diff --git a/log_abundance_char_zero_v2.txt b/log_abundance_char_zero_v2.txt
index 58cf64e..ace7993 100644
--- a/log_abundance_char_zero_v2.txt
+++ b/log_abundance_char_zero_v2.txt
@@ -107,7 +107,7 @@ produce two outputs, in this order.
 [label=(roman*)]
  Smooth canonical nonvanishing (NV_n). A smooth
 projective complex n-fold X with pseudo-effective K_X has
-H^0(X,mK_X)neq0 for some m>0.
+H^0(X,mK_X)ne0 for some m>0.
  Good log minimal models in dimension n for all
 projective lc pairs with effective real boundary and pseudo-effective
 adjoint. On a good model the adjoint is semiample. This is the
@@ -284,7 +284,7 @@ zero (Section~sec:completion).
 Throughout, kappa(L) is the Iitaka dimension of a rational divisor
 L, with kappa(L)=-∞ if every positive integral system is
 empty; nu(L) is the numerical dimension. For nef L we always have
-kappa(L)lenu(L); abundance is the reverse inequality.
+kappa(L)lenu(L); abundance is the reverse ineuality.
 
 ## Consequences covered by this paper
 
@@ -722,7 +722,7 @@ upstairs thus implies generation downstairs.
 
 For a klt source, the coefficient at an extracted prime would be
 strictly negative, an impossibility; the remaining discrepancy
-inequalities make the model klt.
+ineualities make the model klt.
 
 
 [Semiampleness on the reduced boundary]
@@ -780,12 +780,12 @@ the same integer m is Cartier.
 
 
 Proof. 
-Both Delta-ε P and Delta-tfrac12P are effective klt
+Both Delta-ε P and Delta-1/2P are effective klt
 boundaries. If R is a ray negative for L-ε P, nefness of L
-makes R negative for L-tfrac12P. Choose a rational curve C
+makes R negative for L-1/2P. Choose a rational curve C
 spanning R and satisfying the length bound
 [
- -(L-tfrac12P)· Cle2n.
+ -(L-1/2P)· Cle2n.
 ]
 Writing a=L· Cge0 and p=P· C, we have
 [
@@ -807,7 +807,7 @@ separately, that the transformed L is nef and that mL remains
 Cartier with the same m.
 
 The driving boundary remains klt throughout its LMMP. The transformed
-Delta-tfrac12P remains effective and is smaller than the driving
+Delta-1/2P remains effective and is smaller than the driving
 boundary, so it too is klt. The same estimate and the same integer m
 apply inductively at every subsequent step.
 
@@ -838,7 +838,7 @@ minimal model. Fix the support of its boundary, impose the rational
 affine constraints that its coefficient-one components remain equal
 to one, and take a sufficiently small rational polytope around the
 given boundary within those constraints. On a fixed log resolution
-witnessing dlt, the strict discrepancy inequalities remain strict on
+witnessing dlt, the strict discrepancy ineualities remain strict on
 this neighbourhood, so its boundaries remain dlt. The
 rational-polytope theorem for nef adjoints
 cite[Remark~3.1, Proposition~3.2(3)]{Birkar2011II}, applied to all
@@ -1172,7 +1172,7 @@ T having no component in common with S, with the following
 properties.
 [label=(roman*)]
 
-The pair (X,S+T)calX,S+T) is log canonical, and a fixed
+The pair (X,S+T)X,S+T) is log canonical, and a fixed
 isomorphism of reflexive sheaves is given by
 [
  omega_{X}+T)~=OO_{X}
@@ -1187,7 +1187,7 @@ Put J=(Scap T)_{red}.  Both S and J are Du Bois
 on scheme charts, the ideal I_{J/S} is maximal
 Cohen--Macaulay, and
 [
- B:=calB:=
+ B:=B:=
  Hom_S( I_calI_{J/S},omega_S)
  ~=OO_S(aS).
 ]
@@ -1208,11 +1208,11 @@ This is a morphism from the reduced divisor S; no extension
 to a neighborhood of S in X is asserted here.
 Writing h=pcirc g and
 T={P}*OO_S, the morphism g factors
-through a closed embedding T↪calT↪Y,
+through a closed embedding T↪T↪Y,
 and
 [
  g_*OO_S=OO_{T},
- OO_S(S)=g^*C, B=g^*C^a.calB=g^*C^a.
+ OO_S(S)=g^*C, B=g^*C^a.B=g^*C^a.
 ]
 
 For a separated quasi-projective scheme chart
@@ -1270,7 +1270,7 @@ in the finite algebra
  A=⊕_calA=⊕_{i=0}^{q-1}F^{[i]}.
 ]
 Let tau:R^ RcalR be the normalization
-of Spec_{R} A.calA. In codimension one this is
+of Spec_{R} A.A. In codimension one this is
 the ordinary cover of a torsion line bundle and is etale. The
 cover preserves log canonicity, klt ambient singularities, and
 the reduced Cartier boundary. From now on S_+,S_-,S_0 denote
@@ -1284,7 +1284,7 @@ on every atlas.  This equivariance will be needed for the
 quotient below.
 
 smallskipnoindentSeparating the positive and negative parts.
-Let p_1:Rtilde R^calR^dagger be the
+Let p_1:Rtilde R^R^dagger be the
 normalized blowup of the ideal of S_+cap S_-. The spaces
 constructed so far are related by
 [
@@ -1496,7 +1496,7 @@ divisor (SsubsetX has dimension (N=n-1).  If
 (J=(Scap T)_{red}), then (S,J) are Du Bois,
 (I_{J/S}) is maximal Cohen--Macaulay, and
 
- B:=calB:=
+ B:=B:=
  Hom_S( I_calI_{J/S},omega_S)
  ~= OO_S(aS)=g^*C^a,
  
@@ -1521,7 +1521,7 @@ the common underlying topological map; a ringed-space map from a
 thickening to (U) has not yet been constructed.
 
 
-For every such chart (U), every (jgeq0), and every (kgeq1),
+For every such chart (U), every (jge0), and every (kge1),
 the transition
 [
  g_*(I^j/I^{j+k+1})
@@ -1584,14 +1584,14 @@ coherent lifting obstructions. On each chart (U), write
 (J_U=Jtimes_{Y}
 For (j:S_U∖ J_U↪ S_U), put
 [
- A^bullet_U=mathbf DbfD!(
+ A^bullet_U=D!(
              j_!Q^H_{S_U∖ J_U}[N]),
  
  A_{0,U}={}^{p}!H^0A^bullet_U,
  
  M_U={}^{p}!H^1g_*A_{0,U}.
 ]
-Here (mathbf DbfD) is Hodge-module duality, ({}^pH^i) denotes
+Here (D) is Hodge-module duality, ({}^pH^i) denotes
 perverse cohomology, and (g_*) in the definition of (M_U)
 is the derived Hodge-module direct image.  Later, (g_+) will
 denote the corresponding direct image of differential modules.
@@ -1629,7 +1629,7 @@ Coherent duality and the maximal-Cohen--Macaulay property give
 
 Here ([N]) cancels the dimension shift of the dualizing
 complex.  In the smooth case, our convention gives
-(mathbf D(bfD(Q^H[N])=Q^H[N](N)); its right differential module
+(D(D(Q^H[N])=Q^H[N](N)); its right differential module
 has (omega) as its lowest piece, at index zero.
 
 To pass to perverse cohomology, work locally on a fixed chart
@@ -1740,7 +1740,7 @@ before passing to a finite cover.
 
 For right differential modules, use the transfer bimodule
 [
- D_{Y P}calP
+ D_{Y P}P
  =
  OO_{Y} ⊗ _{p^{-1}OO_{P}}
                          p^{-1}D_{P},
@@ -1867,7 +1867,7 @@ kills the conormal error.
 
 Define
 [
- E_{j,k}=g_*(I^j/I^{j+k}), jgeq0, kgeq1.
+ E_{j,k}=g_*(I^j/I^{j+k}), jge0, kge1.
 ]
 The first graded layer is
 [
@@ -1895,7 +1895,7 @@ give additive maps
        → R^1g_*OO_{S_U} ⊗  C^{-j-k}.
 ]
 Together they define (D_k) on the graded algebra
-(⊕_{jgeq0}OO_{T_U} ⊗  C^{-j}), with
+(⊕_{jge0}OO_{T_U} ⊗  C^{-j}), with
 graded shift (k); no (OO_U)-linearity is asserted.
 Its values are computed by choosing local
 lifts and taking Cech differences.  For a product, this
@@ -1974,7 +1974,7 @@ the trace inclusion
 
 The first identity is the binomial expansion of
 ((1+eta_{ij}y_i^k)^{-(a+k)}): all quadratic terms vanish
-because (2kgeq k+1).  In the last two identities,
+because (2kge k+1).  In the last two identities,
 (sigma) is viewed in the dualizing sheaf of the thickening
 through the trace inclusion.  They express Cartier trace and
 the annihilator of the thickening.
@@ -2037,7 +2037,7 @@ the same localization of this support-torsion module, because
 the two translations differ nilpotently on every section.
 
 The change from (h_i) to (h_j) thus has a finite Taylor
-expansion on (b_j).  Again using (2kgeq k+1), products
+expansion on (b_j).  Again using (2kge k+1), products
 of two differences from Equation~(hg:coordinate-errors)
 annihilate (b_j).  The linear terms are determined by
 [
@@ -2283,8 +2283,8 @@ widetilde y gives a frame in which this section is
 widetilde y^ell.  Restricting to the Z_q gives
 compatible trivializations
 [
- N_0|_{widehatwidehat Z}Z~=OO_{widehatwidehat Z}Z,
-  s_0|_{widehatwidehat Z}Z=s^ell.
+ N_0|_{Zhat}Z~=OO_{Zhat}Z,
+  s_0|_{Zhat}Z=s^ell.
 ]
 Full faithfulness algebraizes this line-bundle trivialization
 and its section identity on Z; no algebraic map from Z
@@ -2514,7 +2514,7 @@ Together with ble v, this yields b=v=nu(B,G_B).
 Since G_B is nef on the b-dimensional base, it follows
 that G_B^b>0, so G_B is big.  Pullback of
 sections then gives kappa(X,L)ge b=v.  The
-inequality kappa(X,L)lenu(X,L) for nef divisors
+ineuality kappa(X,L)lenu(X,L) for nef divisors
 provides the reverse bound and proves abundance.
 
