its class in
 N^1(X)_R lies in the closure of the cone generated by effective
 divisors; big if its class lies in the interior of that cone;
-nef if Dcdot Cge0 for every integral curve C.  A divisor
+nef if D· Cge0 for every integral curve C.  A divisor
 is semiample if some positive multiple is globally generated.
 The Iitaka dimension kappa(D) is the maximal dimension of
 the image of a map defined by a linear system |kD|, for k verifying
@@ -316,11 +316,11 @@ countable.
 
 For a nef divisor N on a projective variety, the nef
 reduction [cite] is an almost holomorphic dominant
-rational map picolon Xdashrightarrow Z with connected compact
+rational map pi: Xdashrightarrow Z with connected compact
 general fibres such that:
  N|_Fequiv0 on a very general compact fibre F;
  if C is a curve through a very general point of X not
-contracted by pi, then Ncdot C>0.
+contracted by pi, then N· C>0.
 
 
 The base dimension dim Z is the nef dimension n(X,N).  It
@@ -370,7 +370,7 @@ delta_ito 0.  Each D+delta_iA is big, so there is an effective
 rational divisor E_isim_Q D+delta_iA.  A point outside the
 countable union ∪_iSupp E_i is very general, and every curve
 C through such a point meets no Supp E_i.  Therefore
-(D+delta_iA)cdot Cge0 for every i, and Dcdot Cge0 after
+(D+delta_iA)· Cge0 for every i, and D· Cge0 after
 letting ito∞.
 
 
@@ -454,7 +454,7 @@ For a projective klt Q-pair (X,B) and a nef Q-Cartier divisor
 M on X, write T=K_X+B.  Assume T is pseudo-effective, and
 consider the following condition: for some ainQ_{>0},
 
- (T+aM)cdot C>0
+ (T+aM)· C>0
  quadfor every integral curve C through a very general point of X.
 
 Note that T+aM need not be nef; the condition is on the class it
@@ -472,7 +472,7 @@ pseudo-effective, and suppose (eq:full-condition) holds for some
 rational a>0.  Then T+cM is big for every rational c>0.
 [(Z_n)] Under the same hypotheses on (X,B), M, T but
 without (eq:full-condition), there is a projective birational
-morphism wcolon Wto X from a smooth projective variety such that
+morphism w: Wto X from a smooth projective variety such that
 P_sigma(w^*(T+M)) is rational and numerically equivalent to a
 semiample Q-divisor on W.
 
@@ -515,7 +515,7 @@ nef rational Cartier b-divisor data.  Every
 (K_V+Delta+N_V)-negative extremal ray has an integral rational curve
 generator C with
 [
-  -4nle (K_V+Delta+N_V)cdot C < 0.
+  -4nle (K_V+Delta+N_V)· C < 0.
 ]
 The contraction, and the flip when it is small, can be realized using
 an ordinary klt adjoint negative on the same ray.  Birational
@@ -536,12 +536,12 @@ 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)cdotgammage-tfrac12 Gcdotgamma for every nonzero
+-(G+eta A)·gammage-tfrac12 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.
-Choose a projective log resolution hcolon V'to V carrying
+Choose a projective log resolution h: V'to V carrying
 mathbf NbfN as a locally free sheaf, and write
 [
  K_{V'}+Delta'+N'=h^*G
@@ -565,12 +565,12 @@ negativity lemma.
 The ordinary cone theorem for the klt pair (V,Θ) gives a
 rational curve generator C of the ray with
 [
- 0<-(G+eta A)cdot Cle2n.
+ 0<-(G+eta A)· Cle2n.
 ]
 The chain
--(G+eta A)cdot Cge-tfrac12 Gcdot C
-gives -Gcdot Cle 4n.
-The contraction ccolon Vto S of the ray for G is the contraction
+-(G+eta A)· Cge-tfrac12 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
 cone-contraction uniqueness, and the flip behaves identically: since
 Pic(V/S) is one-dimensional, there is a rational lambda>0 with
@@ -668,14 +668,14 @@ generalized pair with coefficient s is generalized klt.  For a
 ]
 together with nefness of U_i shows the ray is
 (U_i+sN_i)-negative.  Pick its rational curve generator C from
-Lemma~lemma:generalized-ray-bound.  If U_icdot C>0,
-integrality of the free dU_i gives U_icdot Cge1/d, so
+Lemma~lemma:generalized-ray-bound.  If U_i· C>0,
+integrality of the free dU_i gives U_i· Cge1/d, so
 [
- (U_i+s_0N_i)cdot C
+ (U_i+s_0N_i)· C
  ge(1-s_0/s)/(d)-4n(s_0)/(s)>0
 ]
 by (eq:small-nef-coefficient); contradiction.  Thus
-U_icdot C=0.
+U_i· C=0.
 
 The morphism defined by the free dU_i is constant on each fibre of
 the extremal contraction, so it descends to the base; pulling back to
@@ -713,12 +713,12 @@ nonnegative, hence zero, degree along F.  The strict lift
 widetilde Csubset W avoids the effective exceptional divisors
 E_0,E_1.  Both equalities in (eq:two-comparisons) then give
 [
- p^*Tcdotwidetilde C=p^*Mcdotwidetilde C=0.
+ p^*T·widetilde C=p^*M·widetilde C=0.
 ]
 The image p(widetilde C) passes through the chosen very general
-point of X.  Hence (T+aM)cdot p(widetilde C)>0 by
+point of X.  Hence (T+aM)· p(widetilde C)>0 by
 (eq:full-condition), but the two equalities above give
-(T+aM)cdot p(widetilde C)=0.
+(T+aM)· p(widetilde C)=0.
 
 
 
@@ -742,7 +742,7 @@ Proof.
 After a small Q-factorialization, apply
 Lemma~lemma:two-nef-model to obtain the model (Y,B_Y), its
 transforms N, U, and the rational s_0>0.  Let
-hcolon Yto S be the contraction defined by the semiample divisor
+h: Yto S be the contraction defined by the semiample divisor
 U.  Since U is semiample, there is an ample rational divisor
 A_S on S with Usim_Q h^*A_S, and
 dim S=kappa(X,T)>0.  Set Q=U+s_0N.
@@ -878,7 +878,7 @@ numbers with
 ]
 Suppose also A_0=K_Y+D_0+L_0 is big and
 
- A_0cdot P_0^{f-1}le C_1 P_0^f.
+ A_0· P_0^{f-1}le C_1 P_0^f.
 
