diff --git a/abundance_terminal_threefolds_charp_v2.txt b/abundance_terminal_threefolds_charp_v2.txt
index 01deb00..86820fb 100644
--- a/abundance_terminal_threefolds_charp_v2.txt
+++ b/abundance_terminal_threefolds_charp_v2.txt
@@ -103,7 +103,7 @@ Terminal & The mildest kind of singularity for the minimal model program.
 Numerical dimension & The number of directions in which a nef divisor is positive. 
 Semiample & Some positive multiple is generated by global sections. 
 Kodaira dimension & The growth rate of spaces of pluricanonical sections. 
-Frobenius & The map xmapsto x^p in characteristic p. 
+Frobenius & The map x|-> x^p in characteristic p. 
 Cartier descent & Recovering data from its Frobenius pullback. 
 Jet & A section remembered only up to a finite infinitesimal order. 
 Picard group & The group of line bundles up to isomorphism. 
@@ -151,9 +151,9 @@ and K_X is semiample.
 
 Recall that nu(L)=1 for a nef divisor L means
 [
-Lcdot H^2>0,
+L· H^2>0,
 
-L^2cdot H=0
+L^2· H=0
 ]
 for an ample Cartier divisor H. The theorem makes no assumption on the irregularity, the Albanese map, or the first Betti number.
 
@@ -184,9 +184,9 @@ The argument has two halves that meet in a contradiction.
 [
 D=sum_alpha m_alpha D_alpha
 ]
-with connected simple-normal-crossing support, primitive multiplicities, D nef and numerically trivial on each component, cO_D(D) torsion, and a local relation omega_V^{ ⊗  p^b}simeqcO_V(B_*) supported on D. See Proposition~prop:prepared-cycle.
+with connected simple-normal-crossing support, primitive multiplicities, D nef and numerically trivial on each component, O_D(D) torsion, and a local relation omega_V^{ ⊗  p^b}~=O_V(B_*) supported on D. See Proposition~prop:prepared-cycle.
  Slopes. Prove that the tangent bundle T_V is strongly semistable for the slope c_1(-)HD/rk(-). Deduce c_2(V)Dge0 and uniform bounds on polar parts of sections of semistable bundles. See Proposition~prop:strong-semistability, Proposition~prop:polar-bounds, and Corollary~cor:no-scalar-poles.
- Jets. Study finite jets of line bundles along D. A three-row spectral sequence relates scalar, middle, and canonical jets. Prove the scalar last width is asymptotically p/(p+1) and consequently c_2(V)D=0. Prove a canonical formal root widehatomega_VsimeqwidehatcO_V(B_0). See Proposition~prop:scalar-widths and Proposition~prop:canonical-formal-root.
+ Jets. Study finite jets of line bundles along D. A three-row spectral sequence relates scalar, middle, and canonical jets. Prove the scalar last width is asymptotically p/(p+1) and consequently c_2(V)D=0. Prove a canonical formal root widehatomega_V~=widehatO_V(B_0). See Proposition~prop:scalar-widths and Proposition~prop:canonical-formal-root.
  Laurent cohomology. Turn the jet estimates into a criterion: for a bundle satisfying an Euler-characteristic identity, bounded polar parts, and concentration of leading jets, every Laurent v Cech one-cocycle has a primitive. Apply this to one-forms, exact forms, and closed forms. See Theorem~thm:laurent-vanishing and Corollary~cor:form-vanishing.
  Cartier descent. Use the vanishing of Laurent cocycles to give every line bundle on the formal punctured neighborhood an integrable connection whose forms are Cartier-fixed. Cartier descent then shows that every line bundle is a pth power. Hence Pic(|D|,cR) is p-divisible. See Proposition~prop:picard-divisibility.
  Degree contradiction. Orders of Laurent units and intersection with H give a nonzero homomorphism
@@ -215,7 +215,7 @@ Laurent cocycles have primitives
 Pic(|D|,cR)is p-divisible
   arrow[dr]
 && overlinePhi:Pic(|D|,cR)toZ
-             overlinePhi(cO(H))=DH^2>0
+             overlinePhi(O(H))=DH^2>0
   arrow[dl] 
 & Contradiction &
 
@@ -227,22 +227,22 @@ Caption: The maximal-nef-dimension assumption forces the Picard group of the for
 
 
 ## Varieties and divisors
-A variety is an integral separated scheme of finite type over a field. A threefold has dimension three. A divisor is a formal integer combination of prime divisors; it is Cartier if it is locally defined by one equation. We write sim for linear equivalence and equiv for numerical equivalence. A divisor L is nef if Lcdot Cge0 for every integral curve C. It is ample if some positive multiple is very ample. It is semiample if some positive multiple is globally generated.
+A variety is an integral separated scheme of finite type over a field. A threefold has dimension three. A divisor is a formal integer combination of prime divisors; it is Cartier if it is locally defined by one equation. We write sim for linear equivalence and equiv for numerical equivalence. A divisor L is nef if L· Cge0 for every integral curve C. It is ample if some positive multiple is very ample. It is semiample if some positive multiple is globally generated.
 
 For a nef divisor L, the nef dimension n(L) is the dimension of its nef reduction. In particular, n(L)=dim X means that no L-trivial curve passes through a very general point. This is a different invariant from the numerical dimension nu(L).
 
 ## Completion and Laurent sheaves
-Let V be a smooth variety and D an effective Cartier divisor. We write widehatcO_V for the completion along D, regarded as a sheaf on the Zariski space |D|. Set
+Let V be a smooth variety and D an effective Cartier divisor. We write widehatO_V for the completion along D, regarded as a sheaf on the Zariski space |D|. Set
 [
-cR=widehatcO_V(*D).
+R=widehatO_V(*D).
 ]
 Thus a section of cR is a formal function on a formal neighborhood of D that is allowed a finite pole along D. A line bundle on (|D|,cR) is a locally free rank-one cR-module in the Zariski topology. Its Picard group is denoted Pic(|D|,cR).
 
 For a vector bundle E, we write widehatwidehat EE for its completion and
 [
-E_{cR}=widehatwidehat EE ⊗ _{widehatcO_V}cR.
+E_{R}=widehatwidehat EE ⊗ _{widehatO_V}R.
 ]
-A Laurent v Cech cocycle is a v Cech cocycle with values in E_{cR} on a finite affine cover adapted to D and E. A primitive is a cochain whose coboundary is the given cocycle.
+A Laurent v Cech cocycle is a v Cech cocycle with values in E_{R} on a finite affine cover adapted to D and E. A primitive is a cochain whose coboundary is the given cocycle.
 
 ## Frobenius and Cartier
 Let k be perfect of characteristic p>0. Absolute Frobenius is the map
@@ -264,7 +264,7 @@ Let D be an effective Cartier divisor and E a vector bundle. For integers u<i, t
 [
 P_i^s(E)=(E(iD))/(E((i-s)D))
 ]
-is the bundle of width-s jets with upper weight i. Its sections remember the first s infinitesimal layers along D above the weight i. The last width of a line bundle L at weight i is the largest s for which L(iD) is trivial on the s-th thickening sD. We write d_e for the last width of cO_V(M_eD) when M_e=rp^e and r is the order of cO_D(D).
+is the bundle of width-s jets with upper weight i. Its sections remember the first s infinitesimal layers along D above the weight i. The last width of a line bundle L at weight i is the largest s for which L(iD) is trivial on the s-th thickening sD. We write d_e for the last width of O_V(M_eD) when M_e=rp^e and r is the order of O_D(D).
 
 # Birational preparation
 
@@ -290,19 +290,19 @@ such that:
  Supp D is connected and has simple normal crossings. The divisor D is nef, and D|_{D_alpha} is numerically trivial for every alpha.
  No D-trivial curve passes through a very general point of V. Moreover
 [
-Dcdot H^2>0,
+D· H^2>0,
 
-D^2cdot H=0.
+D^2· H=0.
 ]
- cO_D(D) is torsion. Let r be its order.
+ O_D(D) is torsion. Let r be its order.
  There are an integer bge0, an open neighborhood U of |D|, and integers b_alpha such that
 
 omega_V^{ ⊗  p^b}|_U
-simeq cO_U(B_*),
+~= O_U(B_*),
 
 B_*=sum_alpha b_alpha D_alpha.
 
-Consequently K_Vcdot Dcdot A=0 for every Cartier divisor A on V.
+Consequently K_V· D· A=0 for every Cartier divisor A on V.
 
 
 
@@ -315,7 +315,7 @@ By nonvanishing cite[Theorem~1.4]{xu2024}, there is a nonzero effective Cartier
 
 m>0.
 ]
-It is nef and N_0^2cdot H_X=0 for an ample divisor H_X. Every prime component T of N_0 satisfies N_0cdot H_Xcdot T=0, because these numbers are nonnegative and have positive weighted sum zero. In fact N_0|_Tequiv0: for an integral curve Csubset T, take a large multiple of H_X|_T with an effective member containing C; the nef restriction has intersection zero with it, so every component contributes zero.
+It is nef and N_0^2· H_X=0 for an ample divisor H_X. Every prime component T of N_0 satisfies N_0· H_X· T=0, because these numbers are nonnegative and have positive weighted sum zero. In fact N_0|_Tequiv0: for an integral curve Csubset T, take a large multiple of H_X|_T with an effective member containing C; the nef restriction has intersection zero with it, so every component contributes zero.
 
 If n(X,K_X)le2, the low-nef-dimension abundance theorem cite[Theorem~1.8]{xu2024} makes K_X semiample. Since nu(K_X)=1, some multiple has at least two sections, and we are done. Hence we may assume
 [
@@ -395,25 +395,25 @@ Let sigma:Σ'toΣ be the finite S_2-ification, an isomorphism away from finitely
 where (Σ',Delta') is an slc surface pair cite[Lemma~2.19]{xu2026}. The local verification uses the following lemma.
 
 [Exceptional surface intersections]
-Let P be an excellent normal local surface with closed residue field l, and let g:Tto P be a nontrivial projective regular resolution that is an isomorphism away from the closed point. Write G_i for its exceptional integral curves, and take intersection degrees over l. Then G_i^2<0 for every i. More generally, if an exceptional real divisor Z=sum_i z_iG_i has a positive coefficient, there is an index j with z_j>0 and Zcdot G_j<0.
+Let P be an excellent normal local surface with closed residue field l, and let g:Tto P be a nontrivial projective regular resolution that is an isomorphism away from the closed point. Write G_i for its exceptional integral curves, and take intersection degrees over l. Then G_i^2<0 for every i. More generally, if an exceptional real divisor Z=sum_i z_iG_i has a positive coefficient, there is an index j with z_j>0 and Z· G_j<0.
 
 
 Proof. 
-The exceptional fiber is connected. Choose a nonzero nonunit tincO_P and write
+The exceptional fiber is connected. Choose a nonzero nonunit tinO_P and write
 [
 div_T(t)=A+sum_i c_iG_i,
  c_i>0.
 ]
 The strict transform A meets the exceptional fiber. For C=sum_i c_iG_i,
 [
-Ccdot G_i=-Acdot G_ile0.
+C· G_i=-A· G_ile0.
 ]
 Distinct curves have nonnegative intersections. Connectedness implies G_i^2<0: every vertex has either a neighbor or, if it is the only vertex, a positive intersection with A.
 