 
@@ -2562,7 +2562,7 @@ smooth projective and (dim B>0). The map from (B) to a subvariety
 of (Alb(X)) is generically finite. Wedges of invariant one-forms
 on the abelian variety therefore give a nonzero section of (K_B):
 at the generic point, choose (dim B) independent pulled-back
-one-forms. Thus (kappa(B,K_B)geq0).
+one-forms. Thus (kappa(B,K_B)ge0).
 
 For a very general smooth fiber (F), the restriction of the
 pseudo-effective class (K_W) is pseudo-effective. For example, restrict
@@ -2570,9 +2570,9 @@ a positive current representing it to almost every fiber, or use
 effective approximations with arbitrarily small ample error.
 Adjunction identifies this restriction with (K_F). If (F) has
 positive dimension, the induction hypothesis gives
-(kappa(F,K_F)geq0); for a point the same assertion holds by
+(kappa(F,K_F)ge0); for a point the same assertion holds by
 convention. Applying Theorem~asm:iitaka with both boundaries
-empty gives (kappa(W,K_W)geq0), contradicting
+empty gives (kappa(W,K_W)ge0), contradicting
 (ex:counterexample).
 
 This is the only use of Theorem~asm:iitaka in the proof.
@@ -2646,7 +2646,7 @@ that closure to equal (V). At a general
 diagonal point, smoothness of the parameter map identifies its
 vertical tangent with the corresponding web tangent. Every web
 tangent is consequently tangent to (V), and so is every bracket. Hence
-(dim VgeqrkF, while the reverse inequality was proved
+(dim VgerkF, while the reverse ineuality was proved
 above. A plaque is thus open in (V). The equations of (V),
 pulled back to the connected immersed leaf, vanish on a nonempty
 open and hence on the entire leaf. Its closure is exactly (V).
@@ -2699,7 +2699,7 @@ Restricting the ramification formula of its generically finite
 total evaluation shows that (K_V) dominates the restriction of
 the pulled-back (K_X) by an effective divisor. Thus (K_V) is
 pseudo-effective. The induction hypothesis gives a good minimal
-model of (V), so (kappa(V)geq0). Since (V) is not of general
+model of (V), so (kappa(V)ge0). Since (V) is not of general
 type, its general Iitaka fibers have positive dimension and
 Kodaira dimension zero. Spreading these fibers gives another
 covering family with smooth domains (G), of dimension less than
@@ -2788,9 +2788,9 @@ model, hence nonvanishing upstairs, contradicting
 
 
 Let (Y) be a projective (Q)-factorial terminal birational model
-of (X). If a rational divisor (P) has (kappa(Y,P)geq1),
+of (X). If a rational divisor (P) has (kappa(Y,P)ge1),
 then (K_Y+cP) is big for every rational (c>0).
-Consequently, if (alphaneq0) is a supporting functional of the
+Consequently, if (alphane0) is a supporting functional of the
 pseudo-effective cone with (alpha(K_Y)=0), then
 (alpha(P)>0).
 
@@ -2814,7 +2814,7 @@ cone is strictly positive on its interior. Applying it to
 ## Valuations and positive currents
 
 The next two exclusions involve currents whose numerical intersection
-gaps tend to zero. A limiting inequality alone is insufficient: we
+gaps tend to zero. A limiting ineuality alone is insufficient: we
 need one family on which the gap is exactly zero. The ACC statement
 below supplies that step for a fixed current and bounded discrepancies.
 
@@ -2887,7 +2887,7 @@ divisorial valuation,
  ord_EJ(kT)<= knu_E(T).
 
 The local Bergman weight is bounded below by the original weight
-up to a constant; this inequality persists after pullback and
+up to a constant; this ineuality persists after pullback and
 gives (ex:multiplier-order). These are statements for a
 fixed current, not uniform assertions over all currents.
 
@@ -3026,8 +3026,8 @@ pseudo-effective cone at (K_Y), and put
 on higher smooth base models are movable classes by
 pseudo-effective/movable duality cite[Theorem~0.2]{BDPP2013}.
 They annihilate base-exceptional divisors. Since (S) is positive,
-(eta·{S}geq0). Applying (alpha) to
-(ex:descent-domination) gives the reverse inequality,
+(eta·{S}ge0). Applying (alpha) to
+(ex:descent-domination) gives the reverse ineuality,
 because (alpha(K_Y)=0). Hence (eta·{S}=0).
 If (c_D[D]) is a positive Siu component of (S), both
 (S-c_D[D]) and ([D]) are positive. It follows that
@@ -3066,7 +3066,7 @@ theorem of Kov'acs--Patakfalvi
 cite[Theorem~9.9]{KovacsPatakfalvi2017} applies to a log
 canonical fiber space with smooth base, SNC total pair, and
 log-general-type geometric generic fiber; for a rational base
-divisor (M) with (kappa(M)geq0), it gives relative
+divisor (M) with (kappa(M)ge0), it gives relative
 subadditivity with the term (h^*M). Here both spaces are
 smooth projective, the boundary is zero, and we take (M=0).
 Invariance of plurigenera on the smooth locus identifies the
@@ -3082,9 +3082,9 @@ For every sufficiently divisible (m), pushing forward an
 effective divisor
 (div_W(phi)+m(K_W-h^*K_B))
 gives
-(div_Y(phi)+mPgeq0).
+(div_Y(phi)+mPge0).
 The resulting injection of sections preserves their ratios,
-so (kappa(Y,P)geq1). Corollary~ex:supporting now gives
+so (kappa(Y,P)ge1). Corollary~ex:supporting now gives
 (0<alpha(P)=-K_B·eta), proving
 (ex:negative-base).
 
@@ -3098,12 +3098,12 @@ If (F_k) is its divisor, the class
 ]
 is nef. Put (eta_k=T_{B_k}^*alpha). The class (eta_k)
 is movable and (F_k) is effective, so
-(0leq P_k·eta_kleq (L_k·eta)/k).
+(0le P_k·eta_kle (L_k·eta)/k).
 Since (eta·{S}=0) and (L_k-k{S}) stays in a
 bounded set, this gives the first estimate below. The second
 uses the fact that (eta_k) kills exceptional divisors:
 [
- 0leq P_k·eta_k=O(k^{-1}),
+ 0le P_k·eta_k=O(k^{-1}),
  K_{B_k}·eta_k=K_B·eta<0.
 ]
 Fix an ample divisor (H) on (B), and write
@@ -3120,8 +3120,8 @@ Approximate (eta_k) by a positive sum
 of covering-curve classes,
 closely enough that
 [
- Q_k·gamma_kleq C/k,
- -K_{B_k}·gamma_kgeqdelta/2
+ Q_k·gamma_kle C/k,
+ -K_{B_k}·gamma_kgedelta/2
 ]
 for a constant (C) independent of (k). Keep only terms
 with (K_{B_k}· C_i<0). Their total anticanonical
@@ -3148,11 +3148,11 @@ Shrink each parameter space so that the curve class and its
 contacts with the exceptional divisors and the strict transforms
 of the Siu divisors of (S) are constant. The general parametrized
 member is free in characteristic zero. Consequently
-(K_{B_k}· R_kleq-2). Put (barbar R_k=r_R_k=r_{k*}R_k).
+(K_{B_k}· R_kle-2). Put (barbar R_k=r_R_k=r_{k*}R_k).
 Its bounded (H)-degree bounds (K_B·barbar R_kR_k), and
 effectivity of the relative canonical divisor gives
 
- 0leq K_{B_k/B}· R_k
+ 0le K_{B_k/B}· R_k
    =K_{B_k}· R_k-K_B· r_{k*}R_k=O(1).
 
 In particular, the upper bound is independent of (k).
@@ -3252,10 +3252,10 @@ Fix a positive current (T) on a smooth resolution in the
 pullback class of (K_Y). On a common smooth resolution of
 (Ydashrightarrow Y_t), write
 [
- p^*K_Y=p_t^*K_t+J_t, J_tgeq0,
+ p^*K_Y=p_t^*K_t+J_t, J_tge0,
 ]
 where (J_t) is exceptional over (Y_t).
-Then (nu_E(T)geqcoeff_E J_t) for every
+Then (nu_E(T)gecoeff_E J_t) for every
 prime on that resolution.
 
 Proof. 
@@ -3351,17 +3351,17 @@ and calculating the tame ramification of DVRs. It is
 positive and bounded away from zero: (Y) is terminal,
 so (a(E;Y,0)>1), and the index (m) puts these log
 discrepancies in (m^{-1}Z). In particular
-(a(E;Y,0)geq1+1/m). The
+(a(E;Y,0)ge1+1/m). The
 remaining ramification coefficients are nonnegative.
 On the general parameter fiber,
 [
  K_{U_t}· C_t=2g(C_t)-2,
- q_t^*K_Y· C_tgeq0.
+ q_t^*K_Y· C_tge0.
  ]
-The latter inequality follows from pseudoeffectivity and
+The latter ineuality follows from pseudoeffectivity and
 the covering property. Thus
 [
- 0leq (K_{U_t}-q_t^*K_Y)· C_tleq 2g_0-2.
+ 0le (K_{U_t}-q_t^*K_Y)· C_tle 2g_0-2.
 ]
 Each exceptional contact contributes
 ((r_Za(E;Y,0)-1)(Z· C_t)) to this bounded sum.
@@ -3395,7 +3395,7 @@ difference below and is nonnegative. Consequently
 
  
  0&<= q_t^*K_Y· C_t
-       -sum_{Z {rm exceptional}}
+       -sum_{Z {exceptional}}
                 r_Znu_E(T),Z· C_t
   &<= K_t· C_t
    <= (K_t+tA_t)· C_t
@@ -3483,7 +3483,7 @@ Take a small projective (Q)-factorialization
  D_i=K_{S'_i}+B_i=p_i^*((K_{V_i}+Delta_i)|_{S_i}),
   C_i=p_i^*(A_i|_{S_i}).
 ]
-Adjunction gives a dlt pair ((S'_i,B_i)), with (B_i,C_igeq0),
+Adjunction gives a dlt pair ((S'_i,B_i)), with (B_i,C_ige0),
 and ((S'_i,B_i+lambda_i C_i)) is lc. The scaling divisor
 has no floor component, so its restriction is effective.
 The number (lambda_i) is rational, being the ratio of
@@ -3590,7 +3590,7 @@ have Iitaka dimension at least one: by
 Corollary~ex:supporting, (K_W+cP) would be big,
 and
 ((1+c)P=(K_W+cP)+D) would then be big as well.
-Hence (kappa(W,P)leq0).
+Hence (kappa(W,P)le0).
 