 Then
 
@@ -911,18 +911,18 @@ map defined by H is birational onto its image; H is nef and big.
 In the rest of this proof, P_0 and L_0 denote pullbacks to
 widetilde Y.  Effectivity of F and nefness of P_0 give
 
- Hcdot P_0^{f-1}le m_f C_1 P_0^f.
+ H· P_0^{f-1}le m_f C_1 P_0^f.
 
 
-For fge2, put s_i=P_0^{f-i}cdot H^i for 0le ile f.
+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
 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
 
- P_0cdot H^{f-1}
- le P_0^f(frac{Hcdot P_0^{f-1}}{P_0^f})^{f-1}
+ P_0· H^{f-1}
+ le P_0^f(frac{H· P_0^{f-1}}{P_0^f})^{f-1}
  le (m_f C_1)^{f-1} C_0.
 
 Choose a very general point yinwidetilde Y in the isomorphism
@@ -934,16 +934,16 @@ general members meet in an effective curve cycle with class
 H^{f-1}; the component C through y is a curve because the
 local equations are independent.  Nefness of P_0 gives
 [
- P_0cdot Cle P_0cdot H^{f-1}.
+ P_0· Cle P_0· H^{f-1}.
 ]
 The curve C is not contracted by mu, and its image passes
 through the very general point mu(y).  Full nef dimension and
-integrality therefore give L_0cdot Cge1.  Since P_0-rL_0 is
-nef, we obtain P_0cdot Cge r, which together with
+integrality therefore give L_0· Cge1.  Since P_0-rL_0 is
+nef, we obtain P_0· Cge r, which together with
 (j:mixed-degree) proves (j:bounded-curve-obstruction).
 
 If f=1, take C=Y directly; then
-P_0cdot Yge rL_0cdot Yge r, and rle C_0 is the claim.
+P_0· Yge rL_0· Yge r, and rle C_0 is the claim.
 Finally, cite[Theorem~1.3]{BZ} applies to every multiple of its
 prescribed integer; a common multiple of the finitely many integers
 m_1,...,m_n gives the last assertion.
@@ -965,7 +965,7 @@ Let X be a normal integral projective variety that is not
 uniruled, and let L be a nef Cartier divisor of full nef dimension
 on X.  Then there is a complement X^circ of a countable union of
 proper closed subsets of X with the following property.  If Y is
-a smooth integral projective variety and gcolon Yto X is
+a smooth integral projective variety and g: Yto X is
 generically finite onto a positive-dimensional image that meets
 X^circ, then g^*L has full nef dimension, Y is not uniruled,
 and K_Y is pseudo-effective.
@@ -1007,7 +1007,7 @@ subvariety are computed after pullback to its normalization and a
 smooth projective resolution, because the subvariety itself may be
 singular.
 
-A linear system mathcal HcalHsubset H^0(Z,OO_Z(kR)) has a
+A linear system H H^0(Z,OO_Z(kR)) has a
 base ideal mathfrak bfrakb 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
@@ -1015,7 +1015,7 @@ localized ideal mathfrak bfrakbOO_{Z,eta_V} is primary for the
 maximal ideal mathfrak mfrakm 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
-strongly mathcal HcalH vanishes along V at its generic point.
+strongly H vanishes along V at its generic point.
 
 The first input is a numerical version of the
 Kawakita--Ambro adjunction for a lc-trivial fibration:
@@ -1026,12 +1026,12 @@ Let Z be a projective Q-factorial klt variety, let Θge0
 be a rational divisor, and let Vsubset Z be integral of dimension
 f>0.  Suppose (Z,Θ) is log canonical at the generic point of
 V, and admits a divisor of log discrepancy zero centered on V.  If
-P is a nef rational Cartier divisor on V and rhocolon V^*to V
+P is a nef rational Cartier divisor on V and rho: V^*to V
 is a smooth projective resolution factoring through its
 normalization, then
 [
- K_{V^*}cdot(rho^*P)^{f-1}
- le (K_Z+Θ)|_Vcdot P^{f-1}.
+ K_{V^*}·(rho^*P)^{f-1}
+ le (K_Z+Θ)|_V· P^{f-1}.
 ]
 
 
@@ -1055,7 +1055,7 @@ The second input transfers a base component to a genuine pair:
 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
-0nemathcal HcalHsubset H^0(Z,OO_Z(kR)) be a linear subspace with
+0neH H^0(Z,OO_Z(kR)) be a linear subspace with
 proper base locus and base ideal mathfrak b.frakb.  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}.
@@ -1067,9 +1067,9 @@ at eta_V and has a log canonical place centered on V.  The
 degree and canonical bounds are
 
  
- R^fcdot V&le(k/h)^{d-f}R^d,
- K_{V^*}cdot R^{f-1}
-   &le(K_Z+cR)|_Vcdot R^{f-1}(f>0),
+ R^f· V&le(k/h)^{d-f}R^d,
+ K_{V^*}· R^{f-1}
+   &le(K_Z+cR)|_V· R^{f-1}(f>0),
  
 
 where V^* is a smooth projective resolution.  The degree bound
@@ -1094,18 +1094,18 @@ k>0 such that kR is Cartier.  Choose positive real numbers
 tau_0<...<tau_d with common gap delta>0.  For xin
 Z_{sm}, put
 [
- mathcal H_i(x)=H^0(Z,calH_i(x)=H^0(Z,OO_Z(kR)
+ H_i(x)=H^0(Z,OO_Z(kR)
               ⊗ mathfrak m_x^frakm_x^{lceil ktau_irceil}).
 ]
 Suppose that for very general x all these spaces are nonzero, and
-that the base locus of mathcal H_0(x)calH_0(x) has a positive-dimensional
+that the base locus of H_0(x) has a positive-dimensional
 component through x.  If kdeltage4, then, after an algebraic
 parameter extension and restriction to nonempty open subsets, there
 are: an integral parameter variety T with a dominant marked-point
-map xcolon Tto Z_{sm}; an index iin{0,...,d-1}
+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 mathcal H_calH_{i+1}(x(t)).  The incidence
+H_i(x(t))calH_i(x(t)) and H_{i+1}(x(t)).  The incidence
 [
  Gamma={(t,z):zin V_t}subset Ttimes Z
 ]
@@ -1114,19 +1114,19 @@ dominates Z.
 For general t, the dimension f of V_t lies in [1,d-1],
 V_t is an isolated component of the higher base locus at its
 generic point, and the entire higher-kernel base ideal is contained
-there in the ordinary power mathcal I_calI_{V_t}^h, where
+there in the ordinary power I_{V_t}^h, where
 [
  h=lceil kdelta/2rceil, k/hle2/delta.
 ]
 The incidence base ideal is likewise contained in
-mathcal I_calI_Gamma^h on a dense open of Gamma.  These
+I_ on a dense open of Gamma.  These
 containments are taken in the smooth ambient open.  If Z is also
 Q-factorial and klt, then
 