 For the second assertion, put a=max_i(z_i/c_i)>0, and let I be the nonempty set where the maximum is attained. Choose jin I that either meets A or has a neighbor outside I. Such a vertex exists by connectedness. Then
 [
-Zcdot G_j
-=aCcdot G_j+sum_i(z_i-ac_i)G_icdot G_j<0.
+Z· G_j
+=aC· G_j+sum_i(z_i-ac_i)G_i· G_j<0.
 ]
 
 
@@ -424,42 +424,42 @@ A terminal threefold in the present setting has only isolated singularities.
 Proof. 
 Localize at a codimension-two point. This gives an excellent normal local surface P with residue field l and a projective regular resolution g:Tto P. For an integral exceptional curve G, divisor adjunction and curve duality give
 [
-(K_T+G)cdot G
-=2dim_lH^1(G,cO_G)-2dim_lH^0(G,cO_G).
+(K_T+G)· G
+=2dim_lH^1(G,O_G)-2dim_lH^0(G,O_G).
 ]
-If K_Tcdot G<0, set l'=H^0(G,cO_G) and d=[l':l]. Both G^2 and K_Tcdot G are negative multiples of d. The displayed formula and Lemma~lem:exceptional-intersections force
+If K_T· G<0, set l'=H^0(G,O_G) and d=[l':l]. Both G^2 and K_T· G are negative multiples of d. The displayed formula and Lemma~lem:exceptional-intersections force
 [
-H^1(G,cO_G)=0,
+H^1(G,O_G)=0,
 
-G^2=K_Tcdot G=-d.
+G^2=K_T· G=-d.
 ]
-The line bundle cO_G(-G) has degree one over l', and Riemann--Roch gives at least two independent sections over l'. Two independent sections have no common zero, so they define a degree-one finite birational map GtoPP^1_{l'}, hence an isomorphism. In particular GsimeqPP^1_{l'} and cO_G(G)simeqcO_{PP^1_{l'}}(-1).
+The line bundle O_G(-G) has degree one over l', and Riemann--Roch gives at least two independent sections over l'. Two independent sections have no common zero, so they define a degree-one finite birational map GtoPP^1_{l'}, hence an isomorphism. In particular G~=PP^1_{l'} and O_G(G)~=O_{PP^1_{l'}}(-1).
 
 One can contract such a curve while retaining a projective regular resolution. Choose a relatively very ample Cartier divisor A, put a=deg_{l'}(A|_G)>0, and set Q=A+aG. The divisor Q-G is relatively ample. The restriction Q|_G is trivial, and a trivializing section extends successively to nG because
 [
-H^1(G,cO_G(nQ-jG))
-=H^1(PP^1_{l'},cO(j))=0
+H^1(G,O_G(nQ-jG))
+=H^1(PP^1_{l'},O(j))=0
 (jge1).
 ]
 For large n, relative Serre vanishing lifts the sections to T. They generate along G and, away from G, the sections of n(Q-G) generate and separate points and tangent directions. Thus nQ defines a projective contraction whose only nontrivial fiber is G.
 
 Let q be the image of G. Formal functions identify the completed local ring at q with
 [
-varprojlim_n H^0(nG,cO_{nG}).
+varprojlim_n H^0(nG,O_{nG}).
 ]
-The graded pieces are H^0(G,cO_G(-nG))=H^0(PP^1_{l'},cO(n)), and there is no obstruction. The associated graded algebra is l'[x,y], so the completed local ring, and then the local ring at q, is regular.
+The graded pieces are H^0(G,O_G(-nG))=H^0(PP^1_{l'},O(n)), and there is no obstruction. The associated graded algebra is l'[x,y], so the completed local ring, and then the local ring at q, is regular.
 
 Contracting all K_T-negative exceptional curves yields a regular resolution with K_T relatively nef. If any exceptional curves remain, terminality gives
 [
 E=K_T-g^*K_P=sum_i a_iG_i,
  a_i>0.
 ]
-Lemma~lem:exceptional-intersections gives Ecdot G_j<0 for some j, contradicting relative nefness. Thus the codimension-two point was regular. Normality already gives regularity in codimension one, so the singular locus is finite.
+Lemma~lem:exceptional-intersections gives E· G_j<0 for some j, contradicting relative nefness. Thus the codimension-two point was regular. Normality already gives regularity in codimension one, so the singular locus is finite.
 
 
 Using these lemmas one proves that Σ' is nodal in codimension one and that (eq:slc-adjunction) is an equality of log canonical classes. By (eq:dlt-adjunction-input), the right side is numerically trivial. Surface abundance for projective slc surfaces cite[Theorem~3.1]{posva2024} makes it semiample, hence Q-linearly trivial. Therefore some positive Cartier multiple L of f^*S_Y satisfies
 
-cO_{Y_d}(L)|_{Σ∖ Z}simeqcO_{Σ∖ Z}
+O_{Y_d}(L)|_{Σ∖ Z}~=O_{Σ∖ Z}
 
 for a finite set Z of closed points, and L|_{Σ_i}equiv0 for every component.
 
@@ -467,41 +467,41 @@ for a finite set Z of closed points, and L|_{Σ_i}equiv0 for every component.
 The reduced boundary is now under control. We need torsion on the full nonreduced cycle.
 
 [Extending torsion across finitely many points]
-Let Y be a normal projective threefold over an algebraically closed field of characteristic p>0, and let L be an effective Cartier divisor. Suppose L is numerically trivial on every integral component of its support and cO_Y(L)|_{L_{red}∖ Z} is trivial for a finite set of closed points Z. Then cO_L(L) is torsion.
+Let Y be a normal projective threefold over an algebraically closed field of characteristic p>0, and let L be an effective Cartier divisor. Suppose L is numerically trivial on every integral component of its support and O_Y(L)|_{L_{red}∖ Z} is trivial for a finite set of closed points Z. Then O_L(L) is torsion.
 
 
 Proof. 
-Replace Z by Zcap|L|. For zin Z, put R=cO_{Y,z}. Normality gives depth at least two, so H^1_{mathfrak m}frakm(R)=0. Every nonmaximal localization of R is a normal local ring of dimension at most two, hence Cohen--Macaulay. This implies H^2_{mathfrak m}frakm(R) has finite length. Present R=A/I with A regular local of dimension n. The height of I is n-3, and Auslander--Buchsbaum gives projective dimension n-3 at nonmaximal primes. Hence Ext_A^{n-2}(R,A) is supported only at the closed point and has finite length. Local duality identifies H^2_{mathfrak m}frakm(R) with its Matlis dual, up to completion.
+Replace Z by Zcap|L|. For zin Z, put R=O_{Y,z}. Normality gives depth at least two, so H^1_{mathfrak m}frakm(R)=0. Every nonmaximal localization of R is a normal local ring of dimension at most two, hence Cohen--Macaulay. This implies H^2_{mathfrak m}frakm(R) has finite length. Present R=A/I with A regular local of dimension n. The height of I is n-3, and Auslander--Buchsbaum gives projective dimension n-3 at nonmaximal primes. Hence Ext_A^{n-2}(R,A) is supported only at the closed point and has finite length. Local duality identifies H^2_{mathfrak m}frakm(R) with its Matlis dual, up to completion.
 
 Let t_z be a local equation of L. Choose j so large that t_z^{j-1} kills H^2_{mathfrak m}frakm(R) for every zin Z. Then the restriction diagram
 [
 [column sep=small]
 0 arrow[r] &
-cO_Y((u-j)L) arrow[r] arrow[d] &
-cO_Y(uL) arrow[r] arrow[d,equal] &
-cO_{jL}(uL) arrow[r] arrow[d] & 0
+O_Y((u-j)L) arrow[r] arrow[d] &
+O_Y(uL) arrow[r] arrow[d,equal] &
+O_{jL}(uL) arrow[r] arrow[d] & 0
 0 arrow[r] &
-cO_Y((u-1)L) arrow[r] &
-cO_Y(uL) arrow[r] &
-cO_L(uL) arrow[r] & 0
+O_Y((u-1)L) arrow[r] &
+O_Y(uL) arrow[r] &
+O_L(uL) arrow[r] & 0
 
 ]
 shows that the connecting map
 [
-H^1_Z(Y,cO_{jL}(uL))→ H^1_Z(Y,cO_L(uL))
+H^1_Z(Y,O_{jL}(uL))→ H^1_Z(Y,O_L(uL))
 ]
 is zero for every integer u.
 
-The kernel of cO_{jL∖ Z}tocO_{L_{red}∖ Z} is nilpotent. Lift a generator of the trivial line cO_Y(L)|_{L_{red}∖ Z} to local generating sections on a finite open cover of jL∖ Z. On overlaps their differences lie in the nilpotent ideal. For u=p^e sufficiently large, their u-th powers agree, since the p^e-power map is additive in characteristic p and kills that ideal. Thus we get a trivializing section
+The kernel of O_{jL∖ Z}toO_{L_{red}∖ Z} is nilpotent. Lift a generator of the trivial line O_Y(L)|_{L_{red}∖ Z} to local generating sections on a finite open cover of jL∖ Z. On overlaps their differences lie in the nilpotent ideal. For u=p^e sufficiently large, their u-th powers agree, since the p^e-power map is additive in characteristic p and kills that ideal. Thus we get a trivializing section
 [
-s_jin H^0(jL∖ Z,cO_{jL}(uL)).
+s_jin H^0(jL∖ Z,O_{jL}(uL)).
 ]
-Project it to s on L∖ Z. Its obstruction to extending over L is killed by the vanishing above, so s extends to H^0(L,cO_L(uL)).
+Project it to s on L∖ Z. Its obstruction to extending over L is killed by the vanishing above, so s extends to H^0(L,O_L(uL)).
 
-This extension is a generator everywhere. On each integral projective surface component T of L_{red} it is nonzero, since it is a generator off Z. If it vanished somewhere on T, its zero scheme would be a nonempty effective Cartier divisor, which has positive degree against an ample class, whereas cO_T(uL) is numerically trivial. This is impossible. Nonvanishing after reduction implies invertibility on L. Hence cO_L(uL)simeqcO_L.
+This extension is a generator everywhere. On each integral projective surface component T of L_{red} it is nonzero, since it is a generator off Z. If it vanished somewhere on T, its zero scheme would be a nonempty effective Cartier divisor, which has positive degree against an ample class, whereas O_T(uL) is numerically trivial. This is impossible. Nonvanishing after reduction implies invertibility on L. Hence O_L(uL)~=O_L.
 
 
-Applying the lemma to Y_d and L proves that cO_L(L) is torsion.
+Applying the lemma to Y_d and L proves that O_L(L) is torsion.
 