 Run the dlt LMMP for (P) with ample scaling.
 The divisor (P) has a signed rational representative
@@ -3614,7 +3614,7 @@ by Proposition~prop:boundary-restriction.
 Apply Theorem~thm:signed with
 (L=K_Z+D_Z) and (c=1), so that the required
 pseudo-effective class (L-cD_Z) is (K_Z).
-It gives (kappa(Z,L)=nu(Z,L)geq0).
+It gives (kappa(Z,L)=nu(Z,L)ge0).
 The Iitaka dimension is at most zero, as proved above,
 so both dimensions are zero. Intersecting
 (K_Z+D_Zequiv0) with an ample
@@ -3635,7 +3635,7 @@ pullback from (Z), so is supported on the exceptional
 locus. The support theorem makes it effective divisorial.
 Independence of exceptional divisor classes, by negativity,
 identifies its coefficients with those of (E); hence
-(Egeq0). Pushing forward by (a) shows that
+(Ege0). Pushing forward by (a) shows that
 (K_W) is (Q)-linearly effective,
 contradicting (ex:counterexample).
 
@@ -3756,7 +3756,7 @@ All powers of ideals below are ordinary powers.
 ## Numerical subadjunction on the center
 
 The first result converts a generic log canonical center into a
-canonical-intersection inequality. Its testing class is allowed
+canonical-intersection ineuality. Its testing class is allowed
 to live on the center itself. The proof uses dlt adjunction and
 the klt-trivial canonical bundle formula; it does not require the
 auxiliary pair to be log canonical away from the generic center.
@@ -3764,7 +3764,7 @@ auxiliary pair to be log canonical away from the generic center.
 [Numerical generic subadjunction]
 
 Let Z be a projective Q-factorial klt variety over C, let
-Θgeq0 be a rational divisor, and let Vsubset Z be an
+Θge0 be a rational divisor, and let Vsubset Z be an
 integral positive-dimensional subvariety. Suppose that (Z,Θ)
 is lc at the generic point of V and that a divisor of log
 discrepancy zero has center V. Let f=dim V, let P be a nef
@@ -3788,7 +3788,7 @@ birational morphism p:Qto Z with Q Q-factorial and
 
  p^*(K_Z+Θ)=K_Q+B_Q+E_Q,
 
-where (Q,B_Q) is dlt and E_Qgeq0 is the surplus over the
+where (Q,B_Q) is dlt and E_Qge0 is the surplus over the
 non-lc locus. In this construction the coefficients of the strict
 boundary are truncated at one and the relevant extracted divisors
 are reduced; the coefficients left over form E_Q. The surplus
@@ -3823,7 +3823,7 @@ Here q has connected fibers and eta is finite. Put
 D_V=nu^*((K_Z+Θ)|_V) and
 D'=eta^*D_V. We apply the canonical bundle formula on the
 connected-fiber base V', then push the resulting intersection
-inequality through the finite map eta. Its ramification will
+ineuality through the finite map eta. Its ramification will
 contribute an effective term. The final comparison is between the
 canonical cycle on V^nu and the canonical divisor on V^*.
 
@@ -3858,27 +3858,27 @@ non-lc fibers caused by E_Q|_S are permitted.
 The discriminant B_{V'} on this initial base is effective. To
 check this at a prime divisor Fsubset V', work over its generic
 point, where the base is regular. A component of its inverse
-image on S has multiplicity m_Fgeq1 and nonnegative boundary
+image on S has multiplicity m_Fge1 and nonnegative boundary
 coefficient b_F. Its discrepancy bounds the generic lc threshold
 t_F from above by
 [
- t_Fleq(1-b_F)/(m_F)leq1.
+ t_Fle(1-b_F)/(m_F)le1.
 ]
 Thus the discriminant coefficient 1-t_F is nonnegative. If the
 fiber is already non-lc, t_F can be negative; this only increases
 that coefficient.
 
 Let P'=eta^*nu^*P. On a higher projective base
-mu:widehatwidehat V'V'to V' where the moduli divisor is nef, its
-trace satisfies M_{V'}=mu_*M_{widehatwidehat V'}V'. The projection
+mu:Vhat'V'to V' where the moduli divisor is nef, its
+trace satisfies M_{V'}=mu_*M_{Vhat'}V'. The projection
 formula gives
 [
  M_{V'}·(P')^{f-1}
- =M_{widehatwidehat V'}V'·(mu^*P')^{f-1}geq0.
+ =M_{Vhat'}V'·(mu^*P')^{f-1}ge0.
 ]
 Only this intersection is asserted to be nonnegative; the trace
 M_{V'} need not itself be nef. Likewise
-B_{V'}·(P')^{f-1}geq0 by effectivity and nefness.
+B_{V'}·(P')^{f-1}ge0 by effectivity and nefness.
 All intersections with Weil divisors here mean their cycle
 intersections with rational Cartier powers, so they are defined
 even when the separate canonical or discriminant divisor is not
@@ -3889,7 +3889,7 @@ and separable, the codimension-one ramification formula, followed
 by pushforward, is
 [
  eta_*K_{V'}=(degeta)K_{V^nu}+eta_*R_eta,
-  R_etageq0.
+  R_etage0.
 ]
 This identity of Weil cycles uses the discrete valuation rings at
 generic prime divisors and does not require K_{V^nu} to be
@@ -3905,7 +3905,7 @@ Finally the pushforward of K_{V^*} to V^nu is its compatible
 canonical cycle. All additional divisors on the resolution have
 image of codimension at least two, so their intersection with
 f-1 pulled-back Cartier powers is zero. Dividing by
-degeta proves Equation~(common:subadjunction-inequality). When f=1
+degeta proves Equation~(common:subadjunction-ineuality). When f=1
 there are no birational exceptional divisors on the normal curve,
 and the same calculation applies directly.
 
@@ -3922,21 +3922,21 @@ Let Z be a projective Q-factorial klt variety of dimension d,
 let P be a nef rational Cartier divisor, and let k>0 be an
 integer for which kP is Cartier. Let
 0neH H^0(Z,O_Z(kP)) be a linear subspace
-with proper base locus and base ideal mathfrak b.frakb.
+with 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_{rm sm}. Write a=d-f and
-mathfrak m=frakm=mathfrak m_frakm_{O_{Z,eta_V}}. If
+eta_Vin Z_{sm}. Write a=d-f and
+m=m=m_{O_{Z,eta_V}}. If
 [
- mathfrak bfrakbO_{Z,eta_V}subsetmathfrak m^hfrakm^h
+ bO_{Z,eta_V}subsetm^hfrakm^h
  (hinZ_{>0}),
 ]
 then
 
 
- P^f· Vleq (k/h)^aP^d.
+ P^f· Vle (k/h)^aP^d.
 
-There are a rational number c with 0<cleq ak/h and an
+There are a rational number c with 0<cle ak/h and an
 effective rational divisor Θsim_{Q}cP such that
 (Z,Θ) is lc at eta_V and has an lc place with center
 V. If f>0, on a smooth resolution V^* one consequently has
@@ -3951,15 +3951,15 @@ asserted only for positive-dimensional V.
 
 Proof. 
 The local ring R=O_{Z,eta_V} is regular of dimension a.
-Isolation means that mathfrak bRfrakbR is mathfrak m-primary.frakm-primary.
+Isolation means that bR is m-primary.m-primary.
 Choose a general sections of H. Their local equations
-form a system of parameters in R, since an mathfrak m-primaryfrakm-primary
-ideal is contained in no smaller prime. Let mathfrak qfrakq be the
-parameter ideal they generate. It is contained in mathfrak m^h.frakm^h.
+form a system of parameters in R, since an m-primaryfrakm-primary
+ideal is contained in no smaller prime. Let q be the
+parameter ideal they generate. It is contained in m^h.m^h.
 Regularity and the multiplicity comparison give
 [
- length(R/mathfrak q)frakq)
- =e(mathfrak q,R)frakq,R)>= e(mathfrak m^h,R)=h^a.frakm^h,R)=h^a.
+ length(R/q)q)
+ =e(q,R)q,R)>= e(m^h,R)=h^a.m^h,R)=h^a.
 ]
 This is the generic intersection multiplicity along V.
 
@@ -3976,17 +3976,17 @@ P-degree. After a cuts the surviving effective cycle has
 P-degree at most k^aP^d, and contains V with multiplicity at
 least h^a. This proves Equation~(common:base-component-degree).
 