  
- R^fcdot V_t&le(2/delta)^{d-f}R^d,
- K_{V_t^*}cdot R^{f-1}
-   &le(K_Z+cR)|_{V_t}cdot R^{f-1},
+ R^f· V_t&le(2/delta)^{d-f}R^d,
+ K_{V_t^*}· R^{f-1}
+   &le(K_Z+cR)|_{V_t}· R^{f-1},
       0<cle2(d-f)/delta.
  
 
@@ -1147,14 +1147,14 @@ morphism:
 [Restriction of an isolated base ideal,
         {cite[Lemma~7.4]{LA}}]
 
-In the setting of Lemma~j:moving-kernel, let gcolon Zto B
+In the setting of Lemma~j:moving-kernel, let g: Zto B
 be a morphism.  There is a dense open of the incidence with the
 following property.  Choose (t,z) in this open, put b=g(z), and
 suppose V_t is smooth over a smooth open of its actual image at
 z.  Let F be the reduced component through z of the scheme
 fibre of V_tto g(V_t).  Near z in Z_b, the restricted
 higher-kernel base ideal has support exactly F and is contained
-in the ordinary power mathcal I_F^h.calI_F^h.
+in the ordinary power I_F^h.
 
 If F is proper in an integral ambient fibre component containing
 it, the restricted series is nonzero there and F is an isolated
@@ -1177,21 +1177,21 @@ a finite list once the branch divisors do not sweep:
         {cite[Lemma~7.5]{LA}}]
 
 Let Y be a normal integral projective variety, let
-mathcal VcalVto T be a smooth projective family with integral fibres
-over an integral parameter variety, and let gcolonmathcal VcalVto Y
+V T be a smooth projective family with integral fibres
+over an integral parameter variety, and let gcolonV Y
 be a morphism.  Suppose the general fibre maps
-g_tcolon V_tto Y are dominant and generically finite of degree
+g_t: V_tto Y are dominant and generically finite of degree
 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
-mathcal R_1,...,calR_1,...,mathcal R_NcalR_N in mathcal V.calV.  Retain this
+R_1,...,calR_1,...,R_N in V.  Retain this
 notation after the base change.  Let I consist of those j for
-which g_t((mathcal R_j)_t)barcalR_j)_t)bar is a divisor in Y for
+which g_t((R_j)_t)bar is a divisor in Y for
 general t.  Assume
 [
- g(mathcal R_j)barcalR_j)barne Yqquad(jin I).
+ g(R_j)bar Y (jin I).
 ]
 After shrinking again, there is a smooth nonempty open
 Y^circsubset Y over which every general finite normal Stein
@@ -1202,7 +1202,7 @@ finite lists may depend on all fixed parameters of the family.
 
 
 The hypothesis removes Y_{sing} and the finitely many
-proper image closures g(mathcal R_j)barcalR_j)bar for jin I.
+proper image closures g(R_j)bar for jin I.
 Every branch prime of a general Stein cover is accounted for by one
 of these relative Jacobian components, so purity makes the
 restricted covers 'etale.  The fundamental group of the resulting
@@ -1215,7 +1215,7 @@ its restriction to this fixed open.
 
 The base-component estimates require a Q-factorial ambient
 variety.  When the ambient is only klt, we may pass to a small
-projective Q-factorialization sigmacolon Z'to Z.  This
+projective Q-factorialization sigma: Z'to Z.  This
 passage is made only for components that meet the isomorphism open of
 sigma.  The strict transforms then retain the generic
 base-ideal and multiplicity conditions for the pulled-back series,
@@ -1281,8 +1281,8 @@ for all t.  After discarding finitely many t we may assume
 
 Every curve through a very general point of X has positive
 integral L-degree, hence P-degree at least r (on such a curve,
-Pcdot C = r(L+tA)cdot C ge r because Lcdot Cge1 and
-Acdot Cge0).  Therefore a covering family of curves of bounded
+P· C = r(L+tA)· C ge r because L· Cge1 and
+A· Cge0).  Therefore a covering family of curves of bounded
 P-degree will contradict full nef dimension.
 
 We will use the same obstruction on subvarieties and covers, so we
@@ -1295,7 +1295,7 @@ K_W by an effective L-exceptional divisor.  In particular W,
 and therefore X, is not uniruled [cite].
 
 By Lemma~j:very-general-images, if Y is a smooth projective
-variety with a generically finite map gcolon Yto X onto a
+variety with a generically finite map g: Yto X onto a
 positive-dimensional image that meets a sufficiently very general
 point of X, then Y is not uniruled, K_Y is pseudo-effective,
 and the pullback L_Y=g^*L has full nef dimension.  If
@@ -1310,7 +1310,7 @@ Lemma~j:very-general-images.
 Consequently, for f<n it suffices to find a nef testing divisor
 P_Y on such Y with
 [
- 0<P_Y^fle C_0, P_Y-rL_Y nef,qquadand 
+ 0<P_Y^fle C_0, P_Y-rL_Y nef, and 
  K_YP_Y^{f-1}le C, P_Y^f,
 ]
 where C_0 and C are independent of t.  Indeed
@@ -1350,7 +1350,7 @@ powers makes k arbitrarily large and divisible.
 Choose n+1 levels
 [
  tau_i=frac32rho+ideltaquad(0le ile n),
- qquaddelta=(2rho)/(n).
+  delta=(2rho)/(n).
 ]
 These lie strictly between rho and 4rho.  For a sufficiently
 large divisible k, the corresponding full jet kernels are nonzero
@@ -1405,7 +1405,7 @@ Set
 