 ## The tame cover and the primitive cycle
 Choose a smooth projective common resolution V_0 of Y_d and X, isomorphic over the common open set
@@ -533,32 +533,32 @@ Then D is a nonzero primitive effective Cartier divisor. Its support is connecte
 
 The pullback L_V is numerically trivial on its support. On a neighborhood of C, L_V=aD. Consequently D|_{D_alpha}equiv0. For a curve not contained in Supp D, effectivity gives nonnegative intersection with D; for one contained in a component the degree is zero. Hence D is nef. For every very ample H,
 [
-D^2cdot H=0,
+D^2· H=0,
 
-Dcdot H^2>0.
+D· H^2>0.
 ]
 
-If cO_L(tL)simeqcO_L, its pullback is trivial on the Cartier scheme L_V. Near C, the inclusion Dsubset aD=L_V gives
+If O_L(tL)~=O_L, its pullback is trivial on the Cartier scheme L_V. Near C, the inclusion Dsubset aD=L_V gives
 [
-cO_D(taD)simeqcO_D.
+O_D(taD)~=O_D.
 ]
-Thus cO_D(D) is torsion. Finally, the rational canonical section above has, on a neighborhood of C, divisor supported on D. It gives (eq:prepared-canonical). For any Cartier divisor A,
+Thus O_D(D) is torsion. Finally, the rational canonical section above has, on a neighborhood of C, divisor supported on D. It gives (eq:prepared-canonical). For any Cartier divisor A,
 [
-p^bK_Vcdot Dcdot A
-=B_*cdot Dcdot A
-=sum_alpha b_alpha,c_1(cO_{D_alpha}(D))cdot A|_{D_alpha}=0.
+p^bK_V· D· A
+=B_*· D· A
+=sum_alpha b_alpha,c_1(O_{D_alpha}(D))· A|_{D_alpha}=0.
 ]
-In particular K_V^2cdot D=0.
+In particular K_V^2· D=0.
 
 ## Numerical proportionality and the final interface
 Put rho=rho_0circpi and N_{tot}=rho^*N_0. Both D and N_{tot} are nef and nonzero. The latter is numerically trivial on every component of its support. We have
 [
-D^2cdot H=N_{tot}^2cdot H
-=Dcdot N_{tot}cdot H=0,
+D^2· H=N_{tot}^2· H
+=D· N_{tot}· H=0,
 
-Dcdot H^2>0,
+D· H^2>0,
 
-N_{tot}cdot H^2>0.
+N_{tot}· H^2>0.
 ]
 The surface Hodge index theorem gives a positive rational c such that
 
@@ -566,9 +566,9 @@ Dequiv cN_{tot}.
 
 To check this on every curve, take an integral curve C_0subset V and a large a with an integral surface Tin|aH| containing C_0. On a smooth projective resolution widetilde T of T, the pullbacks of D and N_{tot} are nef, isotropic, mutually orthogonal classes. The pullback of H|_T has positive square. The Hodge index form on N^1(widetilde T)_R has signature (1,dim-1), so two nonzero orthogonal isotropic classes with positive intersection against that class are positively proportional. The ratio is
 [
-c=frac{Dcdot H^2}{N_{tot}cdot H^2}.
+c=frac{D· H^2}{N_{tot}· H^2}.
 ]
-Projection of degrees then gives (D-cN_{tot})cdot C_0=0 for every curve.
+Projection of degrees then gives (D-cN_{tot})· C_0=0 for every curve.
 
 Finally a very general point of V lies in the generically finite locus of rho and maps outside the countable union excluded by the nef reduction on X. A curve through such a point has a curve as its image. If it were D-trivial, (eq:prepared-proportionality) and the projection formula would make its image N_0-trivial, contradiction. This finishes Proposition~prop:prepared-cycle.
 
@@ -589,13 +589,13 @@ The divisor D is nef, and no D-trivial curve passes through a very general point
 ## Slope calculus
 For a torsion-free coherent sheaf E of positive rank, set
 [
-deg(E)=c_1(E)cdot Hcdot D,
+deg(E)=c_1(E)· H· D,
 
 mu(E)=(deg(E))/(rk(E)).
 ]
 Stability and semistability refer to this slope. A sheaf is strongly semistable if all its iterated Frobenius pullbacks are semistable.
 
-The usual Harder--Narasimhan theory works for this nef slope. An effective divisor has nonnegative degree, because H and D are nef. Saturating a subsheaf does not decrease its degree. Every torsion-free sheaf embeds in a finite direct sum of copies of cO_V(aH) for some a: dualize a sufficiently positive global-generation map for E^vee and use E↪ E^{veevee}. If Bsubset E has rank b, taking exterior powers and determinants gives a nonzero map from det B to some copy of cO_V(baH), hence
+The usual Harder--Narasimhan theory works for this nef slope. An effective divisor has nonnegative degree, because H and D are nef. Saturating a subsheaf does not decrease its degree. Every torsion-free sheaf embeds in a finite direct sum of copies of O_V(aH) for some a: dualize a sufficiently positive global-generation map for E^vee and use E↪ E^{veevee}. If Bsubset E has rank b, taking exterior powers and determinants gives a nonzero map from det B to some copy of O_V(baH), hence
 [
 mu(B)le aH^2D.
 ]
@@ -626,7 +626,7 @@ B→ (F_V^*E/B) ⊗ Ω_V^1.
 
 
 Proof. 
-The induced map is cO_V-linear by the Leibniz rule. Suppose it vanishes. Then the generic subspace defined by B is horizontal. Let K=k(V) and choose a pulled-back basis. In a Grassmannian chart, represent the subspace by a matrix (Imid M). Horizontality says every entry of M has zero differential. Since K is separably generated over the perfect field k, the kernel of d:KtoΩ_{K/k}^1 is K^p. Taking p-th roots of the entries of M defines a subspace before Frobenius pullback.
+The induced map is O_V-linear by the Leibniz rule. Suppose it vanishes. Then the generic subspace defined by B is horizontal. Let K=k(V) and choose a pulled-back basis. In a Grassmannian chart, represent the subspace by a matrix (Imid M). Horizontality says every entry of M has zero differential. Since K is separably generated over the perfect field k, the kernel of d:KtoΩ_{K/k}^1 is K^p. Taking p-th roots of the entries of M defines a subspace before Frobenius pullback.
 
 Saturate this subspace to a subsheaf B_0subset E. At every codimension-one point the relevant quotients are free over a discrete valuation ring. Frobenius is flat on the smooth variety, so F_V^*B_0 is saturated and agrees with B there. Consequently
 [
@@ -649,13 +649,13 @@ a>0,
 
 mu_{max}(Q)<a.
 ]
-The rank of A is one or two. Work where A and Q are bundles. The bracket modulo A is an cO_V-linear map
+The rank of A is one or two. Work where A and Q are bundles. The bracket modulo A is an O_V-linear map
 [
 ∧^2A→ Q.
 ]
 It vanishes automatically in rank one. In rank two, Q has rank one and slope -2a, whereas ∧^2A has slope 2a. Equation~(eq:slope-hom-vanishing) makes the map zero.
 
-Once bracket closure holds, the p-th power of derivations modulo A defines an cO_V-linear map
+Once bracket closure holds, the p-th power of derivations modulo A defines an O_V-linear map
 
 F_V^*A→ Q.
 
@@ -704,7 +704,7 @@ Vxrightarrow{pi}Wxrightarrow{rho}V^{(1)}
 ]
 of relative Frobenius. On open sets whose complements have codimension at least two,
 
-omega_Vsimeqpi^*omega_W ⊗ (det A)^{ ⊗ (p-1)}.
+omega_V~=pi^*omega_W ⊗ (det A)^{ ⊗ (p-1)}.
 
 
 
@@ -725,7 +725,7 @@ f = sum_{k=0}^{p-1} x^k, g_k,
 Applying this successively to the commuting u_i gives
 
 R = ⊕_{0le I_1,...,I_d<p} P x^I
-simeq P[X_1,...,X_d]/(X_1^p-c_1,...,X_d^p-c_d),
+~= P[X_1,...,X_d]/(X_1^p-c_1,...,X_d^p-c_d),
  c_i = x_i^p in P.
 
 Uniqueness follows by iterating the highest-derivative extraction. In particular pi is locally free of degree p^d on this open set.
@@ -739,7 +739,7 @@ Remove further the singular locus of W and its inverse image; normality and fini
 
 where the last map is evaluation on A. To verify local freeness of the first image, put N=< dc_1,...,dc_d> inside R ⊗ _PΩ_{P/k}^1. Then
 [
-Ω_{R/k}^1 simeq ((R ⊗ _PΩ_{P/k}^1)/N) ⊕ ⊕_{i=1}^d R,dx_i .
+Ω_{R/k}^1 ~= ((R ⊗ _PΩ_{P/k}^1)/N) ⊕ ⊕_{i=1}^d R,dx_i .
 ]
 Smoothness of R makes the first summand locally free of rank 3-d; smoothness of P then makes N a locally free direct summand of rank d in the rank-three bundle.
 
@@ -747,7 +747,7 @@ The transition maps match those of F_V^*(A^vee). Let x'_j be another such set of
 
 Taking determinants in (eq:foliation-cotangent) gives
 [
-pi^*omega_W simeq (det A)^{-p} ⊗ omega_V ⊗ det A,
+pi^*omega_W ~= (det A)^{-p} ⊗ omega_V ⊗ det A,
 ]
 which is (eq:foliation-canonical).
 
@@ -797,9 +797,9 @@ NGamma le (dim Y+1)(NC)/(-K_YC).
 ]
 The statement is valid in arbitrary characteristic. Choose t so large that (eq:bend-break-ratio) is less than 1/4. Applied on W with N=D', it gives at each closed point of C_t a rational curve Gamma with 0le D'Gamma<1. Since D' is Cartier, this degree is an integer, so D'Gamma=0.
 
-A separate very-general-point argument upgrades this to a D-trivial curve through any prescribed very general yin V. Fix the product of the two very ample linear systems and let mathcal TcalT be the nonempty open locus of smooth integral complete intersections avoiding the bad locus. The parameter space mathcal TcalT is irreducible, its universal curve mathcal CcalCtomathcal TcalT is smooth, and the evaluation mathcal CcalCto W is dominant: the ambient incidence of points and pairs of divisors through them is a product of projective bundles over W, hence irreducible.
+A separate very-general-point argument upgrades this to a D-trivial curve through any prescribed very general yin V. Fix the product of the two very ample linear systems and let T be the nonempty open locus of smooth integral complete intersections avoiding the bad locus. The parameter space T is irreducible, its universal curve C TcalT is smooth, and the evaluation C W is dominant: the ambient incidence of points and pairs of divisors through them is a product of projective bundles over W, hence irreducible.
 