-Let lambda=lct_{eta_V}(mathfrak b)frakb) be the generic ideal
+Let lambda=lct_{eta_V}(b)b) be the generic ideal
 threshold. It is positive and rational. The valuation from
 blowing up the smooth generic center has log discrepancy a
 and ideal order at least h, so
 [
- 0<lambdaleq a/h.
+ 0<lambdale a/h.
 ]
-Because mathfrak bRfrakbR is primary, a divisor computing this
+Because bR is primary, a divisor computing this
 threshold has center V. We realize the ideal threshold by a
 divisor rather than apply subadjunction to an ideal formally.
-Take a log resolution of Z and mathfrak b,frakb, so that the
+Take a log resolution of Z and b,b, so that the
 pulled-back system has fixed divisor F and a basepoint-free
 moving part. Choose sufficiently many general members
 D_1,...,D_N of H and put
@@ -4002,7 +4002,7 @@ lc at the generic point of V and has the same lc place there
 as the ideal threshold. Global log canonicity is unnecessary.
 For f>0, Lemma~common:numerical-subadjunction gives
 Equation~(common:base-component-canonical), and
-c=klambdaleq ak/h.
+c=klambdale ak/h.
 
 
 ## Differentiating the full moving jet kernel
@@ -4031,7 +4031,7 @@ subspace of sections of kP vanishing at x to order at least
 lceil ktau_irceil. Suppose that for very general x all these
 subspaces are nonzero and the base locus of H_0(x)
 has a positive-dimensional component through x.
-If kdeltageq4, then, after an algebraic parameter extension
+If kdeltage4, then, after an algebraic parameter extension
 and restriction to nonempty opens, there are an irreducible
 parameter space T, a fixed adjacent pair of levels, and a
 family of integral subvarieties V_t common to the two base
@@ -4046,7 +4046,7 @@ there in
 
 
  I_{V_t}^{h}, h=lceil kdelta/2rceil,
-  k/hleq2/delta.
+  k/hle2/delta.
 
 These are ordinary local ideal powers in the smooth ambient open.
 
@@ -4096,9 +4096,9 @@ We now prove ordinary high-order vanishing. Work on a smooth
 parameter open, and denote the higher-kernel bundle there by
 K. Evaluation on Ttimes Z gives a map from the
 pullback of K to the pullback of O_Z(kP).
-Let mathfrak b_Tfrakb_T be its base ideal. After trivializing the
+Let b_T be its base ideal. After trivializing the
 target line bundle, a local frame of K generates
-mathfrak b_T.frakb_T. We show that every frame section lies in
+b_T.b_T. We show that every frame section lies in
 I_
 Each frame element has the form
 [
@@ -4115,9 +4115,9 @@ the section starts in order lceil ktau_{i+1}rceil. Each parameter
 derivative lowers this order by at most one. A derivative of
 order r therefore belongs to the lower kernel if
 [
- rleqlceil ktau_{i+1}rceil-lceil ktau_irceil.
+ rlelceil ktau_{i+1}rceil-lceil ktau_irceil.
 ]
-The right side is at least kdelta-1. Since kdeltageq4,
+The right side is at least kdelta-1. Since kdeltage4,
 every derivative of order less than h=lceil kdelta/2rceil is
 allowed. It thus vanishes along Gamma, which lies in the
 lower base locus. A change of higher-kernel frame contributes
@@ -4244,8 +4244,8 @@ 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_{rm sm} to divisors
-R_1,...,calR_1,...,R_N in V.
+Jacobian over Y_{sm} to divisors
+R_1,...,R_1,...,R_N in V.
 Retain the notation V,T,gcalV,T,g after this base change.
 Let I be the set of indices j for which the image closure
 g_t((R_j)_t)bar is a divisor in Y for general t.
@@ -4281,7 +4281,7 @@ parameters disappear upon shrinking.
 
 Put
 [
- B=Y_{rm sing}cup∪_{jin I}g(R_j)bar.
+ B=Y_{sing}cup∪_{jin I}g(R_j)bar.
 ]
 Each closed set in this finite union is proper by hypothesis,
 so Bne Y. The finite normal covers are unramified in
@@ -4578,16 +4578,16 @@ canonical singularities are canonical, and the projective bundles
 are klt.  The two disjoint Cartier axes form a plt boundary (D),
 and
 [
- K_{Z_0}+D=K_{rm sum},
- Dsim2xi-mK_{rm sum}.
+ K_{Z_0}+D=K_{sum},
+ Dsim2xi-mK_{sum}.
 ]
-The notation (K_{rm sum}) means (K_1+K_2), or (K) on a slice.
+The notation (K_{sum}) means (K_1+K_2), or (K) on a slice.
 
 For (0<sigmale1), put
 [
  gamma=2sigma+frac{q(1+sigma m)}r,
  Xi_tsim_{Q}xi+
-       ((1+sigma m)t)/(gamma)A_{rm sum}.
+       ((1+sigma m)t)/(gamma)A_{sum}.
 ]
 Both endpoint weight systems are big, so choose an effective rational
 representative of (Xi_t) containing neither axis, and denote the
@@ -4600,7 +4600,7 @@ the coefficient
  a=((1+sigma m)t)/(gamma)
 ]
 lies in ((0,1]).  Thus the numerical classes
-(xi+aA_{rm sum}), and their restrictions to either fixed axis,
+(xi+aA_{sum}), and their restrictions to either fixed axis,
 range over bounded segments independent of (sigma,q,r,t).
 On one fixed smooth resolution the effective pullbacks of these chosen
 divisors have uniformly bounded degree against a fixed very ample class.
@@ -4726,7 +4726,7 @@ Write (d=2n+1). Choose a gap (delta>0) large enough that
 [
  2(n+1)ε^n(2/delta)^n<1.
 ]
-This inequality will exclude a component that is a whole projective-line
+This ineuality will exclude a component that is a whole projective-line
 fiber of a slice. Now choose (q) sufficiently large that
 [
  (((2n+1)!)/(4(n!)^2)qε^{2n})^{1/d}
@@ -4857,10 +4857,10 @@ preserves the order (h). In degree (k), expand each of these sections
 into its toric weights. At the generic point of (W), let (I_l) be
 the ideal generated by the coefficients of the nonzero weight (l)
 over all these sections.  Put (h=lceil kdelta/2rceil).
-All (I_l) lie in (mathfrak m_W^hfrakm_W^h): vanishing along the generic
+All (I_l) lie in (m_W^hfrakm_W^h): vanishing along the generic
 torus fiber says that the coefficient of every Laurent monomial
 vanishes to this order.  Their sum is
-(mathfrak m_Wfrakm_W)-primary.  Otherwise its larger common zero germ,
+(m_W)-primary.  Otherwise its larger common zero germ,
 times the generic torus fiber, would contradict isolation of (F).
 
 Our aim is to construct an effective boundary
@@ -4879,7 +4879,7 @@ ideal pair form the rational polytope
 Here (E) runs through the divisors needed to test log canonicity on
 the simultaneous resolution, and (a(E;Y,0)) is the log discrepancy.
 Include the exceptional divisor of the blowup of the closed point.
-Since every (I_lsubsetmathfrak m_W^hfrakm_W^h), it gives
+Since every (I_lsubsetm_W^hfrakm_W^h), it gives
 (sum_lu_lle c/h). The polytope is compact, and (sum_lu_l)
 has a positive rational maximum on it. Fix a rational maximizing point.
 For each coordinate (l), some equality at this point must involve
@@ -5154,7 +5154,7 @@ integer q>0 and real numbers r>1, ε>0. Suppose
  H^nle(2ε)^n,
  K_WH^{n-1}le2H^n/r,
  (2L+qS)H^{n-1}le4H^n,
- (14ε)^n<tfrac14, 80ε<tfrac34.
+ (14ε)^n<1/4, 80ε<3/4.
  
 
 Assume that the following fixed data exist.
@@ -5165,8 +5165,8 @@ for fixed positive integers l_i, with
 mu_{min}(Ω_W^1|_C)ge0.
  On the projective bundle
 [
- Z=mathbf P(bfP(O_W(S)_1 ⊕ 
-                         O_W(S)_2)→calO_W(S)_2)→ Wtimes W,
+ Z=P(P(O_W(S)_1 ⊕ 
+                         O_W(S)_2)→O_W(S)_2)→ Wtimes W,
   xi=c_1(O_Z(1)), M=L_1+L_2+qxi,
 ]
 use the convention that the direct image of O_Z(a
@@ -5175,19 +5175,19 @@ point z_* in the complement of the two axes and finitely many
 sections of k_0M, for some integer k_0>0 with k_0L Cartier,
 generate jets through order (2n+2)k_0 at z_*.
  A point xin W, its blowup
-pi_x:widehatwidehat 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^*(L+K_W/2+lambda S)·gamma
  le40ε J·gamma (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.
 
@@ -5213,7 +5213,7 @@ nonvanishing at each of the two base coordinates of z_*.
 Fix these finitely many sections. All models, morphisms, divisors,
 points, the flag, the complete-intersection witness for gamma,
 and the jet and filler sections spread over an integral finitely
-generated mathbf Z-algebrabfZ-algebra of characteristic zero. Shrink its
+generated Z-algebrabfZ-algebra of characteristic zero. Shrink its
 spectrum so that the fibers and flag are smooth projective and
 geometrically integral, the birational morphisms remain birational,
 the fixed ample bundles remain ample, and the finite jet
@@ -5294,7 +5294,7 @@ Let F:Wto W' be 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_*O_W(B_p), s=rkE=p^n, R=s^2,
+ E=F_*E=F_*O_W(B_p), s=rkE=p^n, R=s^2,
   U_a=H^0(W,O_W(B_p+paS)).
 ]
 Projection formula gives an evaluation map on W'times W':
@@ -5302,7 +5302,7 @@ Projection formula gives an evaluation map on W'times W':
  ⊕_{substack{a+b=qa,bge0}}
  U_a ⊗  U_b ⊗ 
  O(-aS'_1-bS'_2)
- → E E.calE.
+ → E E.E.
 
