raggedbottom

# Abundance in numerical dimension onefor terminal threefolds in positive characteristic



maketitle

Abstract: 
Let X be a projective Q-factorial terminal threefold over an algebraically closed field of characteristic p>3. If the canonical divisor K_X is nef and has numerical dimension one, then some positive multiple of K_X has at least two independent global sections. Hence K_X is semiample and has Kodaira dimension one.




# Overview
addcontentsline{toc}{section}{Overview}

This paper proves a case of the abundance conjecture in algebraic geometry. The result is advanced, but the shape of the argument is simple enough to explain in ordinary language.

The goal.: 
We have a geometric object X called a threefold: over the complex numbers it looks locally like C^3. To each such object one attaches a line bundle K_X, the canonical bundle, which records how volume and differential forms behave. A divisor is nef if it never has negative intersection with any curve; think of it as a direction in which the space is not bending negatively. We assume K_X is nef and that its numerical dimension is 1. This means K_X is positive in one direction and zero in the other two.

Abundance predicts that some positive multiple mK_X has enough global sections to define a map to a curve. We prove exactly that: there are at least two independent sections of some mK_X. Equivalently, the Kodaira dimension is 1 and K_X is semiample.

Why this is hard.: 
The hypothesis only says K_X is not negative and has one positive direction. It does not directly produce a section. The proof therefore has to rule out a hidden bad case: maximal nef dimension. If that bad case occurred, the positive direction of K_X would be locally invisible---no curve of K_X-degree zero would pass through a very general point. The main body of the paper shows that this bad case leads to a contradiction.

The contradiction in one sentence.: 
Inside the bad case one builds a formal neighborhood of a cycle D on a smooth model V. The geometry of jets and the semistability of the tangent bundle make the Picard group of that formal neighborhood divisible by every power of p. On the other hand, intersection with an ample divisor H gives a nonzero integer-valued degree on that same Picard group. A group that is divisible by every p^n cannot map nontrivially to Z. Contradiction.

A small dictionary.: 

{@{}ll@{}}

Term & Plain meaning 

Threefold & A space of dimension 3; locally like A^3 or C^3. 
Nef & Never negative against curves. 
Ample & Positive enough to embed the space in projective space. 
Cartier divisor & A codimension-one object cut out locally by one equation. 
Q-factorial & Some positive multiple of every divisor is Cartier. 
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 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. 




How the sections fit together.: 
Section~sec:introduction states the theorem and the proof map. Section~sec:conventions fixes notation. Section~sec:preparation performs the birational reduction to a smooth threefold V and a connected cycle D. Section~sec:slopes proves strong semistability of the tangent bundle and derives polar-part bounds. Section~sec:jets computes finite jets along D, proves the limiting ratio p/(p+1), and obtains c_2(V)D=0. Section~sec:laurent proves the Laurent-cohomology vanishing criterion. Section~sec:cartier uses Cartier descent to make the Picard group of the formal neighborhood p-divisible. Section~sec:degree constructs the nonzero integer degree and derives the contradiction. Section~sec:descent descends the result back to the original field.

Readers who want the shortest path can read:
[
Theorem~thm:main
→
Proposition~prop:prepared-cycle
→
Proposition~prop:strong-semistability
→
Proposition~prop:scalar-widths
→
Theorem~thm:laurent-vanishing
→
Proposition~prop:picard-divisibility
→
Proposition~prop:laurent-degree-contradiction.
]
The intervening lemmas explain why each arrow is valid.

# Introduction


## The theorem


Let X be a normal projective Q-factorial threefold over an algebraically closed field k of characteristic p>3. Suppose that X has terminal singularities and that K_X is nef with numerical dimension
[
nu(K_X)=1.
]
Then some positive Cartier multiple of K_X has at least two independent global sections. In particular,
[
kappa(X,K_X)=1
]
and K_X is semiample.


Recall that nu(L)=1 for a nef divisor L means
[
L· H^2>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.