-Let Z_1,Z_2,...subset V be proper closed subsets outside whose union no D-trivial curve passes; take Z_j to be the countable family furnished by the maximal-nef-dimension hypothesis. For each j, the locus in mathcal TcalT of curves not contained in pi(Z_j) is a nonempty open (the image under the open map mathcal CcalCtomathcal TcalT of the complement of ev^{-1}(pi(Z_j))). Over the uncountable field there is a common k-point, whose curve C_t is integral and not contained in any pi(Z_j). Its points outside ∪pi(Z_j) are nonempty.
+Let Z_1,Z_2,...subset V be proper closed subsets outside whose union no D-trivial curve passes; take Z_j to be the countable family furnished by the maximal-nef-dimension hypothesis. For each j, the locus in T of curves not contained in pi(Z_j) is a nonempty open (the image under the open map C TcalT of the complement of ev^{-1}(pi(Z_j))). Over the uncountable field there is a common k-point, whose curve C_t is integral and not contained in any pi(Z_j). Its points outside ∪pi(Z_j) are nonempty.
 
 Choose a closed point xin C_t outside that union, and let yin V be the unique inverse-image point. The rational curve Gamma through x has D'Gamma=0. Its reduced inverse image contains a curve widetildeGammasubset V through y, finite and surjective over Gamma. Then
 [
@@ -811,7 +811,7 @@ contradicting the choice of y. Hence T_V is semistable.
 Strong semistability follows by Frobenius iteration. Suppose ege1 is the first integer for which F_V^{e*}T_V is unstable, with first HN term A' and quotient Q'=F_V^{e*}T_V/A'. Applying Lemma~lem:connection-destabilization to the semistable sheaf F_V^{(e-1)*}T_V gives a nonzero map
 [
 A'→ Q' ⊗ Ω_V^1,
-qquadequivalentlyqquad
+ equivalently 
 A' ⊗ (Q')^vee→Ω_V^1
 ]
 on a big open set. One of the tensor factors has rank one. Semistability of Ω_V^1 (equally, of T_V) and the rank-one twisting rule give
@@ -845,7 +845,7 @@ A projective integral curve in S through x (e.g. a component of a sufficiently a
 [Hom bound]
 If G_1,G_2 are torsion-free semistable sheaves of slope zero, then
 [
-dim_kHom(G_1,G_2)lerk G_1cdotrk G_2.
+dim_kHom(G_1,G_2)lerk G_1·rk G_2.
 ]
 
 
@@ -868,9 +868,9 @@ For each fixed J=G or J=G^{ ⊗ 2}, the increasing spaces H^0(V,J(iD)/J) for i=1
 
 
 Proof. 
-Apply Lemma~lem:semistable-hom-bound to cO_V, G(iD) and to G^vee, G(iD), using
+Apply Lemma~lem:semistable-hom-bound to O_V, G(iD) and to G^vee, G(iD), using
 [
-H^0(G(iD))=Hom(cO_V, G(iD)),
+H^0(G(iD))=Hom(O_V, G(iD)),
 
 H^0(G^{ ⊗ 2}(iD))=Hom(G^vee, G(iD)).
 ]
@@ -886,21 +886,21 @@ More explicitly, choose i_0 after stabilization. Every global polar part, being
 
 
 [No scalar polar parts]
-For every i>0, H^0(V,cO_V(iD)/cO_V)=0.
+For every i>0, H^0(V,O_V(iD)/O_V)=0.
 
 
 Proof. 
-Apply Proposition~prop:polar-bounds to G=cO_V. A nonzero polar part eta has genuine negative valuation at some component of D: if all component valuations were nonnegative, normality and the absence of poles away from D would make the local representatives regular, causing the quotient section to vanish.
+Apply Proposition~prop:polar-bounds to G=O_V. A nonzero polar part eta has genuine negative valuation at some component of D: if all component valuations were nonnegative, normality and the absence of poles away from D would make the local representatives regular, causing the quotient section to vanish.
 
-For every ege0, local representatives f^{p^e} define a global polar part eta^{p^e}in H^0(V,cO_V(p^eiD)/cO_V): well defined because two representatives differing by a regular function have p^e-th powers differing by a regular function. At the chosen component the negative valuation is multiplied by p^e, so these pole orders are unbounded. This contradicts the uniform bound of Proposition~prop:polar-bounds.
+For every ege0, local representatives f^{p^e} define a global polar part eta^{p^e}in H^0(V,O_V(p^eiD)/O_V): well defined because two representatives differing by a regular function have p^e-th powers differing by a regular function. At the chosen component the negative valuation is multiplied by p^e, so these pole orders are unbounded. This contradicts the uniform bound of Proposition~prop:polar-bounds.
 
 
 # Jets and the three rows
 
 
-We retain the prepared data of Proposition~prop:prepared-cycle: V is a smooth projective threefold, D=sum_alpha m_alpha D_alpha is an effective Cartier divisor with connected support, gcd_alpha m_alpha=1, and D|_{D_alpha} is numerically trivial for every alpha. In particular D^2A=0 for every divisor A. The line bundle cO_D(D) is torsion. Near D there is an isomorphism
+We retain the prepared data of Proposition~prop:prepared-cycle: V is a smooth projective threefold, D=sum_alpha m_alpha D_alpha is an effective Cartier divisor with connected support, gcd_alpha m_alpha=1, and D|_{D_alpha} is numerically trivial for every alpha. In particular D^2A=0 for every divisor A. The line bundle O_D(D) is torsion. Near D there is an isomorphism
 
-omega_V^{ ⊗  p^b}simeqcO_V(B_*),
+omega_V^{ ⊗  p^b}~=O_V(B_*),
 
 B_*=sum_alpha b_alpha D_alpha,
 
@@ -908,13 +908,13 @@ b_alphainZ,
 
 and K_V^2D=0. We use c_2(V)Dge0 from Proposition~prop:strong-semistability and the absence of scalar polar parts from Corollary~cor:no-scalar-poles.
 
-The argument computes finite jets and their ranks. Its two outputs are (a) a trivialization of the canonical line on the completion up to an integral divisor supported on D, and (b) the equality c_2(V)D=0 together with the limit p/(p+1) for the normalized last trivialization widths. The width at period rp^e is the largest integer s for which cO(rp^eD) is trivial on sD. The only passage to a formal limit is an explicit inverse-limit argument for finite-dimensional spaces of sections.
+The argument computes finite jets and their ranks. Its two outputs are (a) a trivialization of the canonical line on the completion up to an integral divisor supported on D, and (b) the equality c_2(V)D=0 together with the limit p/(p+1) for the normalized last trivialization widths. The width at period rp^e is the largest integer s for which O(rp^eD) is trivial on sD. The only passage to a formal limit is an explicit inverse-limit argument for finite-dimensional spaces of sections.
 
 ## Finite pages and duality
 
-Write widehatcO_V for the completion along D, regarded as a sheaf on |D|, and set
+Write widehatO_V for the completion along D, regarded as a sheaf on |D|, and set
 [
-cR=widehatcO_V(*D).
+R=widehatO_V(*D).
 ]
 Choose a finite cover by the restrictions to |D| of affine open subsets of V, refined so that D is principal and the bundles under consideration are free on each member. All intersections of these affine opens are affine, since V is separated. For a vector bundle E, let C_i^bullet(E) be the resulting v{C}ech complex with coefficients in the completion of E(iD). These complexes give an increasing filtration, indexed by iinZ, on the Laurent v{C}ech complex.
 
@@ -943,7 +943,7 @@ g_s^a(i;E)=dim_k E_s^{i,a}(E).
 Here E_s^{i,a}(E) is a page space, not a new bundle.
 
 [Finite-page calculus]
-The operation xmapsto∂ x induces maps
+The operation x|->∂ x induces maps
 [
 d_s:E_s^{i,a}(E)→ E_s^{i-s,a+1}(E)
 ]
@@ -957,7 +957,7 @@ g_{s+1}^1(i;E)
 &=g_s^1(i;E)-delta_s^0(i+s)-delta_s^2(i-s).
 
 
-If M>0 and cO(MD)|_{sD} is trivial, the page-s ranks are periodic in i modulo M, and
+If M>0 and O(MD)|_{sD} is trivial, the page-s ranks are periodic in i modulo M, and
 
 sum_{ibmod M}(g_s^0(i;E)-g_s^1(i;E)+g_s^2(i;E))
 =sum_{ibmod M}chi(P_i^1(E)).
@@ -987,7 +987,7 @@ There is no outgoing differential in degree two. The kernel calculation above an
 
 The two-term locally free presentation of P_j^s(E) gives
 [
-mathcal EcalE xt^1(P_j^s(E),omega_V)=(E^!((s-j)D))/(E^!(-jD))=P_{s-j}^s(E^!),
+E xt^1(P_j^s(E),omega_V)=(E^!((s-j)D))/(E^!(-jD))=P_{s-j}^s(E^!),
 ]
 and all other sheaf Ext groups vanish. Serre duality on the smooth projective threefold consequently identifies H^2(P_j^s(E))^vee with H^0(P_{s-j}^s(E^!)). Under this identification, the dual of the displayed inclusion is
 [
@@ -995,7 +995,7 @@ H^0(P_{1-i}^s(E^!))→ H^0(P_{1-i}^1(E^!)),
 ]
 and the map is the projection. This proves (eq:top-endpoint). The adjacent differentials to the middle term have ranks delta_s^0(i+s) and delta_s^2(i-s), giving (eq:middle-recurrence).
 
-Triviality of cO(MD) on sD identifies each quotient of width tle s with its translate by M, compatibly with layer maps. Equations (eq:bottom-endpoint) and (eq:top-endpoint) therefore give periodicity of both outer rows at every such width; page one is periodic in all degrees; (eq:middle-recurrence) gives periodicity of the middle row by induction.
+Triviality of O(MD) on sD identifies each quotient of width tle s with its translate by M, compatibly with layer maps. Equations (eq:bottom-endpoint) and (eq:top-endpoint) therefore give periodicity of both outer rows at every such width; page one is periodic in all degrees; (eq:middle-recurrence) gives periodicity of the middle row by induction.
 
 For (eq:periodic-euler), let e_t(i)=g_t^0(i)-g_t^1(i)+g_t^2(i). The recurrence yields
 [
@@ -1009,35 +1009,35 @@ For the last assertion, set A_s=H^0(P_i^s(E)) and B_s=∩_{tge s}im(A_tto A_s).
 ## Scalar jets and their last widths
 
 [The scalar jet rule]
-Let u<i be integers and xiin H^0(cO(iD)/cO(uD)) nonzero. There is an integer t with u<tle i such that local lifts of xi have pole order exactly tm_alpha at every D_alpha. Those lifts trivialize cO(tD) on (t-u)D. In particular,
+Let u<i be integers and xiin H^0(O(iD)/O(uD)) nonzero. There is an integer t with u<tle i such that local lifts of xi have pole order exactly tm_alpha at every D_alpha. Those lifts trivialize O(tD) on (t-u)D. In particular,
 
-g_s^0(i;cO_V)=
+g_s^0(i;O_V)=
 
-1,&cO(iD)|_{sD}simeqcO_{sD},
+1,&O(iD)|_{sD}~=O_{sD},
 0,&otherwise.
 