 
 
@@ -5369,7 +5369,7 @@ On the diagonal of W'times W', every source twist of
 Equation~(common:two-slot-evaluation) becomes -qS'.
 Finite base change and 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 diagonal point
@@ -5487,7 +5487,7 @@ The divisors B_p-pL=T_0-(p-k_0N_p)L range over a fixed finite
 list.  Combining Equation~(common:intersection-bounds), Equation~(common:flag-degrees), Lemma~common: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,
 
@@ -5519,8 +5519,8 @@ 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.
@@ -5530,12 +5530,12 @@ 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_i h_i|_C)).
+ h^0(C,G_j|_C(-G_j|_C(-sum_i b_i h_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.
+to the displayed lattice sum preserves the ineuality.
 
 A summand in Equation~(common:flag-lattice-sum) vanishes when
 sum_i b_i(h_i· C)>M_p, because then its maximum slope is
@@ -5562,7 +5562,7 @@ the number of graded pieces.  Finally
 [
  7^nH^nle(14ε)^n<1/4
 ]
-by Equation~(common:intersection-bounds), proving the strict inequality
+by Equation~(common:intersection-bounds), proving the strict ineuality
 for all large p.
 
 
@@ -5572,7 +5572,7 @@ Choose R generically independent columns of
 Equation~(common: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
@@ -5581,7 +5581,7 @@ slot i. The target determinant is
 chosen source determinant is
 O(-m_1S')boxtimesO(-m_2S'). Thus
 [
- Q_i=( E)^calE)^{ ⊗  s}
+ Q_i=( E)^E)^{ ⊗  s}
                   ⊗ O_{W'}(m_iS'),
  lambda_i=m_i/R.
 ]
@@ -5603,10 +5603,10 @@ 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 gives
+For 0lelambdale q, the fixed movable ineuality gives
 [
  (L+K_W/2+lambda S)·gammale40ε J·gamma,
 ]
@@ -5685,12 +5685,12 @@ specialized to positive characteristic.
 Assume the lower-dimensional real-boundary good-model hypothesis of
 Assumption~mmp:lower-models.
 If (X) is a smooth projective complex variety of dimension (n) and
-(K_X) is pseudo-effective, then (kappa(X,K_X)geq0).
+(K_X) is pseudo-effective, then (kappa(X,K_X)ge0).
 
 
 In dimension zero the assertion is immediate.  On a smooth curve,
-pseudo-effectivity of (K_X) gives (g(X)geq1), hence
-(h^0(X,K_X)=g(X)>0). We therefore suppose (ngeq2) and argue by
+pseudo-effectivity of (K_X) gives (g(X)ge1), hence
+(h^0(X,K_X)=g(X)>0). We therefore suppose (nge2) and argue by
 contradiction. We use the late terminal model (Y), the divisors
 (K=K_Y) and (A=A_Y), and the scaling models of the preceding
 section. Fix (m>0) with (mK) Cartier, and write
@@ -5708,7 +5708,7 @@ so that
  (14ε)^n<frac14,
  80ε<frac34.
 
-The strict inequalities leave room for the errors that tend to zero
+The strict ineualities leave room for the errors that tend to zero
 with the characteristic.  We next fix a sufficiently large integer
 (q) as in Proposition~prop:large-seshadri, and then choose
 a positive rational (t) small enough for the estimates below and for
@@ -5726,7 +5726,7 @@ Choose a smooth common resolution (W) of (Y) and the scaling model
 for (K+tA). In this application (K,A,L) also denote their pullbacks to
 (W), and
 [
- K_W=K+E, Egeq0.
+ K_W=K+E, Ege0.
 ]
 Let (H_0) be the pullback of the nef semiample positive part (N_0) of
 (L) on its scaling model. The comparison of canonical pullbacks and
@@ -5743,7 +5743,7 @@ Every divisor exceptional over (Y) is therefore also exceptional
 over (Y_t).  The differences discarded when passing to the moving
 parts are exceptional over the scaling model, and the transform of
 (A) is effective up to rational linear equivalence.  Intersecting
-with (H_0^{n-1}) gives the displayed equalities and inequality.
+with (H_0^{n-1}) gives the displayed equalities and ineuality.
 In particular, since (qm/r<1),
 [
  (2L+qmK)H_0^{n-1}
@@ -5757,9 +5757,9 @@ with (K_WH_0^{n-1}le H_0^n/r), allow us to choose (eta)
 small enough that continuity gives
 
 
- H^nleq(2ε)^n,
+ H^nle(2ε)^n,
  K_WH^{n-1}<=(2H^n)/(r),
- (2L+qmK)H^{n-1}leq4H^n.
+ (2L+qmK)H^{n-1}le4H^n.
 
 
 We next choose the flag for the section estimates.  The canonical
@@ -5778,7 +5778,7 @@ classes, and choose a smooth integral complete-intersection flag
 for which
 
 
- mu_{min}(Ω_W^1|_C)geq0.
+ mu_{min}(Ω_W^1|_C)ge0.
 
 Here and below slopes on (C) mean degree divided by rank. One may
 obtain the flag by applying restriction to the finitely many
@@ -5788,7 +5788,7 @@ Harder--Narasimhan quotients of (Ω_W^1).
 
 For the vanishing-order estimate, choose a very general point
 (xin W), away from the exceptional loci, and let
-(pi_x:widehatwidehat WWto W) be its blowup, with exceptional
+(pi_x:WWtohat W) be its blowup, with exceptional
 divisor (J). Set
 [
  D^+=2r(K+2tA)+2E.
@@ -5798,19 +5798,19 @@ Proposition~prop:scalar-jets: pushing to (Y) identifies the
 section spaces, since the added exceptional part does not change them.
 If (rho=vol(2r(K+2tA))^{1/n}), then
 [
- rholeq4rmu_tleq8ε
+ rhole4rmu_tle8ε
 ]
 for our sufficiently small (t). Thus every section of a sufficiently
 divisible multiple of (D^+) has normalized order at (x) at most
-(4rholeq32ε).
+(4rhole32ε).
 
-We convert this order bound into a curve inequality.
+We convert this order bound into a curve ineuality.
 It follows that (pi_x^*D^+-40ε J) is not pseudo-effective.
 Otherwise its convex combination with the big class (pi_x^*D^+)
 would make (pi_x^*D^+-cJ) big for some rational
 (32ε<c<40ε), giving a section of excessive order.
 Movable-curve duality cite[Theorem~2.2]{BDPP2013} therefore supplies an
-actual covering curve class (gamma) on (widehatwidehat WW) such that
+actual covering curve class (gamma) on (WWhat) such that
 
 
  (pi_x^*D^+)·gamma
@@ -5820,21 +5820,21 @@ The strictness lets us fix an algebraic family that will survive
 reduction.  More precisely, the negative pairing can first be detected
 by a strongly movable curve: take the pushforward of a general complete
 intersection of very ample divisors on one fixed smooth birational
-model of (widehatwidehat WW). Fix this model, its divisors, and the resulting
+model of (WWhat). Fix this model, its divisors, and the resulting
 covering family.  Thus the choice consists of finite algebraic data,
 rather than a limiting real movable class.  Positivity of
-(J·gamma) follows from the strict inequality and
+(J·gamma) follows from the strict ineuality and
 pseudo-effectivity of (pi_x^*D^+).
 
 To turn this curve into the witness required by
 Theorem~common:finite-data-frobenius, set S=mK on W;
-it is an integral Cartier divisor. We verify all its weight inequalities
+it is an integral Cartier divisor. We verify all its weight ineualities
 before proceeding to the jet construction. Put
 Q_lambda=L+K_W/2+lambda S for 0lelambdale q.
 The equality K_W=K+E gives
 
  D^+-Q_lambda
- =(r-tfrac12-lambda m)K+3rtA+tfrac32E.
+ =(r-1/2-lambda m)K+3rtA+3/2E.
 
 This class is pseudo-effective in characteristic zero: r>qm+1,
 K is pseudo-effective, and A,E are effective up to the stated
@@ -5845,10 +5845,10 @@ equivalences. Pairing with the fixed strongly movable class therefore gives
  <40ε J·gamma,
   J·gamma>0.
 ]
-These are the required movable inequalities. Only the two endpoint
-inequalities need to be retained as finite data, since the pairing is
+These are the required movable ineualities. Only the two endpoint
+ineualities need to be retained as finite data, since the pairing is
 affine in lambda. The common theorem spreads these fixed numerical
-inequalities and the actual birational complete-intersection witness;
+ineualities and the actual birational complete-intersection witness;
 it does not spread the pseudo-effective assertion used to obtain them.
 
 ### A finite two-slot jet system
@@ -5888,7 +5888,7 @@ intersection and flag hypotheses of
 Theorem~common:finite-data-frobenius are
 Equations~(fr:epsilon), (fr:intersection-bounds),
 and~(fr:generic-nef). The curve construction gives its movable
-witness and both endpoint inequalities, and the finite jet system
+witness and both endpoint ineualities, and the finite jet system
 gives the required surjection at z_*. This point need not lie over
 the separately chosen testing point x. The proof of the common
 theorem also fixes its auxiliary divisor T_0 and finitely many
@@ -5902,7 +5902,7 @@ jets give full evaluation rank (R), while the small polarization
 bounds the rank on the diagonal by (R/4). Thus a nonzero maximal
 minor vanishes to order at least (3R/4) at ((x',x')), where (x')
 is the Frobenius image of the testing point (x). Its line bundle
-is an actual external product. Applying the fixed curve inequality
+is an actual external product. Applying the fixed curve ineuality
 to each factor bounds the same order above by
 (R(80ε+O(p^{-1}))), a contradiction for large (p).
 The theorem proves these determinant estimates for the fixed witnesses
@@ -6042,7 +6042,7 @@ resolution (p:Wto X), (q:Wto V), compatible canonical divisors
 satisfy
 [
  p^*K_X=q^*K_V+E,
-  Egeq0quadand E is q-exceptional.
+  Ege0quadand E is q-exceptional.
 ]
 In particular (kappa(X,K_X)=kappa(V,K_V)).
 