  Z=PP(OO_X(L)_1 ⊕ OO_X(L)_2)
  xrightarrow{pi}Xtimes X,
- qquadxi=c_1(OO_Z(1)),
+  xi=c_1(OO_Z(1)),
   R=P_1+P_2+qxi,
 
 where q is a positive integer to be fixed shortly.  We use the
@@ -1467,7 +1467,7 @@ pi_*(xi^j)=L^{j-1}.
 For a nef rational Cartier divisor R and a smooth point z, its
 Seshadri constant is
 [
- ε(R;z)=inf_{Cni z}(Rcdot C)/(mult_z C),
+ ε(R;z)=inf_{Cni z}(R· C)/(mult_z C),
 ]
 where C ranges over integral curves through z.  A lower bound
 for this constant controls the degree of curves through z relative
@@ -1513,16 +1513,16 @@ nonzero for sufficiently large divisible k, because the highest
 level has d-th power smaller than R^d.  More explicitly: the
 length of the quotient by jets of order <ktau is
 binom{ktau+d}{d}sim(ktau)^d/d!, and the number of sections is
-chi(OO(kR))sim(kR)^d/d!=(ktau)^d/d!cdot (R/tau)^d, which
+chi(OO(kR))sim(kR)^d/d!=(ktau)^d/d!· (R/tau)^d, which
 dominates as long as tau^d<R^d.  For our chosen levels
 tau_i<(c_nqε^{2n})^{1/d}le (R^d)^{1/d}, this holds.
 
 Failure of (j:large-seshadri) supplies, through each very
-general point, a curve C with Rcdot C/mult_zC smaller than the
+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 Rcdot C<tau_0mult_zC and the order bound ktau_0.
+inequality 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.
 
@@ -1532,13 +1532,13 @@ components Vsubset Z whose incidence dominates Z.  For general
 members of dimension f, the estimates are
 
  
- 0<R^fcdot V&le(2/delta)^{d-f}R^d,
- K_{V^*}R^{f-1}&le(K_Z+cR)|_Vcdot R^{f-1},
+ 0<R^f· V&le(2/delta)^{d-f}R^d,
+ K_{V^*}R^{f-1}&le(K_Z+cR)|_V· R^{f-1},
   0<cle 2(d-f)/delta.
  
 
 Here and below, classes on a resolution are understood by pullback.
-The positivity R^fcdot V>0 follows from ampleness of R.  General
+The positivity R^f· V>0 follows from ampleness of R.  General
 members meet the torus open, so (j:axes-canonical) makes the
 ambient canonical contribution nonpositive.  All constants in these
 bounds are uniform once q,ε,delta have been fixed.
@@ -1563,18 +1563,18 @@ le n.  The ambient fibre of either projection Zto X over a
 smooth point is the slice Z_{sl} of dimension n+1.
 Lemma~j:fiber-restriction gives a nonzero restricted subseries
 with F an isolated base component, whose base ideal is contained in
-mathcal I_F^hcalI_F^h with h=lceil kdelta/2rceil.  Generic smoothness
+I_F^h with h=lceil kdelta/2rceil.  Generic smoothness
 is imposed over the actual image of V, which is why a general fibre
 through the incidence point was chosen; the chosen point is smooth
 in the ambient slice by the very general choice.  Applying
 Lemma~j:base-component on that slice gives
 
  
- 0<R_{sl}^{f_{sl}}cdot F
+ 0<R_{sl}^{f_{sl}}· F
  &le(2/delta)^{n+1-f_{sl}}
   R_{sl}^{n+1},
  K_{F^*} R_{sl}^{f_{sl}-1}
- &le(K_{Z_{sl}}+cR_{sl})|_F
+ &le(K_{Z_{sl}}+R_{sl})|_F
   R_{sl}^{f_{sl}-1},
  
 
@@ -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}cdot F=q.  The first inequality in
+R_{sl}· F=q.  The first inequality in
 (j:slice-bounds), together with (j:bundle-volumes) and
 (j:slice-gap), gives
 [
@@ -1618,13 +1618,13 @@ R_{sl}cdot F=q.  The first inequality in
 impossible.
 
 Suppose w>0.  Resolve W_0 to W_0^*, and take as a resolution
-of F the induced projective bundle p_0colon F^*to W_0^*.  Its
+of F the induced projective bundle p_0: F^*to W_0^*.  Its
 relative canonical divisor is -2xi+p_0^*L, exactly the
 restriction of K_{Z_{sl}}.  Cancelling this term in
 (j:slice-bounds) gives
 
  p_0^*K_{W_0^*}R_{sl}^w
- le cR_{sl}^{w+1}cdot F.
+ le R_{sl}^{w+1}· F.
 
 The divisor K_{W_0^*} is pseudo-effective by the very general point
 condition.  On expanding the left side of (j:saturated-canonical)
@@ -1633,18 +1633,18 @@ wqK_{W_0^*}P^{w-1}; every other nonzero term is nonnegative, since
 it pairs the pseudo-effective K_{W_0^*} with nef classes.  On the
 right, the expansion used for (j:bundle-volumes) gives
 
- (w+1)qP^wcdot W_0
- le R_{sl}^{w+1}cdot F
- le C'_n qP^wcdot W_0,
+ (w+1)qP^w· W_0
+ le R_{sl}^{w+1}· F
+ 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
-(j:slice-bounds) then bounds 0<P^wcdot W_0 uniformly from
+(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)
 together give
 [
- K_{W_0^*}P^{w-1}le(cC'_n)/(w)P^wcdot W_0.
+ K_{W_0^*}P^{w-1}le(cC'_n)/(w)P^w· W_0.
 ]
 Now apply (j:lower-big) on W_0^*, with the pullback of P as
 the testing class and L as the full-nef-dimension divisor.
@@ -1664,10 +1664,10 @@ associated rational section, or equivalently its projection, gives
 
 For either axis S, the classes R_{sl}-qS and
 R_{sl}-P are nef.  Intersecting with the effective cycle
-Scdot F and then using nefness on F yields
+S· F and then using nefness on F yields
 
- qD_iP^{n-1}le qScdot Fcdot R_{sl}^{n-1}
- le R_{sl}^ncdot F.
+ qD_iP^{n-1}le qS· F· R_{sl}^{n-1}
+ le R_{sl}^n· F.
 