 
 
 
 Proof. 
-Choose local algebraic lifts f_lambda of xi on a finite affine cover. Membership in cO(uD) is checked by valuations at the prime divisors of the smooth, hence normal, V. Some component therefore detects the nonzero quotient: the maximum over components of the ratios -ord_{D_alpha}(f_lambda)/m_alpha that exceed u is a well-defined rational number tin(u,i]. Changing a lift adds a section of cO(uD) and cannot alter such a valuation.
+Choose local algebraic lifts f_lambda of xi on a finite affine cover. Membership in O(uD) is checked by valuations at the prime divisors of the smooth, hence normal, V. Some component therefore detects the nonzero quotient: the maximum over components of the ratios -ord_{D_alpha}(f_lambda)/m_alpha that exceed u is a well-defined rational number tin(u,i]. Changing a lift adds a section of O(uD) and cannot alter such a valuation.
 
-Choose a positive integer n with ntinZ. Then f_lambda^nincO(ntD), and these sections agree on D_{red}: the difference f_lambda-f_mu lies in cO(uD), and in the factorization of f_lambda^n-f_mu^n each term has one factor of pole ratio at most u and n-1 factors of pole ratio at most t. After normalization in cO(ntD) its valuation at every D_alpha is at least (t-u)m_alpha>0. They define a section of cO(ntD)|_{D_{red}}, nonzero on a component where the maximum is attained.
+Choose a positive integer n with ntinZ. Then f_lambda^ninO(ntD), and these sections agree on D_{red}: the difference f_lambda-f_mu lies in O(uD), and in the factorization of f_lambda^n-f_mu^n each term has one factor of pole ratio at most u and n-1 factors of pole ratio at most t. After normalization in O(ntD) its valuation at every D_alpha is at least (t-u)m_alpha>0. They define a section of O(ntD)|_{D_{red}}, nonzero on a component where the maximum is attained.
 
 The restriction of this line bundle to every integral component is numerically trivial. A nonzero section of a numerically trivial line bundle on a projective integral surface has no zeros: a nonempty effective Cartier zero divisor has positive intersection with an ample divisor. Hence the section is nowhere zero on that component. At an intersection with another component its common value is nonzero; connectedness of |D| propagates this to all components.
 
 It follows that ord_{D_alpha}(f_lambda)=-tm_alpha for every alpha. In particular every tm_alpha is an integer. Primitivity of the cycle (gcd_alpha m_alpha=1) gives tinZ.
 
-The original lifts are sections of the honest line bundle cO(tD), units in it along D, agreeing modulo cO(uD). Since cO(tD)/cO(uD)=cO(tD)|_{(t-u)D}, they give the asserted trivialization.
+The original lifts are sections of the honest line bundle O(tD), units in it along D, agreeing modulo O(uD). Since O(tD)/O(uD)=O(tD)|_{(t-u)D}, they give the asserted trivialization.
 
 A jet of width s at upper weight i has nonzero leading image exactly when the uniform ratio is t=i, giving existence in (eq:scalar-endpoint) in both directions. For dimension one, take two such jets; their normalized unit sections on D_{red} are proportional (a connected reduced projective scheme over k has only constant global functions). Subtracting the appropriate multiple leaves a leading image that would give a unit on D_{red} with ratio i, contradicting the cancellation.
 
 
-Let r be the order of cO_D(D). For ege0 put
+Let r be the order of O_D(D). For ege0 put
 
 M_e=rp^e,
 
-d_e=max{sge1:cO(M_eD)|_{sD}simeqcO_{sD}}.
+d_e=max{sge1:O(M_eD)|_{sD}~=O_{sD}}.
 
 These maxima exist and satisfy
 
@@ -1047,7 +1047,7 @@ d_{e+1}ge p d_e,
 
 (d_e)/(M_e)↗ a_*in(0,1].
 
-A trivialization on M_eD would give, by Lemma~lem:scalar-jets, a nonzero section of cO(M_eD)/cO, contrary to Corollary~cor:no-scalar-poles. Triviality at width one follows from the definition of r. Taking pth powers of local trivializing sections multiplies the gluing precision by p and gives d_{e+1}ge pd_e. Division by M_{e+1}=pM_e proves monotonicity and the positive limit. If r=1, the inequality 1le d_0<M_0=1 is a contradiction; hence r>1.
+A trivialization on M_eD would give, by Lemma~lem:scalar-jets, a nonzero section of O(M_eD)/cO, contrary to Corollary~cor:no-scalar-poles. Triviality at width one follows from the definition of r. Taking pth powers of local trivializing sections multiplies the gluing precision by p and gives d_{e+1}ge pd_e. Division by M_{e+1}=pM_e proves monotonicity and the positive limit. If r=1, the inequality 1le d_0<M_0=1 is a contradiction; hence r>1.
 
 We will repeatedly use the following consequence of nilpotence. If a line bundle L on a finite thickening T of D is trivial on D, then its class has p-power order. Indeed its transition functions can be chosen in 1+J, where J is the nilpotent ideal of D in T. For p^N at least a nilpotence exponent, (1+x)^{p^N}=1 for xin J, so L^{ ⊗  p^N} is trivial. Multiplication by an integer prime to p is injective on such classes by B'ezout and surjective on their cyclic subgroups. It follows that
 
@@ -1058,24 +1058,24 @@ This includes negative j. If rnmid i, no scalar leading jet occurs even at width
 
 Finally, Hirzebruch--Riemann--Roch on the smooth threefold gives
 
-chi(P_i^1(cO_V))
-=chi(cO(iD))-chi(cO((i-1)D))
+chi(P_i^1(O_V))
+=chi(O(iD))-chi(O((i-1)D))
 =(Dc_2(V))/(12)=:gammage0.
 
 The terms involving D^3, D^2K_V, and DK_V^2 vanish, leaving the displayed linear term in the threefold Riemann--Roch polynomial.
 
 ## Canonical jets and a formal canonical root
 
-For E=cO_V, (eq:top-endpoint) computes the top-row rank at weight i from leading jets of omega_V at weight 1-i. We need to describe that row as explicitly as the scalar bottom row. First, canonical jets must survive to arbitrarily large widths.
+For E=O_V, (eq:top-endpoint) computes the top-row rank at weight i from leading jets of omega_V at weight 1-i. We need to describe that row as explicitly as the scalar bottom row. First, canonical jets must survive to arbitrarily large widths.
 
 
 There are nonzero leading jets of omega_V of arbitrarily large width, with upper weight allowed to vary.
 
 
 Proof. 
-Suppose instead that g_s^0(l;omega_V)=0 for every l and every sge S, for some S. By (eq:top-endpoint), the top row for E=cO_V then vanishes at all these widths. Take e large enough that s_e=d_{e-1}+1ge S. At page s_e the bottom row is nonzero precisely at multiples of M_e (by (eq:scalar-last-width) and s_ele d_e). This page is periodic modulo M_e. Equations (eq:periodic-euler) and (eq:scalar-euler) give
+Suppose instead that g_s^0(l;omega_V)=0 for every l and every sge S, for some S. By (eq:top-endpoint), the top row for E=O_V then vanishes at all these widths. Take e large enough that s_e=d_{e-1}+1ge S. At page s_e the bottom row is nonzero precisely at multiples of M_e (by (eq:scalar-last-width) and s_ele d_e). This page is periodic modulo M_e. Equations (eq:periodic-euler) and (eq:scalar-euler) give
 [
-sum_{ibmod M_e}g_{s_e}^1(i;cO_V)
+sum_{ibmod M_e}g_{s_e}^1(i;O_V)
 =1-M_egammale1.
 ]
 At differential length d_e, the bottom entries at jM_e with pnmid j each drop by one. By (eq:middle-recurrence) these drops consume one middle-row dimension at -d_e+jM_e. Middle-row dimensions can only decrease as the page increases. Periodicity and the preceding upper bound therefore force the initial middle row to consist exactly of one entry at -d_epmod{M_e}.
@@ -1086,7 +1086,7 @@ Between pages s_e and s_{e+1}=d_e+1 these are the only drops: there is no top ro
 [The canonical formal root]
 Every coefficient of B_* in (eq:canonical-p-power) is divisible by p^b. Writing B_*=p^bB_0 with B_0 integral, there is an isomorphism on the completion along D,
 
-widehatomega_VsimeqwidehatcO_V(B_0).
+widehatomega_V~=widehatO_V(B_0).
 
 
 
@@ -1099,7 +1099,7 @@ B=P(l-s)D+B_*.
 ]
 The lower pole bound is multiplied by P exactly because P is a power of the characteristic. Set beta_alpha=b_alpha/m_alpha and choose an integer c_*gemax_alphabeta_alpha. Enlarging both bounds gives a section of
 [
-(cO((Pl+c_*)D))/(cO((P(l-s)+c_*)D)),
+(O((Pl+c_*)D))/(O((P(l-s)+c_*)D)),
 ]
 nonzero for all sufficiently large s, uniformly in l. Indeed, at a component detecting the original leading layer, a local coefficient f in an ordinary frame of omega_V has -ord_{D_alpha}(f)/m_alpha>l-1. Its powered scalar coefficient has pole ratio
 [
@@ -1112,11 +1112,11 @@ Lemma~lem:scalar-jets shows that all these powered coefficients have one integra
 
 |t-Pl|le C,
 
-cO(tD) is trivial to a width at least Ps-C.
+O(tD) is trivial to a width at least Ps-C.
 
 For each fixed J, widths greater than d_{J-1} can occur only at multiples of M_J (zero is allowed as a multiple). Since the widths in (eq:canonical-ratio-bound) tend to infinity, we can assign integers Jto∞ along this sequence such that M_Jmid t.
 
-Fix a component D_alpha. If f is the original local coefficient in a frame of omega_V, the local generator of cO(B_*) has order -b_alpha, so
+Fix a component D_alpha. If f is the original local coefficient in a frame of omega_V, the local generator of O(B_*) has order -b_alpha, so
 
 b_alpha-tm_alpha=Pord_{D_alpha}(f).
 
@@ -1126,11 +1126,11 @@ For the same sequence put h_0=t/P. The order identity shows that the original lo
 [
 L_{h_0}=omega_V(-B_0+h_0D)
 ]
-near D: their coefficient orders are exactly those required. They have no additional poles near D, since the original representatives were sections of omega_V(lD) and membership in these divisor lattices is checked on the normal variety in codimension one. Moreover L_{h_0}^{ ⊗  P}simeqcO(tD), and the Pth powers of these sections are units along all of D by Lemma~lem:scalar-jets. The original sections are therefore themselves units along D. They agree modulo the original lower lattice omega_V((l-s)D), and so trivialize L_{h_0} on the effective thickness
+near D: their coefficient orders are exactly those required. They have no additional poles near D, since the original representatives were sections of omega_V(lD) and membership in these divisor lattices is checked on the normal variety in codimension one. Moreover L_{h_0}^{ ⊗  P}~=O(tD), and the Pth powers of these sections are units along all of D by Lemma~lem:scalar-jets. The original sections are therefore themselves units along D. They agree modulo the original lower lattice omega_V((l-s)D), and so trivialize L_{h_0} on the effective thickness
 [
 T=(h_0-l+s)D-B_0.
 ]
-By (eq:canonical-ratio-bound), |h_0-l| is bounded independently of the sequence. Thus T contains nD for integers nto∞. Also M_Jmid t implies M_{J-b}mid h_0 once Jge b. The line bundle cO(h_0D) is therefore trivial on d_{J-b}D, and d_{J-b}to∞ by (eq:width-bounds). Removing this twist shows that L=omega_V(-B_0) is trivial on arbitrarily large finite thickenings nD.
+By (eq:canonical-ratio-bound), |h_0-l| is bounded independently of the sequence. Thus T contains nD for integers nto∞. Also M_Jmid t implies M_{J-b}mid h_0 once Jge b. The line bundle O(h_0D) is therefore trivial on d_{J-b}D, and d_{J-b}to∞ by (eq:width-bounds). Removing this twist shows that L=omega_V(-B_0) is trivial on arbitrarily large finite thickenings nD.
 