@@ -6141,10 +6141,10 @@ is an effective rational divisor, and put (S=floor(Delta)).
 Assume that (K_X+Delta) is nef and that, for an effective rational
 divisor (D),
 [
- K_X+Deltasim_{Q}Dgeq0,
+ K_X+Deltasim_{Q}Dge0,
   S⊆Supp D.
 ]
-Then, for every integer (mgeq2) for which (m(K_X+Delta)) is Cartier,
+Then, for every integer (mge2) for which (m(K_X+Delta)) is Cartier,
 the restriction map
 [
  H^0(X,OO_X(m(K_X+Delta)))
@@ -6159,7 +6159,7 @@ rational divisor, and (floor( S+B)=S). Since a dlt pair is log
 canonical, Theorem~thm:main makes its nef adjoint semiample.
 The pair and the displayed effective representative therefore satisfy the
 hypotheses of cite[Proposition~5.12]{FujinoGongyo2014}, which gives the
-stated restriction surjectivity for every Cartier multiple with (mgeq2).
+stated restriction surjectivity for every Cartier multiple with (mge2).
 This is the supported extension formulation in
 cite[Conjecture~5.8]{FujinoGongyo2014}.
 
@@ -6192,7 +6192,7 @@ the source need not be projective over (C).
 For an integral Weil divisor (G) on a normal variety (X),
 (OO_X(G)) denotes its divisorial sheaf. For a rational divisor (D)
 on (X) and nonnegative integers (m,n), the inclusions
-(floor( mD)+floor( nD)leqfloor((m+n)D))
+(floor( mD)+floor( nD)lefloor((m+n)D))
 give the usual multiplication on the section algebras below.
 
 
@@ -6202,20 +6202,20 @@ leavevmode
 [label=(roman*)]
  Let (k) be an algebraically closed field of characteristic zero,
 let ((X,Delta)) be a normal projective log canonical pair over (k), and assume
-that (Deltageq0) is rational and (D=K_X+Delta) is (Q)-Cartier.
+that (Deltage0) is rational and (D=K_X+Delta) is (Q)-Cartier.
 Then
 [
- R(X,D)=⊕_{mgeq0}
+ R(X,D)=⊕_{mge0}
  H^0(X,OO_X(floor( mD)))
 ]
 is a finitely generated (k)-algebra.
  Let ((X,Delta)) be a log canonical pair over (C), with
-(Deltageq0) rational and (K_X+Delta) (Q)-Cartier. For every
+(Deltage0) rational and (K_X+Delta) (Q)-Cartier. For every
 proper morphism (f:Xto S) onto an algebraic variety (S), the relative
 log canonical algebra
 [
- R(X/S,K_X+calR(X/S,K_X+Delta)
- =⊕_{mgeq0}f_*OO_X(floor( m(K_X+Delta)))
+ R(X/S,K_X+R(X/S,K_X+Delta)
+ =⊕_{mge0}f_*OO_X(floor( m(K_X+Delta)))
 ]
 is a finitely generated (OO_S)-algebra.
 
@@ -6226,14 +6226,14 @@ Proof.
 In dimension zero, (X) is a point, and in (ii) so is (S).
 The boundary and canonical divisor are zero, and the two algebras are
 (k[t]) and (OO_S[t]), respectively, with (deg t=1).
-We therefore assume (dim Xgeq1).
+We therefore assume (dim Xge1).
 
 We first prove (i). If (D) is not pseudo-effective, a nonzero
 section (s) in degree (m>0) would give
 [
  div(s)+mD
  =(div(s)+floor( mD))
-   +(mD-floor( mD))geq0.
+   +(mD-floor( mD))ge0.
 ]
 This would make (D) pseudo-effective. Thus all positive graded pieces
 vanish in this case. Since (H^0(X,OO_X)=k), we have (R(X,D)=k).
@@ -6249,7 +6249,7 @@ Theorem~thm:main makes the rational adjoint (D_Y) semiample.
 On a common smooth resolution (p:Wto X), (q:Wto Y), choose
 compatible canonical divisors. Then
 
- p^*D=q^*D_Y+E, Egeq0, E is q-exceptional.
+ p^*D=q^*D_Y+E, Ege0, E is q-exceptional.
 
 Indeed, the coefficient of (E) at a prime divisor (F) is
 (a(F;Y,Delta_Y)-a(F;X,Delta)), which is nonnegative by the discrepancy
@@ -6259,22 +6259,22 @@ pushed-forward boundary gives equality of these discrepancies. This proves
 the asserted support.
 
 For any normal variety (Z), rational Cartier divisor (G), and
-(mgeq0), the nonzero global sections of (OO_Z(floor( mG))) are
+(mge0), the nonzero global sections of (OO_Z(floor( mG))) are
 exactly the rational functions (s) satisfying
 [
- div_Z(s)+mGgeq0.
+ div_Z(s)+mGge0.
 ]
-This is equivalent to the usual inequality with (floor( mG))
+This is equivalent to the usual ineuality with (floor( mG))
 because the coefficients of (div_Z(s)) are integers.
-If the inequality holds on (X), pull back its effective (Q)-Cartier
+If the ineuality holds on (X), pull back its effective (Q)-Cartier
 left side to (W) and push forward by (q). Equation~(eq:ring-comparison)
-and (q_*E=0) give the inequality on (Y). Conversely, pull back the
-inequality on (Y), add (mE), and push forward by (p). Thus, inside
+and (q_*E=0) give the ineuality on (Y). Conversely, pull back the
+ineuality on (Y), add (mE), and push forward by (p). Thus, inside
 the common function field,
 [
  H^0(X,OO_X(floor( mD)))
  =H^0(Y,OO_Y(floor( mD_Y)))
- (mgeq0).
+ (mge0).
 ]
 These identifications preserve multiplication, so they identify the full
 graded rings.
@@ -6284,17 +6284,17 @@ It remains to use semiampleness on (Y). Choose (r>0) such that
 defines (psi:YtoPP^N_k) with
 (OO_Y(rD_Y)~=psi^*OO_{PP^N}(1)). Let
 (A=k[T_0,...,T_N]) act on (R(Y,D_Y)) by pulling back the coordinate
-sections, each in degree (r). For (0leq i<r), put
+sections, each in degree (r). For (0le i<r), put
 (F_i=*OO_Y(floor( iD_Y))), which is coherent on
 (PP^N_k) because (psi) is proper. Since
 [
  floor((ell r+i)D_Y)
- =ell(rD_Y)+floor( iD_Y) (ellgeq0),
+ =ell(rD_Y)+floor( iD_Y) (ellge0),
 ]
 projection formula identifies the part in degrees congruent to (i)
 modulo (r), as an (A)-module, with
 [
- ⊕_{ellgeq0}H^0(PP^N_k,F_i(
+ ⊕_{ellge0}H^0(PP^N_k,F_i(
 ]
 This is a finite (A)-module by the standard finite-generation theorem
 for section modules cite[Lemma~30.14.1(5)]{StacksSectionModules}.
@@ -6354,7 +6354,7 @@ For positive-dimensional (X), the theorem of
 Boucksom--Demailly--P{u a}un--Peternell
 cite[Corollary~0.3 and Theorem~2.6]{BDPP2013} says that (X) is
 non-uniruled exactly when (K_X) is pseudo-effective.
-Corollary~cor:smooth-good-model then gives (kappa(X)geq0):
+Corollary~cor:smooth-good-model then gives (kappa(X)ge0):
 the semiample canonical divisor on the good model has a nonzero section
 in a positive multiple, and the pluricanonical section spaces agree.
 Conversely, a nonzero pluricanonical section makes (K_X) pseudo-effective. This proves
@@ -6388,7 +6388,7 @@ equivalence and is non-uniruled unless it is a point. We use
  For every smooth connected projective complex variety (X),
 [
  dim R(X)=
- max_{substack{pinZ_{geq0},,kinZ_{>0}
+ max_{substack{pinZ_{ge0},,kinZ_{>0}
  H^0(X,Sym^k(Ω_X^p))ne0}} p.
 ]
  A compact connected K"ahler manifold (X) is rationally
@@ -6423,7 +6423,7 @@ Birational invariance of global covariant tensor differentials on smooth
 projective varieties gives the same vanishing on (X). This is the
 upper-bound argument of cite[Section~3]{BrunebarbeCampana2015}.
 
-If (r>0), the smooth non-uniruled base has (kappa(R(X))geq0) by
+If (r>0), the smooth non-uniruled base has (kappa(R(X))ge0) by
 Corollary~cor:uniruled-mumford. A nonzero section of (kK_{R(X)})
 for some (k>0) pulls back to a nonzero section of
 (Sym^k(Ω_X^r)). The pullback is taken on (X')
@@ -6448,7 +6448,7 @@ MRC base is a point and (X) is rationally connected.
 We next use ordinary cotangent invariants, with no boundary. If
 (L is a coherent rank-one subsheaf of a vector bundle on a
 smooth projective variety, its double dual is a line bundle; put
-(kappa(L)= L^calL^{**})). Define
+(kappa(L)= L^L^{**})). Define
 [
  omega(X)=
  max_{substack{m>0
@@ -6471,7 +6471,7 @@ also put
  kappa(detF).
 ]
 Here (p) is an integer, (F is coherent, and
-(detF=(∧^calF=(∧^{rkF} F)^calF)^{**}). These are the
+(detF=(∧^F=(∧^{rkF} F)^F)^{**}). These are the
 invariants used in Campana's Appendix~A to Taji [cite].
 
 [Cotangent invariants and the MRC base]
@@ -6486,7 +6486,7 @@ Let (X) be a smooth connected projective complex variety. Then
 ]
 In particular, if (X) is positive-dimensional and non-uniruled, then
 (omega(X)=kappa(X)). If (X) is positive-dimensional and
-(kappa(X)geq0), then
+(kappa(X)ge0), then
 [
  kappa^+(X)=kappa(X).
 ]
@@ -6503,7 +6503,7 @@ resolving the MRC map. If (R(X)) is a point, the displayed formula
 therefore follows without changing our convention (kappa(pt)=0).
 