 Thus both axis-image degrees are uniformly bounded.
 
@@ -1683,7 +1683,7 @@ would give an ordinary MMP ending at a good model Y whose
 transformed adjoint is rationally linearly trivial.  The transform of
 K_X is then linearly trivial as well, so the effective transform
 of G must vanish: every component of G has been contracted.
-On a common resolution acolon Uto X and b_Ycolon Uto Y,
+On a common resolution a: Uto X and b_Y: Uto Y,
 relation (j:signed-L) then expresses a^*L up to rational
 linear equivalence as a signed b_Y-exceptional divisor E.
 Divisors exceptional over X remain exceptional over Y, because
@@ -1702,7 +1702,7 @@ The already proved positive-Iitaka-dimension case
 Its full-positivity hypothesis holds because L has full nef
 dimension and K_X+bG is pseudo-effective.
 
-Take a log resolution acolon Yto X of G, and let D_0 be the
+Take a log resolution a: Yto X of G, and let D_0 be the
 reduced simple normal crossing divisor consisting of the strict
 support of G together with every exceptional divisor.  Decrease b
 if necessary; Proposition~prop:full-positive still applies.
@@ -1750,9 +1750,9 @@ For the remaining V: dimension n, dominant over both factors of
 Xtimes X.  Equation~(j:total-component-bounds) gives uniform
 bounds
 
- R^ncdot Vle C,
+ R^n· Vle C,
  
- K_{V^*}R^{n-1}le cR^ncdot Vle C,
+ K_{V^*}R^{n-1}le cR^n· Vle C,
 
 after increasing C.  Each projection has bounded degree:
 R-P_i is nef, and P^n is bounded below.
@@ -1776,14 +1776,14 @@ positive because its image meets X_{sm} generically.  The
 difference R-P_i is nef, so (j:correspondence-bounds) and the
 projection formula bound
 [
- 0<d_J:=P^{n-1}cdot Jle C.
+ 0<d_J:=P^{n-1}· Jle C.
 ]
 On a resolution J^*, Lemma~j:subadjunction, applied to
 (X,J) at the generic point of J, gives
 
  
  K_{J^*}P^{n-2}
- &le(K_X+J)cdot Jcdot P^{n-2}
+ &le(K_X+J)· J· P^{n-2}
  &=J^2P^{n-2}
  lefrac{(JP^{n-1})^2}{P^n}le C' d_J.
  
@@ -1858,7 +1858,7 @@ therefore return to an original member V^* of this pair, which
 retains (j:correspondence-bounds), and resolve further to
 obtain a birational morphism
 [
- p_0colon V^*→ A_0
+ p_0: V^*→ A_0
 ]
 to an abelian variety.  Higher resolutions do not change the
 canonical intersection with the pulled-back (n-1)-st power of R.
@@ -1976,12 +1976,12 @@ fixed positive integers l_i for which l_iH is Cartier, with
 mu_{min}(Ω_W^1|_C)ge0.
  On the projective bundle
 [
- Z=PP(mathcal O_W(S)_1calO_W(S)_1 ⊕ mathcal O_W(S)_2)calO_W(S)_2)
+ Z=PP(O_W(S)_1 ⊕ O_W(S)_2)
  → Wtimes W,
-  xi=c_1(mathcal O_Z(1)),calO_Z(1)),
+  xi=c_1(O_Z(1)),
   M=D_1+D_2+qxi,
 ]
-use the convention that the direct image of mathcal O_Z(acalO_Z(axi) is
+use the convention that the direct image of O_Z(a is
 the a-th symmetric power of the displayed sum for age0; subscripts
 indicate pullback from the corresponding factor of Wtimes W.  The
 two axes are the sections defined by the two summands.  A smooth point
@@ -1994,12 +1994,12 @@ curve class
 [
  gamma=f_*(A_1... A_{n-1}),
 ]
-where fcolon Vtowidehatwidehat WW is one fixed birational morphism,
+where f: Vtowidehatwidehat WW is one fixed birational morphism,
 V is smooth integral projective, and the A_i are ample integral
-Cartier divisors, such that Jcdotgamma>0 and
+Cartier divisors, such that J·gamma>0 and
 
- pi_x^*(D+K_W/2+lambda S)cdotgamma
- le 40ε, Jcdotgammaquad(0lelambdale q).
+ 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.
 
@@ -2024,7 +2024,7 @@ Put Λ=prod_{i=1}^{n-1}l_i.  The complete-intersection flag
 satisfies
 
  [C]=Λ H^{n-1},
-  h_icdot C=l_iΛ H^n.
+  h_i· C=l_iΛ H^n.
 
 Choose one sufficiently ample integral Cartier divisor T_0 so that
 every T_0+aS, for integers 0le ale(k_0-1)q, has a section
@@ -2069,7 +2069,7 @@ specialization of the characteristic-zero effective cone is used.
 
 For a sufficiently large residual characteristic p, set
 [
- N_p=lfloor p/k_0rfloor,
+ N_p=floor( p/k_0),
   B_p=k_0N_pD+T_0.
 ]
 Then the sections of pqxi+B_{p,1}+B_{p,2} generate jets through
@@ -2115,12 +2115,12 @@ cite[proof of Proposition~2.12, Equation~(2.8)]{MustataSchwede}.
 ## Full rank from the two-slot jets
 
 
-Let Fcolon Wto W' be the relative Frobenius to the base-field
+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
 [
- mathcal E=F_*calE=F_*OO_W(B_p),
-  s=rkmathcal E=p^n,calE=p^n,
+ E=F_*calE=F_*OO_W(B_p),
+  s=rkE=p^n,
   R=s^2.
 ]
 For integers 0le ale q, put
@@ -2130,7 +2130,7 @@ evaluation map on W'times W':
  ⊕_{substack{a+b=qa,bge0}}
  U_a ⊗  U_b ⊗ 
  OO(-aS'_1-bS'_2)
- → mathcal EcalEboxtimesmathcal E.calE.
+ → E E.calE.
 