 Finally, choose the trivializations compatibly. Apply the stabilized-image argument in the proof of Lemma~lem:page-calculus to the spaces H^0(nD,L|_{nD}), additionally imposing restriction to D_{red}. The latter line bundle is trivial and its section space is one dimensional because D_{red} is connected, reduced, and projective. The image from every sufficiently large thickening contains a unit, hence is that full one-dimensional space. The surjective maps between stabilized images therefore lift a chosen nonzero section on D_{red} compatibly to all thickenings. Each lift is a unit (as may be checked after reduction), and their inverse limit trivializes widehatwidehat L.L. This gives (eq:canonical-formal-root).
 
@@ -1139,11 +1139,11 @@ Finally, choose the trivializations compatibly. Apply the stabilized-image argum
 
 Write B_0=sum_alpha b_{0,alpha}D_alpha, and define
 
-h=-Bigllfloormin_alphafrac{b_{0,alpha}}{m_alpha}Bigrrfloor,
+h=-Biglfloor(min_alphafrac{b_{0,alpha}}{m_alpha}Bigr),
 
 q=1-h.
 
-The formal isomorphism of Proposition~prop:canonical-formal-root identifies the finite canonical jet lattices with scalar lattices having upper and lower bounds lD+B_0 and (l-s)D+B_0. All these finite quotients are algebraic sheaves supported on finite thickenings of D: the localized widehatcO_V-linear isomorphism respects every nested divisor lattice, and each width-s quotient is killed by cI_D^s. Since widehatcO_V/cI_D^swidehatcO_V=cO_{sD}, the induced map is cO_{sD}-linear and compatible with the leading-layer projections.
+The formal isomorphism of Proposition~prop:canonical-formal-root identifies the finite canonical jet lattices with scalar lattices having upper and lower bounds lD+B_0 and (l-s)D+B_0. All these finite quotients are algebraic sheaves supported on finite thickenings of D: the localized widehatO_V-linear isomorphism respects every nested divisor lattice, and each width-s quotient is killed by cI_D^s. Since widehatO_V/cI_D^swidehatO_V=O_{sD}, the induced map is O_{sD}-linear and compatible with the leading-layer projections.
 
 The exact canonical counterpart of the scalar rule is: for all sufficiently large s, a canonical jet at weight l with nonzero leading image has uniform integral scalar pole ratio
 
@@ -1153,17 +1153,17 @@ To prove this, enlarge the two bounds to uniform multiples of D as in the preced
 [
 l-1+beta_0<tle l+beta_0.
 ]
-Its unique integral solution is t=l+lfloorbeta_0rfloor=l-h.
+Its unique integral solution is t=l+floor(beta_0)=l-h.
 
-The normalized unit sections of cO(tD) glue on the irregular thickness
+The normalized unit sections of O(tD) glue on the irregular thickness
 
 T_s=(s-h)D-B_0,
 
-because this is the difference between tD and the lower bound (l-s)D+B_0. Conversely a trivialization of cO(tD) on T_s, with t=l-h, includes in the canonical quotient and has nonzero leading image. The leading image has dimension one: normalized unit sections are proportional on D_{red}, and their difference cannot retain the same leading ratio after reduction cancels. We have proved
+because this is the difference between tD and the lower bound (l-s)D+B_0. Conversely a trivialization of O(tD) on T_s, with t=l-h, includes in the canonical quotient and has nonzero leading image. The leading image has dimension one: normalized unit sections are proportional on D_{red}, and their difference cannot retain the same leading ratio after reduction cancels. We have proved
 
 g_s^0(l;omega_V)=
 
-1,&cO((l-h)D)|_{T_s}simeqcO_{T_s},
+1,&O((l-h)D)|_{T_s}~=O_{T_s},
 0,&otherwise,
 
 (s sufficiently large).
@@ -1177,7 +1177,7 @@ Then, coefficient by coefficient,
 
 (s-C_0)Dle T_sle sD.
 
-For e sufficiently large, let d'_e be the last width s for which cO(M_eD)|_{T_s} is trivial. The thicknesses are nested as s increases. Equation~(eq:thickness-comparison) and the definition of d_e give
+For e sufficiently large, let d'_e be the last width s for which O(M_eD)|_{T_s} is trivial. The thicknesses are nested as s increases. Equation~(eq:thickness-comparison) and the definition of d_e give
 
 d_ele d'_ele d_e+C_0.
 
@@ -1199,7 +1199,7 @@ For all sufficiently large e set
 s_e=max(d_{e-1},d'_{e-1})+1.
 ]
 Equations (eq:width-bounds) and (eq:canonical-width-comparison) make these strictly increasing, with
-s_elemin(d_e,d'_e). Work with the three rows for E=cO_V. At page s_e all three rows are periodic modulo M_e, and each outer row has a single entry per period. The bottom entry is at 0, by (eq:scalar-last-width). For the top entry, (eq:top-endpoint) and (eq:canonical-scalar-ratio) associate to a top weight i the scalar ratio t=1-i-h=q-i. Its entry is therefore at qpmod{M_e}.
+s_elemin(d_e,d'_e). Work with the three rows for E=O_V. At page s_e all three rows are periodic modulo M_e, and each outer row has a single entry per period. The bottom entry is at 0, by (eq:scalar-last-width). For the top entry, (eq:top-endpoint) and (eq:canonical-scalar-ratio) associate to a top weight i the scalar ratio t=1-i-h=q-i. Its entry is therefore at qpmod{M_e}.
 
 As the page runs from s_e through s_{e+1}, the only outer drops are those with last-width index e. The bottom drops, of length d_e, consume middle-row dimensions at
 [
@@ -1261,7 +1261,7 @@ M_e=rp^e,
 
 c_2(V)D=0,
 
-with cO_V(M_eD) trivial through width d_e. Choose real numbers
+with O_V(M_eD) trivial through width d_e. Choose real numbers
 
 1-frac2p<c_0<c<2a_*-1.
 
@@ -1269,23 +1269,23 @@ Such choices exist since
 [
 2a_*-1-(1-frac2p)=(2)/(p(p+1))>0.
 ]
-As pge5, we also have 1/2<c_0<c<a_*. Put u_e=lfloor cM_erfloor. The lower bound on c_0 permits using only first and second tensor powers in Lemma~lem:phase-bound; the upper bound on c leaves the precision required for matching jets in Theorem~thm:laurent-vanishing.
+As pge5, we also have 1/2<c_0<c<a_*. Put u_e=floor( cM_e). The lower bound on c_0 permits using only first and second tensor powers in Lemma~lem:phase-bound; the upper bound on c leaves the precision required for matching jets in Theorem~thm:laurent-vanishing.
 
 ## Concentration of leading jets
 
 The first step restricts the positions at which a long jet of a semistable bundle can have a nonzero leading term.
 
 
-Let G be a vector bundle that is semistable of slope zero for the degree c_1(-)cdot Hcdot D. There is a constant A_G such that, for every sufficiently large e,
+Let G be a vector bundle that is semistable of slope zero for the degree c_1(-)· H· D. There is a constant A_G such that, for every sufficiently large e,
 [
-g^0_{lfloor c_0M_erfloor}(i;G)>0
+g^0_{floor( c_0M_e)}(i;G)>0
 ⇒
 dist(i,M_eZ)le A_G.
 ]
 
 
 Proof. 
-Write s=lfloor c_0M_erfloor. For large e we have sle d_e. A unit of cO(M_eD)|_{sD} translates the upper weight of a jet by M_e. We can therefore replace i by a centered representative v, with |v|le M_e/2, without changing the existence of its leading class.
+Write s=floor( c_0M_e). For large e we have sle d_e. A unit of O(M_eD)|_{sD} translates the upper weight of a jet by M_e. We can therefore replace i by a centered representative v, with |v|le M_e/2, without changing the existence of its leading class.
 
 Suppose no uniform bound on |v| exists. Passing to a sequence, we may assume that vto+∞ or vto-∞.
 
@@ -1307,18 +1307,18 @@ Here jto∞ as |v|to∞. After a subsequence the ratios tend to a number yin[lam
 
 Indeed, if y>1-c_0, take m=1; then yle plambda<1. Otherwise take m=2; its lower bound follows from 2yge2lambda>1-c_0, and its upper bound from 2yle2(1-c_0)<4/p<1.
 
-Restrict the original jet to width lfloor c_0M_jrfloor, take its mth tensor power, and multiply by a unit section of cO(M_jD) to that width. Tensoring keeps at least the same width: in a difference of two tensor products, every summand contains a factor from the lower lattice. The resulting jet is a jet of G^{ ⊗  m} at upper weight
+Restrict the original jet to width floor( c_0M_j), take its mth tensor power, and multiply by a unit section of O(M_jD) to that width. Tensoring keeps at least the same width: in a difference of two tensor products, every summand contains a factor from the lower lattice. The resulting jet is a jet of G^{ ⊗  m} at upper weight
 [
 w=M_j+mv.
 ]
-By (eq:phase-tensor-choice), wto+∞ and w<lfloor c_0M_jrfloor with a margin tending to infinity. It therefore gives a global polar part of G^{ ⊗  m}.
+By (eq:phase-tensor-choice), wto+∞ and w<floor( c_0M_j) with a margin tending to infinity. It therefore gives a global polar part of G^{ ⊗  m}.
 
 This polar part has unbounded actual order. At some component of D an original coefficient has pole ratio greater than v-1. The corresponding pure tensor coefficient, after multiplication by the unit, has pole ratio greater than M_j+m(v-1). This tends to infinity. Proposition~prop:polar-bounds applies to both G and G ⊗  G, giving the required contradiction.
 
 
 ## A vanishing criterion
 
-We use widehatwidehat EE for the completion of a bundle E along D, and E_{cR}=widehatwidehat EE ⊗ _{widehatcO}cR. The finite-cover, bounded-pole interpretation is the one used above.
+We use widehatwidehat EE for the completion of a bundle E along D, and E_{R}=widehatwidehat EE ⊗ _{widehatcO}R. The finite-cover, bounded-pole interpretation is the one used above.
 
 
 For the prepared data and periods fixed above, let E be a vector bundle on V satisfying:
@@ -1332,7 +1332,7 @@ dist(i,M_eZ)le A_J
 ]
 for all sufficiently large e.
 