 Suppose that (R=R(X)) has positive dimension. It is non-uniruled, so
-(kappa(R)geq0) by Corollary~cor:uniruled-mumford.
+(kappa(R)ge0) by Corollary~cor:uniruled-mumford.
 To apply Campana's abundance deduction
 cite[Conjecture~A.2 and Remark~A.3]{Taji2014}, take a resolved Iitaka
 fibration (R'to Z), with (R') and (Z) smooth and projective,
@@ -6517,17 +6517,17 @@ cite[Appendix~A, Examples~A.1(2)]{Taji2014} says that the existence of
 such a normal birational model implies (omega(F)=0); it does not
 require (K_F) itself to be trivial. Property~4 of the same appendix
 and birational invariance now give
-(omega(R)=omega(R')leqdim Z=kappa(R)).
+(omega(R)=omega(R')ledim Z=kappa(R)).
 If (F) is a point, this upper bound is simply
-(omega(R)leqdim R=kappa(R)). The reverse inequality
-(omega(R)geqkappa(R)) is one of the defining cotangent inequalities
+(omega(R)ledim R=kappa(R)). The reverse ineuality
+(omega(R)gekappa(R)) is one of the defining cotangent ineualities
 recorded in the same appendix. This proves the piecewise formula.
 A non-uniruled (X) has (R(X)) birational to (X), giving the
 first stated specialization.
 
 Campana also records
-(omega(X)geqkappa^+(X)geqkappa(X))
-cite[Appendix~A, property~2]{Taji2014}. When (kappa(X)geq0),
+(omega(X)gekappa^+(X)gekappa(X))
+cite[Appendix~A, property~2]{Taji2014}. When (kappa(X)ge0),
 Corollary~cor:uniruled-mumford makes (X) non-uniruled, and the
 two outer terms are equal by the formula just proved. Thus
 (kappa^+(X)=kappa(X)). This is the abundance consequence recalled
@@ -6703,7 +6703,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   journal = {Journal of Differential Geometry},
 %   volume = {42}, number = {2}, year = {1995}, pages = {193--219},
 %   doi = {10.4310/jdg/1214457231},
-%   url = {https://www.math.stonybrook.edu/~roblaz/Reprints/Ein.Kuchle.Laz.LPALB.pdf},
+%   url = {./pdf/unknown-nd-url.pdf},
 %   note = {Proposition 2.3}
 % }
 
@@ -6724,7 +6724,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   year = {2013},
 %   pages = {207--221},
 %   doi = {10.1515/crelle.2012.017},
-%   url = {https://math.univ-cotedazur.fr/~hoering/articles/a13-universal.pdf}
+%   url = {./pdf/unknown-nd-url.pdf}
 % }
 
 % @article{Hashizume2018,
@@ -6739,7 +6739,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   eprint = {1609.00121},
 %   archivePrefix = {arXiv},
 %   primaryClass = {math.AG},
-%   url = {https://arxiv.org/abs/1609.00121v4},
+%   url = {./pdf/unknown-nd-1609.00121v4-url.pdf},
 %   note = {Theorem 1.4 in arXiv:1609.00121v4}
 % }
 
@@ -6756,7 +6756,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   eprint = {1302.5194},
 %   archivePrefix = {arXiv},
 %   primaryClass = {math.AG},
-%   url = {https://www.math.kyoto-u.ac.jp/~fujino/log-ring-aspm.pdf},
+%   url = {./pdf/unknown-nd-url.pdf},
 %   note = {Conjectures A--C, Theorem 1.1, Corollary 1.4, and Lemma 3.5}
 % }
 
@@ -6771,7 +6771,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   eprint = {0907.4170},
 %   archivePrefix = {arXiv},
 %   primaryClass = {math.AG},
-%   url = {https://arxiv.org/abs/0907.4170v1},
+%   url = {./pdf/unknown-nd-0907.4170v1-url.pdf},
 %   note = {Remark 3.1 and Proposition 3.2(3) in arXiv:0907.4170v1}
 % }
 
@@ -6787,7 +6787,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   eprint = {0706.1792},
 %   archivePrefix = {arXiv},
 %   primaryClass = {math.AG},
-%   url = {https://arxiv.org/abs/0706.1792v3},
+%   url = {./pdf/unknown-nd-0706.1792v3-url.pdf},
 %   note = {Lemma 3.6 in arXiv:0706.1792v3}
 % }
 
@@ -6814,7 +6814,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   year = {2014},
 %   pages = {593--620},
 %   doi = {10.1112/S0010437X13007495},
-%   url = {https://www.math.kyoto-u.ac.jp/~fujino/fg-comp-final.pdf},
+%   url = {./pdf/unknown-nd-url.pdf},
 %   note = {Theorem 1.5, restated as Theorem 4.3}
 % }
 
@@ -6852,7 +6852,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   eprint = {0907.1506},
 %   archivePrefix = {arXiv},
 %   primaryClass = {math.AG},
-%   url = {https://www.math.kyoto-u.ac.jp/~fujino/MMP21.pdf},
+%   url = {./pdf/unknown-nd-url.pdf},
 %   note = {Version 6.01, January 8, 2009; Theorem 3.74(3), p. 109}
 % }
 
@@ -6862,7 +6862,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   journal = {Compositio Mathematica},
 %   year = {2013}, volume = {149}, number = {2}, pages = {295--308},
 %   doi = {10.1112/S0010437X12000553},
-%   url = {https://www.ms.u-tokyo.ac.jp/~gongyo/papers/redmapsmmp-final.pdf},
+%   url = {./pdf/unknown-nd-url.pdf},
 %   note = {Read with the numerical-dimension erratum on Lehmann's research page}
 % }
 
@@ -6882,7 +6882,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   journal = {Journal of the American Mathematical Society},
 %   year = {2017}, volume = {30}, number = {4}, pages = {959--1021},
 %   doi = {10.1090/jams/871},
-%   url = {https://sites.math.washington.edu/~kovacs/current/papers/Kovacs_Patakfalvi__Projectivity.pdf},
+%   url = {./pdf/unknown-nd-url.pdf},
 %   note = {Theorem 9.9 in the author version}
 % }
 
@@ -6893,7 +6893,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   year = {2013}, volume = {22}, number = {2}, pages = {201--248},
 %   doi = {10.1090/S1056-3911-2012-00574-8},
 %   eprint = {math/0405285}, archivePrefix = {arXiv}, primaryClass = {math.AG},
-%   url = {https://arxiv.org/abs/math/0405285},
+%   url = {./pdf/unknown-nd-math0405285-url.pdf},
 %   note = {Theorem locators refer to arXiv version 1}
 % }
 
@@ -6904,7 +6904,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   year = {2017}, volume = {60}, number = {6}, pages = {1019--1028},
 %   doi = {10.1007/s11425-016-9039-2},
 %   eprint = {1608.04706}, archivePrefix = {arXiv}, primaryClass = {math.AG},
-%   url = {https://arxiv.org/abs/1608.04706},
+%   url = {./pdf/unknown-nd-1608.04706-url.pdf},
 %   note = {Theorem locators refer to arXiv version 1}
 % }
 
@@ -6913,7 +6913,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   title = {Fundamental group and pluridifferentials on compact {K{"a}hler} manifolds},
 %   year = {2015},
 %   eprint = {1510.07922}, archivePrefix = {arXiv}, primaryClass = {math.AG},
-%   url = {https://arxiv.org/abs/1510.07922},
+%   url = {./pdf/unknown-nd-1510.07922-url.pdf},
 %   note = {Version 2}
 % }
 
@@ -6930,7 +6930,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   journal = {Annales de l'Institut Fourier},
 %   year = {2015}, volume = {65}, number = {2}, pages = {835--861},
 %   doi = {10.5802/aif.2945},
-%   url = {https://www.numdam.org/item/AIF_2015__65_2_835_0/}
+%   url = {./pdf/unknown-nd-url.pdf}
 % }
 
 % @article{Langer2004,
@@ -6949,7 +6949,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   year = {2008}, volume = {173}, number = {2}, pages = {427--447},
 %   doi = {10.1007/s00222-008-0125-y},
 %   eprint = {math/0611360v2}, archivePrefix = {arXiv}, primaryClass = {math.AG},
-%   url = {https://arxiv.org/abs/math/0611360v2},
+%   url = {./pdf/unknown-nd-math0611360-url.pdf},
 %   note = {Theorem 3.7 and Lemma 4.2 in arXiv:math/0611360v2}
 % }
 
@@ -6969,7 +6969,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   journal = {Annales de l'Institut Fourier},
 %   year = {2014}, volume = {64}, number = {4}, pages = {1721--1735},
 %   doi = {10.5802/aif.2894},
-%   url = {https://www.numdam.org/item/AIF_2014__64_4_1721_0/}
+%   url = {./pdf/unknown-nd-url.pdf}
 % }
 
 % @incollection{HMX2018,
@@ -6980,7 +6980,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   volume = {97.1}, pages = {309--348},
 %   publisher = {American Mathematical Society}, year = {2018},
 %   doi = {10.1090/pspum/097.1/01677},
-%   url = {https://www.claymath.org/wp-content/uploads/2022/03/Hacon-AG2015.pdf},
+%   url = {./pdf/unknown-nd-url.pdf},
 %   note = {Published Theorem 4.0.1; Theorem 3.0.1 in arXiv version 2}
 % }
 
@@ -7000,7 +7000,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   year = {2000}, volume = {316}, number = {2}, pages = {283--331},
 %   doi = {10.1007/s002080050014},
 %   eprint = {math/9906088}, archivePrefix = {arXiv},
-%   url = {https://arxiv.org/abs/math/9906088v1},
+%   url = {./pdf/unknown-nd-math9906088-url.pdf},
 %   note = {Theorem 0.2, Corollary 0.3 and Proposition 5.3 in arXiv:math/9906088v1}
 % }
 