 
 
@@ -2182,32 +2182,32 @@ The Frobenius fiber product has the cartesian square
 
 Its reduced subscheme is the diagonal W.  Set
 [
- mathcal L_p=calL_p=OO_{Delta_F}(B_{p,1}+B_{p,2}+pqS_1).
+ L_p={Delta_F}(B_{p,1}+B_{p,2}+pqS_1).
 ]
-Write h=Fcircpr_1=Fcircpr_2colon
+Write h=Fcircpr_1=Fcircpr_2:
 Delta_Fto W'.  On this fiber product,
 [
- OO(pS_1)simeq h^*OO_{W'}(S')simeqOO(pS_2).
+ OO(pS_1)~= h^*OO_{W'}(S')~=OO(pS_2).
 ]
 Consequently every pair of weights a+b=q gives the same line bundle:
 [
  OO_{Delta_F}(B_{p,1}+B_{p,2}+paS_1+pbS_2)
- simeqmathcal L_p.calL_p.
+ ~=L_p.
 ]
 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
 [
- (mathcal EcalE ⊗ mathcal E)(qS')calE)(qS')simeq h_*mathcal L_p.calL_p.
+ (E ⊗ E)(qS')calE)(qS')~= h_*L_p.
 ]
 The product sections defining the columns restrict to global
-sections of this one bundle mathcal L_p.calL_p.  Their values at a
+sections of this one bundle L_p.  Their values at a
 diagonal point therefore lie in the image of
-H^0(Delta_F,mathcal L_p)calL_p) in the corresponding fibre of
-h_*mathcal L_p.calL_p.  This proves that the rank at every diagonal point
+H^0(Delta_F,L_p) in the corresponding fibre of
+h_*L_p.  This proves that the rank at every diagonal point
 is at most
 
- h^0(Delta_F,mathcal L_p).calL_p).
+ h^0(Delta_F,L_p).
 
 
 For a rank-n bundle V in characteristic p, denote by A_j(V)
@@ -2220,10 +2220,10 @@ of V^{ ⊗  j}.  Set
  u=n(p-1), e_j=rk A_j(V),
  sum_{j=0}^u e_j=p^n.
 ]
-Filtering mathcal L_pcalL_p by powers of the ideal of the reduced
+Filtering L_p by powers of the ideal of the reduced
 diagonal in Delta_F gives the graded bundles
 
- mathcal G_j=calG_j=OO_W(2B_p+pqS) ⊗  A_j(Ω_W^1),
+ G_j=+pqS) ⊗  A_j(Ω_W^1),
   0le jle u.
 
 Indeed, in smooth local coordinates, the algebra is generated by
@@ -2233,7 +2233,7 @@ be performed 'etale locally or in completions and identifies the
 global associated graded with (fr:diagonal-graded).  This is
 the diagonal ideal form of the canonical Frobenius filtration
 cite[Corollary~3.5]{KS}.  The same calculation, using only the
-bundle from the second factor, filters F^*mathcal EcalE with pieces
+bundle from the second factor, filters F^*E with pieces
 cite[Theorem~3.7]{Sun}
 
  OO_W(B_p) ⊗  A_j(Ω_W^1).
@@ -2241,7 +2241,7 @@ cite[Theorem~3.7]{Sun}
 
 Complementary multiplication is a perfect pairing:
 
- A_{u-j}(V)simeq A_j(V)^vee ⊗ (det V)^{p-1}.
+ A_{u-j}(V)~= A_j(V)^vee ⊗ (det V)^{p-1}.
 
 On monomials, each exponent vector is paired with its complement to
 (p-1,...,p-1).  The top line transforms by the character
@@ -2263,7 +2263,7 @@ There is a constant cge0, independent of the sufficiently large
 residual characteristic p and of j, such that for
 V=Ω_W^1|_C one has
 [
- mu_{max}(A_j(V))le(p-1)K_Wcdot C+nc.
+ mu_{max}(A_j(V))le(p-1)K_W· C+nc.
 ]
 
 
@@ -2312,8 +2312,8 @@ pairing (fr:socle-pairing) gives
 
  mu_{max}(A_j(V))
  &= (p-1)degdet V-mu_{min}(A_{u-j}(V))
- &le(p-1)K_Wcdot C+((u-j)c)/(p-1)
- &le(p-1)K_Wcdot C+nc.
+ &le(p-1)K_W· C+((u-j)c)/(p-1)
+ &le(p-1)K_W· C+nc.
 
 
 
@@ -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}(mathcal G_j|_C)calG_j|_C)
+ mu_{max}(G_j|_C)calG_j|_C)
  &le p(4+2/r)Λ H^n+O(1)notag
  &le M_p:=7pΛ H^n+c_0,
 
@@ -2332,7 +2332,7 @@ harmless room in this common bound.
 
 For all sufficiently large p,
 [
- h^0(Delta_F,mathcal L_p)calL_p)
+ h^0(Delta_F,L_p)
  le R(7^nH^n+O(1/p))<R/4.
 ]
 The error constant is independent of the filtration index j.
@@ -2348,21 +2348,21 @@ Write the fixed flag as
 At step k, fix the preceding nonnegative integers
 b_1,...,b_{k-1} and put
 [
- mathcal F_k=calF_k=mathcal G_j|_calG_j|_{W_{n-k+1}}(-sum_{i<k}b_ih_i).
+ F_k= G_j|_calG_j|_{W_{n-k+1}}(-sum_{i<k}b_ih_i).
 ]
 For each b_kge0, the divisor restriction sequence is
 [
- 0→mathcal F_k(-(b_k+1)h_k)calF_k(-(b_k+1)h_k)
- →mathcal F_k(-b_kh_k)calF_k(-b_kh_k)
- →mathcal F_k(-b_kh_k)|_calF_k(-b_kh_k)|_{W_{n-k}}
+ 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)
+ →F_k(-b_kh_k)|_calF_k(-b_kh_k)|_{W_{n-k}}
  →0.
 ]
 Iterating it for b_k=0,1,... and then proceeding to the next
 member of the flag bounds the section space by
 
- h^0(W,mathcal G_j)calG_j)le
+ h^0(W,G_j)
  sum_{b_1,...,b_{n-1}ge0}
- h^0(C,mathcal G_j|_C(-calG_j|_C(-sum_i b_ih_i|_C)).
+ h^0(C,G_j|_C(-calG_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
@@ -2370,13 +2370,13 @@ in the preceding twists: enlarging the nonnegative finite sums to
 the displayed lattice sum preserves the inequality.
 