-Then global sections of E_{cR} form a finite-dimensional k-vector space. Every Laurent v Cech 1-cocycle on a finite affine cover adapted to the bundles and the divisor has an actual Laurent primitive. The same assertion holds after refining such a cover.
+Then global sections of E_{R} form a finite-dimensional k-vector space. Every Laurent v Cech 1-cocycle on a finite affine cover adapted to the bundles and the divisor has an actual Laurent primitive. The same assertion holds after refining such a cover.
 
 
 Proof. 
@@ -1352,11 +1352,11 @@ d_ele sle M_e+C
 
 for a fixed C, with multiplicities at each length independent of j among integers prime to p.
 
-No drop occurs before length d_e: translation by a unit of cO(M_eD)|_{d_eD} identifies leading images with those at the unshifted stable position v. Choose local formal unit lifts t_lambda of this trivializing section of cO(M_eD)|_{d_eD}, and write on overlaps
+No drop occurs before length d_e: translation by a unit of O(M_eD)|_{d_eD} identifies leading images with those at the unshifted stable position v. Choose local formal unit lifts t_lambda of this trivializing section of O(M_eD)|_{d_eD}, and write on overlaps
 [
 (t_lambda)/(t_mu)=1+b_{lambdamu},
- b_{lambdamu}incI_D^{d_e}widehatcO_V,
- cI_D=cO_V(-D).
+ b_{lambdamu}incI_D^{d_e}widehatO_V,
+ cI_D=O_V(-D).
 ]
 Fix a constant C, and pick d_e<kle M_e+C. For large e, k<2d_e by (eq:laurent-width-input). Normalize local representatives sigma_lambda of a width-k jet at v+jM_e by putting alpha_lambda=sigma_lambda/t_lambda^j. These are local sections of E(vD)hat and satisfy
 
@@ -1409,7 +1409,7 @@ H_e(i)ge#{tin T_e:tequiv ipmod{M_e}}.
 ]
 Summing over a period gives equality of the two sides by the Euler identity of Lemma~lem:page-calculus and property (i). The inequality is therefore equality at every residue. Equation~(eq:middle-distribution) now says H_{e+1} is exactly the distribution of T_e modulo pM_e. For each fixed integer i, the bands (eq:drop-bands) and d_eto∞ imply
 
-g^1_{u_e}(i;E)=0qquadfor all sufficiently large e.
+g^1_{u_e}(i;E)=0 for all sufficiently large e.
 
 
 smallskipnoindentConvergence of primitives.
@@ -1447,7 +1447,7 @@ Write F for absolute Frobenius and Ω=Ω_V^1. Let B^1,Z^1subset F_*Ω be the she
 
 On a smooth affine chart with 'etale coordinates x_1,x_2,x_3 and a local equation f for D, put R=widehatwidehat A[1/f].A[1/f]. The monomials x^I=x_1^{I_1}x_2^{I_2}x_3^{I_3} with 0le I_j<p form a basis of R over R^p. Ordinary differentials and the Cartier isomorphism extend to this ring. The algebraic bundles B^1,Z^1 and their defining exact sequences extend to the corresponding Laurent exact and closed forms, with their Frobenius module structure. In particular,
 
-0→cO_V→ F_*cO_V
+0→O_V→ F_*O_V
 → B^1→0,
 
 0→ B^1→ Z^1
@@ -1455,7 +1455,7 @@ xrightarrow{C}Ω→0,
 
 are sequences of vector bundles and remain exact on Laurent charts. The images and kernels in the Frobenius de Rham complex are locally free, as are the relevant quotients. Moreover,
 
-(F_*cO_V)^vee ⊗ omega_Vsimeq F_*omega_V.
+(F_*O_V)^vee ⊗ omega_V~= F_*omega_V.
 
 
 
@@ -1491,9 +1491,9 @@ Every Laurent v Cech 1-cocycle with values in Ω, B^1, or Z^1 has a Laurent prim
 
 
 Proof. 
-For E=Ω, both E and E^!=T_V ⊗ omega_V are semistable of slope zero by Proposition~prop:strong-semistability and the canonical intersection identities. Proposition~prop:polar-bounds and Lemma~lem:phase-bound give conditions (ii) and (iii) of Theorem~thm:laurent-vanishing; passing from width lfloor c_0M_erfloor to u_e can only decrease the leading images. Hirzebruch--Riemann--Roch gives condition (i): the degree-two terms in ch(Ω)td(T_V) are linear combinations of K_V^2 and c_2(V), both killed by intersection with D, and every term involving D^2 is also zero.
+For E=Ω, both E and E^!=T_V ⊗ omega_V are semistable of slope zero by Proposition~prop:strong-semistability and the canonical intersection identities. Proposition~prop:polar-bounds and Lemma~lem:phase-bound give conditions (ii) and (iii) of Theorem~thm:laurent-vanishing; passing from width floor( c_0M_e) to u_e can only decrease the leading images. Hirzebruch--Riemann--Roch gives condition (i): the degree-two terms in ch(Ω)td(T_V) are linear combinations of K_V^2 and c_2(V), both killed by intersection with D, and every term involving D^2 is also zero.
 
-For E=B^1, the first sequence in (eq:cartier-sequences) gives condition (i). By projection formula, a single cell for F_*cO_V has the Euler characteristic of p successive scalar cells, each zero by Proposition~prop:scalar-widths. Frobenius changes the k-module action but not these finite dimensions.
+For E=B^1, the first sequence in (eq:cartier-sequences) gives condition (i). By projection formula, a single cell for F_*O_V has the Euler characteristic of p successive scalar cells, each zero by Proposition~prop:scalar-widths. Frobenius changes the k-module action but not these finite dimensions.
 
 There are subbundle inclusions with locally free quotients
 
@@ -1505,7 +1505,7 @@ The first follows from Lemma~lem:laurent-cartier-algebra; the second follows by
 
 The same subbundle property gives condition (iii), with a small precision margin. A width-u_e jet of Esubset F_*G with nonzero leading term maps to a width-pu_e jet of G at upper weight pi, with nonzero leading term in the top p ordinary layers. Choose its first nonzero ordinary layer, at a weight a with |a-pi|<p; it has width at least pu_e-p. Since c>c_0 and M_{e+1}=pM_e, for large e
 [
-pu_e-pgelfloor c_0M_{e+1}rfloor.
+pu_e-pgefloor( c_0M_{e+1}).
 ]
 Lemma~lem:phase-bound at period M_{e+1} implies that a, and hence pi, is at bounded distance from M_{e+1}Z. Dividing by p gives the required bounded distance from M_eZ.
 
@@ -1517,14 +1517,14 @@ Theorem~thm:laurent-vanishing now applies to Ω and B^1, including finite-dimens
 
 We work on the Zariski topological space |D|, with
 [
-widehatcO=widehatcO_V,
+widehatcO=widehatO_V,
 
-cR=widehatcO_V(*D).
+R=widehatO_V(*D).
 ]
-Here and throughout, a line bundle means a locally free module of rank one for the indicated sheaf of rings. Write Ω_{cR}^j for the actual Laurent differential forms, and B^1subset Z^1subsetΩ_{cR}^1 for the additive sheaves of exact and closed one-forms. Lemma~lem:laurent-cartier-algebra identifies these with the Laurent extensions of the algebraic Frobenius modules. In particular there is an exact sequence of additive sheaves
+Here and throughout, a line bundle means a locally free module of rank one for the indicated sheaf of rings. Write Ω_{R}^j for the actual Laurent differential forms, and B^1subset Z^1subsetΩ_{R}^1 for the additive sheaves of exact and closed one-forms. Lemma~lem:laurent-cartier-algebra identifies these with the Laurent extensions of the algebraic Frobenius modules. In particular there is an exact sequence of additive sheaves
 
 0→ B^1→ Z^1
-xrightarrow{ C }Ω_{cR}^1→0,
+xrightarrow{ C }Ω_{R}^1→0,
 
 where C is inverse-Frobenius-semilinear for the ordinary scalar structures:
 [
@@ -1539,9 +1539,9 @@ Corollary~cor:form-vanishing supplies the Laurent cocycle vanishings and the fin
 
 Every global Laurent one-form is closed. On the finite-dimensional k-vector space
 [
-W=H^0(|D|,Ω^1_{cR}),
+W=H^0(|D|,Ω^1_{R}),
 ]
-Cartier is a bijective inverse-Frobenius-semilinear map, and C-idcolon Wto W is surjective as an additive map. Moreover, every additive Z^1-valued v Cech one-cocycle is a coboundary after refining its open cover.
+Cartier is a bijective inverse-Frobenius-semilinear map, and C-id: Wto W is surjective as an additive map. Moreover, every additive Z^1-valued v Cech one-cocycle is a coboundary after refining its open cover.
 
 
 Proof. 
@@ -1549,18 +1549,18 @@ Corollary~cor:form-vanishing gives finite-dimensionality of W and actual Laurent
 
 To lift a global form through C, choose local lifts in Z^1 using (eq:laurent-cartier-sequence). Their differences form a B^1-cocycle. Subtracting a coboundary makes the lifts agree, so
 [
-Ccolon H^0(|D|,Z^1)→ W
+C: H^0(|D|,Z^1)→ W
 ]
 is surjective. Its source is a vector subspace of W. Since the ground field is perfect, a semilinear map has the same dimension behaviour as a linear map. Finite-dimensionality therefore implies H^0(Z^1)=W and bijectivity of C.
 
-For the surjectivity of C-id, set F=C^{-1}, a bijective p-semilinear map. Choose coordinates on Wsimeq k^n. Then F(x)=Ax^{[p]} with AinGL_n(k), where x^{[p]} means componentwise p-th powers. The polynomial map F-id extends to
+For the surjectivity of C-id, set F=C^{-1}, a bijective p-semilinear map. Choose coordinates on W~= k^n. Then F(x)=Ax^{[p]} with AinGL_n(k), where x^{[p]} means componentwise p-th powers. The polynomial map F-id extends to
 
 Psi:PP^n→PP^n,
 
 [x:t]⟼
 [Ax^{[p]}-t^{p-1}x:t^p].
 
-There is no base point: if t=0, the first coordinates vanish only when x=0. Also Psi^*cO_{PP^n}(1)=cO_{PP^n}(p). A positive-dimensional fiber would contain a projective curve on which this line has both degree zero and positive degree, contradiction. Hence Psi is finite. Its image is closed of dimension n, so it is all of PP^n. The inverse image of the standard affine chart is exactly that chart, proving surjectivity of F-id over the algebraically closed field. Finally, C-id is obtained from F-id by the substitution (C-id)beta=etaiff(F-id)beta=-Feta, so it is surjective as well.
+There is no base point: if t=0, the first coordinates vanish only when x=0. Also Psi^*O_{PP^n}(1)=O_{PP^n}(p). A positive-dimensional fiber would contain a projective curve on which this line has both degree zero and positive degree, contradiction. Hence Psi is finite. Its image is closed of dimension n, so it is all of PP^n. The inverse image of the standard affine chart is exactly that chart, proving surjectivity of F-id over the algebraically closed field. Finally, C-id is obtained from F-id by the substitution (C-id)beta=etaiff(F-id)beta=-Feta, so it is surjective as well.
 