@@ -7014,7 +7014,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   year = {2007},
 %   pages = {1--32},
 %   note = {Theorem 1.10 in the author's March 23, 2005 notes},
-%   url = {https://indico.ictp.it/event/a04204/session/10/contribution/7/material/0/0.pdf}
+%   url = {./pdf/unknown-nd-url.pdf}
 % }
 
 % @book{SGA1,
@@ -7026,7 +7026,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   year = {1971},
 %   eprint = {math/0206203v2},
 %   archivePrefix = {arXiv},
-%   url = {https://arxiv.org/abs/math/0206203v2},
+%   url = {./pdf/unknown-nd-math0206203-url.pdf},
 %   note = {Expos{'e} X, Theorem 3.1, and Expos{'e} XII, Theorem 5.1}
 % }
 
@@ -7037,7 +7037,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   year = {2022}, volume = {10},
 %   doi = {10.1017/fmp.2022.15},
 %   note = {Paper No. e22; Theorem C and Proposition 13.1},
-%   url = {https://people.math.harvard.edu/~mpopa/papers/Filtration_LC_Pi.pdf}
+%   url = {./pdf/unknown-nd-url.pdf}
 % }
 
 % @article{KollarKovacs2010,
@@ -7046,7 +7046,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   journal = {Journal of the American Mathematical Society},
 %   year = {2010}, volume = {23}, number = {3}, pages = {791--813},
 %   doi = {10.1090/S0894-0347-10-00663-6},
-%   url = {https://sites.math.washington.edu/~kovacs/current/papers/Kollar_Kovacs__Log_canonical_singularities_are_Du_Bois.pdf}
+%   url = {./pdf/unknown-nd-url.pdf}
 % }
 
 % @incollection{BauerEtAl2002,
@@ -7056,7 +7056,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   publisher = {Springer}, address = {Berlin},
 %   year = {2002}, pages = {27--36},
 %   doi = {10.1007/978-3-642-56202-0_2},
-%   url = {https://arxiv.org/abs/math/0106147v1},
+%   url = {./pdf/unknown-nd-math0106147-url.pdf},
 %   note = {Theorem 2.1 in arXiv:math/0106147v1}
 % }
 
@@ -7095,7 +7095,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   volume = {94}, year = {2017}, pages = {17--30},
 %   publisher = {American Mathematical Society},
 %   doi = {10.1090/pspum/094/01620},
-%   url = {https://algebra.dmi.unibas.ch/blanc/articles/structurebir.pdf},
+%   url = {./pdf/unknown-nd-url.pdf},
 %   note = {Theorems 3.8--3.9}
 % }
 
@@ -7115,7 +7115,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   publisher = {International Press and Higher Education Press},
 %   address = {Somerville, MA, and Beijing},
 %   isbn = {978-1-57146-234-3},
-%   url = {https://www-fourier.univ-grenoble-alpes.fr/~demailly/manuscripts/analmeth_book.pdf},
+%   url = {./pdf/unknown-nd-url.pdf},
 %   note = {Theorem numbering refers to the author manuscript dated July 2011}
 % }
 
@@ -7133,7 +7133,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   title = {{Siu}'s invariance of plurigenera: a one-tower proof},
 %   journal = {Journal of Differential Geometry},
 %   year = {2007}, volume = {76}, number = {3}, pages = {485--493},
-%   url = {https://archive.ymsc.tsinghua.edu.cn/pacm_download/21/685-JDG-76-3-485-493.pdf},
+%   url = {./pdf/unknown-nd-url.pdf},
 %   note = {Theorem 1}
 % }
 
@@ -7142,7 +7142,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   title = {Abundance conjecture for 3-folds: case {nu=1}},
 %   journal = {Compositio Mathematica},
 %   volume = {68}, number = {2}, year = {1988}, pages = {203--220},
-%   url = {https://www.numdam.org/item/CM_1988__68_2_203_0/}
+%   url = {./pdf/unknown-nd-url.pdf}
 % }
 
 % @article{Kawamata1992,
@@ -7177,7 +7177,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   title = {A correction to the paper ``Log abundance theorem for threefolds''},
 %   year = {2003},
 %   eprint = {math/0302360v1}, archivePrefix = {arXiv}, primaryClass = {math.AG},
-%   url = {https://arxiv.org/abs/math/0302360v1},
+%   url = {./pdf/unknown-nd-math0302360-url.pdf},
 %   note = {arXiv:math/0302360v1}
 % }
 
@@ -7187,7 +7187,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   journal = {Duke Mathematical Journal},
 %   volume = {102}, number = {3}, year = {2000}, pages = {513--532},
 %   doi = {10.1215/S0012-7094-00-10237-2},
-%   url = {https://www.math.kyoto-u.ac.jp/~fujino/Abundance.pdf}
+%   url = {./pdf/unknown-nd-url.pdf}
 % }
 
 % @article{Gongyo2013,
@@ -7197,7 +7197,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   volume = {22}, number = {3}, year = {2013}, pages = {549--564},
 %   doi = {10.1090/S1056-3911-2012-00593-1},
 %   eprint = {1005.2796v2}, archivePrefix = {arXiv}, primaryClass = {math.AG},
-%   url = {https://arxiv.org/abs/1005.2796v2}
+%   url = {./pdf/unknown-nd-1005.2796v2-url.pdf}
 % }
 
 % @article{DemaillyHaconPaun2013,
@@ -7206,7 +7206,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   journal = {Acta Mathematica},
 %   volume = {210}, number = {2}, year = {2013}, pages = {203--259},
 %   doi = {10.1007/s11511-013-0094-x},
-%   url = {https://www-fourier.univ-grenoble-alpes.fr/~demailly/source_files/dhp/dhp_final_corr.pdf}
+%   url = {./pdf/unknown-nd-url.pdf}
 % }
 
 % @book{EsnaultViehweg1992,
@@ -7215,7 +7215,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   series = {DMV Seminar}, volume = {20}, year = {1992},
 %   publisher = {Birkh{"a}user}, address = {Basel},
 %   doi = {10.1007/978-3-0348-8600-0},
-%   url = {https://page.mi.fu-berlin.de/esnault/books/esvibuch.pdf}
+%   url = {./pdf/unknown-nd-url.pdf}
 % }
 
 % @article{Deligne1971,
@@ -7224,7 +7224,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   journal = {Publications Math{'e}matiques de l'IH{'E}S},
 %   volume = {40}, year = {1971}, pages = {5--57},
 %   doi = {10.1007/BF02684692},
-%   url = {https://www.numdam.org/item/10.1007/BF02684692.pdf}
+%   url = {./pdf/unknown-nd-url.pdf}
 % }
 
 % @misc{OpenAIMinimalModels2026,
@@ -7241,7 +7241,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   title = {Non-vanishing implies numerical dimension one abundance},
 %   year = {2025},
 %   eprint = {2505.05250v2}, archivePrefix = {arXiv}, primaryClass = {math.AG},
-%   url = {https://arxiv.org/abs/2505.05250v2},
+%   url = {./pdf/unknown-nd-2505.05250v2-url.pdf},
 %   note = {arXiv:2505.05250v2, July 30, 2025; Theorems 3.1, 5.1 and 5.4}
 % }
 
@@ -7262,7 +7262,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   volume = {358}, number = {3-4}, year = {2014}, pages = {861--878},
 %   doi = {10.1007/s00208-013-0976-4},
 %   eprint = {1203.1081v1}, archivePrefix = {arXiv}, primaryClass = {math.AG},
-%   url = {https://arxiv.org/abs/1203.1081v1},
+%   url = {./pdf/unknown-nd-1203.1081v1-url.pdf},
 %   note = {Section 2 and Proposition 2.12 in arXiv:1203.1081v1}
 % }
 
@@ -7280,7 +7280,7 @@ boundaries or semi-log-canonical analytic pairs.
 %  author = {Chan, Tsz On Mario and Choi, Young-Jun},
 %  title = {Extension with log-canonical measures and an improvement to the plt extension of {Demailly--Hacon--Pu{a}un}},
 %  year = {2021}, eprint = {1912.08076v5}, archivePrefix = {arXiv},
-%  url = {https://arxiv.org/abs/1912.08076v5},
+%  url = {./pdf/unknown-nd-1912.08076v5-url.pdf},
 %  doi = {10.1007/s00208-021-02152-3},
 %  note = {arXiv:1912.08076v5; Theorem 1.6.1}
 % }
@@ -7314,7 +7314,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   title = {On finiteness of relative log pluricanonical representations},
 %   year = {2025},
 %   howpublished = {Author manuscript},
-%   url = {https://www.math.kyoto-u.ac.jp/~fujino/analytic-abundance7.pdf},
+%   url = {./pdf/unknown-nd-url.pdf},
 %   note = {Version 0.14, May 29, 2025; Theorem 1.10 and its proof on p. 30}
 % }
 
@@ -7323,7 +7323,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   title = {Minimal model program for projective morphisms between complex analytic spaces},
 %   year = {2022},
 %   eprint = {2201.11315v1}, archivePrefix = {arXiv}, primaryClass = {math.AG},
-%   url = {https://arxiv.org/abs/2201.11315v1},
+%   url = {./pdf/unknown-nd-2201.11315v1-url.pdf},
 %   note = {Definitions 2.28, 2.32 and 2.48}
 % }
 
@@ -7333,7 +7333,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   year = {2016},
 %   eprint = {1504.01095v3},
 %   archivePrefix = {arXiv},
-%   url = {https://arxiv.org/abs/1504.01095v3},
+%   url = {./pdf/unknown-nd-1504.01095v3-url.pdf},
 %   note = {Version 3; Theorem 1.1 gives the abelian-base case}
 % }
 
@@ -7343,7 +7343,7 @@ boundaries or semi-log-canonical analytic pairs.
 %   year = {2017},
 %   eprint = {1611.08768v2},
 %   archivePrefix = {arXiv},
-%   url = {https://arxiv.org/abs/1611.08768v2},
+%   url = {./pdf/unknown-nd-1611.08768v2-url.pdf},
 %   note = {Version 2; Theorem 1.1}
 % }
 %