 A summand in (fr:flag-lattice-sum) vanishes when
-sum_i b_i(h_icdot C)>M_p, because then its maximum slope is
+sum_i b_i(h_i· C)>M_p, because then its maximum slope is
 negative.  Each nonzero summand has at most e_j(M_p+1) sections.
-Indeed, evaluation at lfloor M_prfloor+1 distinct points is
+Indeed, evaluation at floor( M_p)+1 distinct points is
 injective; the kernel has negative maximum slope after subtracting
 those points.  By (fr:flag-degrees) it follows that
 
- h^0(W,mathcal G_j)calG_j)
+ h^0(W,G_j)
  &le e_j(M_p+1)
    prod_{i=1}^{n-1}(1+(M_p)/(l_iΛ H^n))notag
  &=e_jp^n(7^nH^n+O(1/p)).
@@ -2403,15 +2403,15 @@ 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',mathcal Q_1calQ_1boxtimesmathcal Q_2)calQ_2)
+ 0nesigmain H^0(W'times W',Q_1 Q_2)calQ_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
 i.  The target determinant is
-(detmathcal E)^scalE)^sboxtimes(detmathcal E)^s,calE)^s, and the chosen source
+(detE)^s, and the chosen source
 determinant is OO(-m_1S')boxtimesOO(-m_2S').  Thus
 [
- mathcal Q_i=(calQ_i=(detmathcal E)^calE)^{ ⊗  s}
+ Q_i=( E)^calE)^{ ⊗  s}
               ⊗ OO_{W'}(m_iS'),
   lambda_i=m_i/R.
 ]
@@ -2420,8 +2420,8 @@ independently of the columns chosen.
 
 Under the numerical identification with the base-field twist,
 
- frac{c_1(mathcal Q_i)}calQ_i)R
- =frac{c_1(mathcal E)}calE)s+lambda_iS
+ frac{c_1(Q_i)}
+ =frac{c_1(E)}+lambda_iS
  =frac{B_p}p+(p-1)/(2p)K_W+lambda_iS,
   0lelambda_ile q.
 
@@ -2433,13 +2433,13 @@ 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^*mathcal 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,calE)=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)
 gives
 [
- (D+K_W/2+lambda S)cdotgammale40ε Jcdotgamma,
+ (D+K_W/2+lambda S)·gammale40ε J·gamma,
 ]
 where pullback by pi_x is understood.  This concerns only
 intersections of fixed divisors and the fixed complete-intersection
@@ -2454,10 +2454,10 @@ T_0-aD-K_W/2, ain{0,...,k_0-1}.  The resulting error after
 intersection with gamma is therefore O(p^{-1}), uniformly over
 the selected determinant columns.
 
-Let x' be the twisted point, and let 0netau_iin H^0(W',mathcal Q_i).calQ_i).
+Let x' be the twisted point, and let 0netau_iin H^0(W',Q_i).
 The strict transform of the effective divisor of tau_i on the
 blowup at x' pairs nonnegatively with the twist of gamma.  Since
-Jcdotgamma>0, it follows that
+J·gamma>0, it follows that
 
  frac{ord_{x'}(tau_i)}Rle40ε+O(1/p).
 
@@ -2466,7 +2466,7 @@ argument tests sections on the reduction itself; it does not require
 them to lift to characteristic zero.
 
 K"unneth identifies the space containing sigma with
-H^0(W',mathcal Q_1)calQ_1) ⊗  H^0(W',mathcal Q_2).calQ_2).  Choose bases
+H^0(W',Q_1) ⊗  H^0(W',Q_2).  Choose bases
 adapted to the orders of vanishing at x' in each factor.  In each
 degree, leading terms are linearly independent.  Tensor products of
 these leading terms remain independent in each bidegree, since the
@@ -2522,7 +2522,7 @@ and the projective bundle in (eq:bundle).  Every remaining
 choice is made over C before reducing to positive characteristic.
 
 The divisors and the flag.:   Choose a smooth projective
-resolution wcolon Wto X and set
+resolution w: Wto X and set
 [
  D=w^*P, S=w^*L, H_0=w^*P.
 ]
@@ -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}),
-  fcolon V→widehatwidehat W,W,
+  f: V→widehatwidehat W,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
@@ -2637,10 +2637,10 @@ generating strongly movable class with negative pairing, then
 approximate its ample real classes by rational ones and clear
 denominators.  We have
 [
- (pi_x^*D^+)cdotgamma<40ε Jcdotgamma.
+ (pi_x^*D^+)·gamma<40ε J·gamma.
 ]
 The left side is nonnegative, because pi_x^*D^+ is
-pseudo-effective, so Jcdotgamma>0.
+pseudo-effective, so J·gamma>0.
 
 For every 0lelambdale q,
 [
@@ -2650,9 +2650,9 @@ 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
 gamma therefore gives
 [
- pi_x^*(D+K_W/2+lambda S)cdotgamma
- le(pi_x^*D^+)cdotgamma
- <40ε Jcdotgamma.
+ pi_x^*(D+K_W/2+lambda S)·gamma
+ le(pi_x^*D^+)·gamma
+ <40ε J·gamma.
 ]
 This proves Hypothesis~(iii).  All hypotheses of
 Theorem~thm:finite-data now hold for one fixed collection of
@@ -2687,7 +2687,7 @@ divisible multiple is linearly trivial.
 
 Take the global index-one cover
 [
- picolon X^{ind}→ X
+ pi: X^{ind}→ X
 ]
 associated with a trivialization of the smallest positive multiple
 of K_X that is linearly trivial.  This is a finite quasi-'etale
@@ -2701,7 +2701,7 @@ canonical.  By the crepant terminalization theorem
 cite[Corollary~1.4.3]{BCHM}, there is a projective birational
 morphism
 [
- hcolon widetilde X→ X^{ind}
+ h: widetilde X→ X^{ind}
 ]
 with widetilde X terminal and Q-factorial and
 K_{widetilde X}=h^*K_{X^{ind}}sim 0.
@@ -2730,8 +2730,8 @@ Choose a rational alpha>0 sufficiently small that
 ]
 Then T_* is pseudo-effective and kappa(T_*)ge0.  The full
 condition holds for T_* and M: at a curve C through a point
-outside Supp B, T_*cdot C=alpha Bcdot Cge0, while
-Mcdot C>0 if that point is very general.
+outside Supp B, T_*· C=alpha B· Cge0, while
+M· C>0 if that point is very general.
 