 
 
@@ -1653,7 +1653,7 @@ by closedness, and the commutator with x_j follows directly from the Leibniz rul
 [Horizontal expansion]
 Let M be a free rank-one R-module with a connection whose operators satisfy (eq:horizontal-relations) and commute. Its horizontal module
 [
-H_M=∩_iker(D_icolon Mto M)
+H_M=∩_iker(D_i: Mto M)
 ]
 is a finite projective rank-one S-module, and multiplication gives an isomorphism
 
@@ -1699,7 +1699,7 @@ Together with (eq:cartier-pbasis), this proves (eq:horizontal-descent).
 
 As an S-module, M is free of rank p^3. The image of the idempotent P is therefore finite projective. The extension Ssubset R is finite free and faithfully flat; (eq:horizontal-descent) shows that this projective module has rank one.
 
-For restriction, let Rto R' be the homomorphism of Laurent chart rings and S'=(R')^p. The shared monomial bases give S' ⊗ _S Rsimeq R'. The restricted module is S' ⊗ _S M, its operators are 1 ⊗  D_i, and its projector is 1 ⊗  P. Taking the image of a split idempotent commutes with every scalar extension, so its horizontal module is S' ⊗ _S H_M. Since horizontality means nabla m=0, the construction is independent of the coordinate system and the chosen line frame.
+For restriction, let Rto R' be the homomorphism of Laurent chart rings and S'=(R')^p. The shared monomial bases give S' ⊗ _S R~= R'. The restricted module is S' ⊗ _S M, its operators are 1 ⊗  D_i, and its projector is 1 ⊗  P. Taking the image of a split idempotent commutes with every scalar extension, so its horizontal module is S' ⊗ _S H_M. Since horizontality means nabla m=0, the construction is independent of the coordinate system and the chosen line frame.
 
 
 ## The formal Zariski topology and Picard divisibility
@@ -1715,9 +1715,9 @@ The Noetherian completion theorem gives that B is Noetherian and flat over A, an
 
 Every maximal ideal mathfrak nfraknsubset B corresponds to a maximal ideal mathfrak mfrakmsubset A containing f, with mathfrak n=frakn=mathfrak mB.frakmB. For each s, completion modulo mathfrak m^sfrakm^s does nothing, since f^s already vanishes there. Hence
 [
-B/mathfrak n^sfrakn^ssimeq A/mathfrak m^s,frakm^s,
+B/mathfrak n^sfrakn^s~= A/mathfrak m^s,frakm^s,
 
-widehatwidehat{B_B_{mathfrak n}frakn}simeqwidehatwidehat{A_A_{mathfrak m}frakm}.
+widehatwidehat{B_B_{mathfrak n}frakn}~=widehatwidehat{A_A_{mathfrak m}frakm}.
 ]
 The latter ring is regular, as A_{mathfrak m}frakm is a smooth local ring. A Noetherian local ring is regular if and only if its maximal-ideal completion is regular cite[Lemma~15.44.4]{stacks}, so every B_{mathfrak n}frakn is regular, and localization gives regularity at every prime of B.
 
@@ -1733,12 +1733,12 @@ Multiplication by p on Pic(|D|,cR) is surjective.
 Proof. 
 Give an arbitrary cR-line bundle cL the connection from Lemma~lem:cartier-fixed-connection. On a chart with a line frame, Lemma~lem:cartier-curvature gives commuting nilpotent connection operators, and Lemma~lem:horizontal-expansion makes the horizontal module finite projective of rank one over R^p. Its restriction compatibility and intrinsic definition glue these modules into a sheaf cH=kernabla over cR^p. Lemma~lem:formal-line-extension shows that cH is locally free of rank one in the required formal Zariski topology. The local multiplication isomorphisms glue to
 [
-cR ⊗ _{cR^p}cHsimeqcL.
+cR ⊗ _{cR^p}cH~=cL.
 ]
 
-Transport cH through the sheaf isomorphism cRtocR^p, amapsto a^p, to obtain a line bundle cN over cR. Extension from cR^p to cR is its absolute Frobenius pullback. In local frames, that pullback raises the transition units of cN to their p-th powers, which are also the transition units of cN^{ ⊗  p}. Thus
+Transport cH through the sheaf isomorphism cRtocR^p, a|-> a^p, to obtain a line bundle cN over R. Extension from cR^p to cR is its absolute Frobenius pullback. In local frames, that pullback raises the transition units of cN to their p-th powers, which are also the transition units of cN^{ ⊗  p}. Thus
 [
-cLsimeqcN^{ ⊗  p}.
+cL~=cN^{ ⊗  p}.
 ]
 This is a global root on the stated ringed space, proving the surjectivity assertion.
 
@@ -1751,9 +1751,9 @@ Infinite Frobenius divisibility forces integral numerical invariants to vanish;
 ## Orders of Laurent units
 
 
-Let iota_alphacolon D_alpha↪|D| be the inclusion of a prime component. There is an exact sequence of abelian sheaves
+Let iota_alpha: D_alpha↪|D| be the inclusion of a prime component. There is an exact sequence of abelian sheaves
 
-1→cOhat^{,times}
+1→Ohat^{,times}
 →cR^times
 xrightarrow{ ord }
 ⊕_alpha(iota_alpha)_*Z
@@ -1761,15 +1761,15 @@ xrightarrow{ ord }
 
 Consequently restriction gives a surjection
 [
-Pic(|D|,cOhat)→Pic(|D|,cR),
+Pic(|D|,Ohat)→Pic(|D|,cR),
 ]
-whose kernel is generated by the completed restrictions of cO_V(D_alpha).
+whose kernel is generated by the completed restrictions of O_V(D_alpha).
 
 
 Proof. 
 Work near a point on an affine chart as above, small enough that each component through it has a prime equation f_alpha, and write f as a unit times prod_alpha f_alpha^{m_alpha}. Every f_alpha remains a non-zero-divisor in B by flatness of completion. It also remains prime, because
 
-B/(f_alpha)simeq A/(f_alpha).
+B/(f_alpha)~= A/(f_alpha).
 
 Indeed, the induced f-adic topology on A/(f_alpha) is the zero-ideal topology, so completion leaves this integral ring unchanged.
 
@@ -1780,7 +1780,7 @@ u=vprod_alpha f_alpha^{n_alpha},
 
 Uniqueness follows from the same prime cancellation. The exponents agree on smaller charts and are locally constant along the respective D_alpha. They define the order map. Its kernel consists exactly of formal units, and the local equations give its local surjectivity, proving (eq:laurent-unit-orders).
 
-On an irreducible space with its Zariski topology, the constant sheaf Z is flasque: every nonempty open is irreducible, its locally constant integer-valued functions are constant, and all restriction maps are surjective. Each pushed-forward sheaf (iota_alpha)_*Z, and their finite direct sum, is therefore flasque. Its first cohomology vanishes. Line bundles on a ringed space are classified by the first cohomology of its unit sheaf. The cohomology sequence of (eq:laurent-unit-orders) consequently proves surjectivity of the Picard restriction map. The image of the connecting map from global integer sections is the subgroup generated by cO_V(D_alpha), as is seen from their local equations. Exactness proves the kernel assertion.
+On an irreducible space with its Zariski topology, the constant sheaf Z is flasque: every nonempty open is irreducible, its locally constant integer-valued functions are constant, and all restriction maps are surjective. Each pushed-forward sheaf (iota_alpha)_*Z, and their finite direct sum, is therefore flasque. Its first cohomology vanishes. Line bundles on a ringed space are classified by the first cohomology of its unit sheaf. The cohomology sequence of (eq:laurent-unit-orders) consequently proves surjectivity of the Picard restriction map. The image of the connecting map from global integer sections is the subgroup generated by O_V(D_alpha), as is seen from their local equations. Exactness proves the kernel assertion.
 
 
 ## The degree contradiction
@@ -1793,11 +1793,11 @@ Proof.
 For a line bundle cL on the completion, its restriction to each integral projective surface D_alpha is an ordinary line bundle. Define the integer
 
 Phi(cL)=sum_alpha m_alpha
-(c_1(cL|_{D_alpha})cdot H|_{D_alpha}).
+(c_1(cL|_{D_alpha})· H|_{D_alpha}).
 
 Intersection with the Cartier class H gives an integral degree on each surface, so Phi is an additive homomorphism to Z. The numerical properties of the prepared cycle give
 [
-Phi(cO_V(D_beta))
+Phi(O_V(D_beta))
 =sum_alpha m_alpha D_alpha D_beta H
 =DD_beta H=0.
 ]
@@ -1807,7 +1807,7 @@ overlinePhi:Pic(|D|,cR)→Z.
 ]
 It is nonzero, since
 
-overlinePhi(cO_V(H)|_{cR})=DH^2>0.
+overlinePhi(O_V(H)|_{R})=DH^2>0.
 
 
 On the other hand, Proposition~prop:picard-divisibility implies that every class in this Picard group is divisible by p^n for every positive integer n. Its image under overlinePhi must therefore be an integer divisible by every p^n, hence zero. This contradicts (eq:positive-laurent-degree).
@@ -1820,13 +1820,13 @@ The assumption of maximal nef dimension in Proposition~prop:prepared-cycle suppl
 # Descent to the original field
 
 
-We now finish over the original field. Denote it by k_0 and the extension by k. Restore X for the original threefold and write X_k=Xtimes_{k_0}k. Choose a globally generated Cartier multiple of K_{X_k}, then take a further positive multiple divisible by the Cartier index of K_X over k_0. Call the resulting integer m. The line bundle cO_{X_k}(mK_{X_k}) is still globally generated and is not numerically trivial, since nu(K_{X_k})=1. It has at least two independent sections: if a globally generated line bundle has only one, that section trivializes it, contradicting its nonzero numerical class.
+We now finish over the original field. Denote it by k_0 and the extension by k. Restore X for the original threefold and write X_k=Xtimes_{k_0}k. Choose a globally generated Cartier multiple of K_{X_k}, then take a further positive multiple divisible by the Cartier index of K_X over k_0. Call the resulting integer m. The line bundle O_{X_k}(mK_{X_k}) is still globally generated and is not numerically trivial, since nu(K_{X_k})=1. It has at least two independent sections: if a globally generated line bundle has only one, that section trivializes it, contradicting its nonzero numerical class.
 
 Compatibility of the canonical Cartier multiple with field extension and proper section-space base change give
 [
-H^0(X,cO_X(mK_X)) ⊗ _{k_0}k
-simeq
-H^0(X_k,cO_{X_k}(mK_{X_k})).
+H^0(X,O_X(mK_X)) ⊗ _{k_0}k
+~=
+H^0(X_k,O_{X_k}(mK_{X_k})).
 ]
 Their dimensions are equal, proving the required inequality over k_0. In particular kappa(X,K_X)ge1.