 We do not use (P_n) here; the argument uses only
 Lemma~lemma:two-nef-model and the B=0 case just proved.
@@ -2778,8 +2778,8 @@ The difference D:=E_1-E_0=p^*(sM)-q^*(sN) is q-exceptional:
 its pushforward under q is sN-sN=0.  We claim D is also
 q-nef.  Indeed, if C is any curve contracted by q, then
 [
- Dcdot C=p^*(sM)cdot C-q^*(sN)cdot C
-      =sMcdot p_*C-0ge0,
+ D· C=p^*(sM)· C-q^*(sN)· C
+      =sM· p_*C-0ge0,
 ]
 because M is nef and q^*N is numerically trivial on the
 q-contracted C.  The negativity lemma then gives Dge0.
@@ -2819,14 +2819,14 @@ to X itself, using that X has rational singularities.
 
 
 The positive part P_sigma satisfies the following properties.
- If pcolon Vto X is a projective birational morphism from a
+ If p: Vto X is a projective birational morphism from a
 smooth projective V to a normal projective X, and D is a
 pseudo-effective rational divisor on X, and Ege0 is
 p-exceptional, then
 [
  P_sigma(p^*D+E)=P_sigma(p^*D).
 ]
- If fcolon Vto S is a surjective morphism of smooth
+ If f: Vto S is a surjective morphism of smooth
 projective varieties, D is pseudo-effective on S, and
 P_sigma(D) is nef, then
 [
@@ -2859,7 +2859,7 @@ rational singularities.  Recall that klt varieties have rational
 singularities.
 
 
-Let pcolon Wto X be a projective birational morphism from a
+Let p: Wto X be a projective birational morphism from a
 smooth projective W to a normal projective X with rational
 singularities.  Let D be a Q-Cartier divisor on X.  If
 p^*D is numerically equivalent to a semiample Q-divisor on
@@ -2874,22 +2874,22 @@ the corresponding line bundles is numerically trivial.  Numerically
 trivial line bundles modulo Pic^0(W) are torsion.  Thus, after
 raising a positive integer r, there is
 [
- OO_W(rA_W)simeq p^*OO_X(rD) ⊗  P_W,
+ OO_W(rA_W)~= p^*OO_X(rD) ⊗  P_W,
   P_WinPic^0(W).
 ]
 Rational singularities identify Pic^0(X) with Pic^0(W) by
 pullback.  Write P_W=p^*P_X for P_XinPic^0(X), and set
-mathcal A_X=calA_X=OO_X(rD) ⊗  P_X.  Then
+A_X= ⊗  P_X.  Then
 [
- p^*mathcal A_XcalA_XsimeqOO_W(rA_W).
+ p^*A_X~=OO_W(rA_W).
 ]
 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 mathcal A_XcalA_X is
+A_X.calA_X.  These sections generate downstairs.  Hence A_X is
 globally generated after a sufficiently high multiple, so the
-divisor representing mathcal A_X,calA_X, divided by r, is a semiample
+divisor representing A_X, divided by r, is a semiample
 Q-divisor on X numerically equivalent to D.
 
 
@@ -2906,13 +2906,13 @@ summand.
 
 
 There are an integer m_0>0, a birational contraction
-varphicolon Xdashrightarrow Y between klt Q-factorial
+phi: Xdashrightarrow Y between klt Q-factorial
 projective varieties, and a common smooth model W with
-pcolon Wto X and qcolon Wto Y, satisfying the following
+p: Wto X and q: Wto Y, satisfying the following
 properties.
  The transform T' of T=K_X+B is pseudo-effective, N (the
 transform of M) is nef, and T'+m_0 N is nef on Y.
- The contraction varphi is M-trivial: on W,
+ The contraction phi is M-trivial: on W,
 p^*M=q^*N.
  There is an effective q-exceptional rational divisor Ege0
 on W with p^*T=q^*T'+E.
@@ -2938,12 +2938,12 @@ program; we verify M-triviality.
 Consider a (T+m_0M)-negative extremal ray, and let C be a
 rational curve generating it.  By the length bound
 (Lemma~lemma:generalized-ray-bound), we may choose C with
-0<-Tcdot Cle2n.  If Mcdot C>0, then m_0Mcdot Cge1 by
+0<-T· Cle2n.  If M· C>0, then m_0M· Cge1 by
 integrality (the divisor m_0M is Cartier).  Hence
 [
- (T+m_0M)cdot Cge -2n +1>0,
+ (T+m_0M)· Cge -2n +1>0,
 ]
-a contradiction if m_0>2n.  So Mcdot C=0 on every
+a contradiction if m_0>2n.  So M· C=0 on every
 (T+m_0M)-negative extremal ray, and M therefore descends
 through the contraction by Cartier descent in the ordinary cone
 and contraction theorem.  The same reasoning applies to a small
@@ -3008,7 +3008,7 @@ case.
 ## Descent along a nef reduction
 
 Now suppose T+mL does not have full nef dimension.  Let
-hcolon Xdashrightarrow Z_1 be the nef reduction of T+mL
+h: Xdashrightarrow Z_1 be the nef reduction of T+mL
 (Definition at the start of Ssec:preliminaries), so
 dim Z_1=r<n.  Fix a general compact fibre F.
 
@@ -3110,7 +3110,7 @@ The pair (Z_2,B_{Z_2}) is klt of dimension dim Z_2<n.  Let
 M_{Z_2}=j^*M_1 be the pullback of M_1 to Z_2: it is nef
 (random check: j is a morphism so j^*M_1 is nef).  The
 induction hypothesis (Z_{dim Z_2}) applies and gives a smooth
-model ncolon W_{Z_2}to Z_2 with
+model n: W_{Z_2}to Z_2 with
 [
  P_sigma(n^*(K_{Z_2}+B_{Z_2}+M_{Z_2}))
 ]
@@ -3158,7 +3158,7 @@ With both inductions established, we can now prove
 Theorem~thm:main over C.  Let (X,B) be a projective klt
 Q-pair of dimension n, M a nef Q-Cartier divisor, and
 K_X+B pseudo-effective.  The sum D=K_X+B+M is nef by
-hypothesis.  On any smooth birational model wcolon Wto X,
+hypothesis.  On any smooth birational model w: Wto X,
 [
  P_sigma(w^*D)=w^*D,
 ]