Geometrically, semiampleness gives a morphism
[
f:X→ C
]
with connected fibers to a normal projective curve C, together with an ample divisor A on C such that
[
mK_Xsim f^*A
]
for some m>0. The image of the map defined by a globally generated pluricanonical multiple is a curve because nu(K_X)=1; the Stein factorization produces f.

## Background and context

Over C, Miyaoka proved this case of threefold abundance by studying effective pluricanonical divisors and their primitive connected cycles [cite]. Kawamata gave another proof using the log minimal model program and formal thickenings [cite]. Over fields of positive characteristic, the minimal model program for threefolds is now available in the necessary generality: Hacon--Xu [cite] and Birkar [cite] established flips and minimal models in large characteristic, and the relative log MMP was proved by Hashizume--Nakamura--Tanaka [cite] and Hacon--Witaszek [cite].

Nonvanishing and abundance in neighboring cases are due to Xu--Zhang [cite], Witaszek [cite], Zhang [cite], Xu [cite], and Posva [cite]. The remaining gap addressed here is a possible effective canonical divisor of numerical dimension one whose nef dimension is maximal. Theorem~thm:main rules out that possibility.

The proof is in the tradition of the positive-characteristic foliation method of Xu--Zhang [cite]. It uses Langer's Bogomolov ineuality [cite], surface abundance [cite], and Katz's Cartier descent [cite].

## The proof map

The argument has two halves that meet in a contradiction.

[label=Step arabic*.]
 Birational preparation. Assume the theorem is false in the worst way: the nef dimension n(X,K_X) is 3. After a birational modification and a finite tame cover, produce a smooth projective threefold V, a very ample divisor H, and a nonzero effective Cartier divisor
[
D=sum_alpha m_alpha D_alpha
]
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 omegahat_V~=Ohat_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
[
overlinePhi:Pic(|D|,cR)→Z.
]
A p-divisible group admits no nonzero homomorphism to Z. Contradiction. See Proposition~prop:laurent-degree-contradiction.
 Descent. The contradiction proves abundance over an uncountable algebraic closure. Standard base-change and Cartier-index arguments descend the conclusion to the original field. See Section~sec:descent.


Figure~fig:proof-map records the structure.

[ht]
centering
small
[column sep=small,row sep=large]
& Prepared cycle (V,D)
  arrow[dl,"slopes and jets"']
  arrow[dr,"Cartier descent"] & 
Strong semistability
Laurent cocycles have primitives
  arrow[d,"Cartier descent"']
&& Orders of Laurent units
             Intersection with H
  arrow[d] 
Pic(|D|,cR)is p-divisible
  arrow[dr]
&& overlinePhi:Pic(|D|,cR)toZ
             overlinePhi(O(H))=DH^2>0
  arrow[dl] 
& Contradiction &

Caption: The maximal-nef-dimension assumption forces the Picard group of the formal neighborhood to be p-divisible, while intersection theory supplies a nonzero integer degree.



# Conventions and background


## 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 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 Ohat_V for the completion along D, regarded as a sheaf on the Zariski space |D|. Set
[
R=Ohat_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 EEhat for its completion and
[
E_{R}=EEhat ⊗ _{Ohat_V}R.
]
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
[
F:V→ V,

F^sharp(a)=a^p.
]
The inverse Cartier operator is the map
[
C^{-1}:Ω^1_V→Ω^1_V/(exact forms),

C^{-1}(a^p db)=a,db.
]
It is inverse-Frobenius-semilinear: C^{-1}(a^palpha)=aC^{-1}(alpha). On a smooth chart with 'etale coordinates x_1,x_2,x_3, the monomials x^I with 0le I_j<p form a basis over pth powers. The de Rham complex decomposes accordingly, and the Cartier operator can be computed monomial by monomial.

## Jets and widths
Let D be an effective Cartier divisor and E a vector bundle. For integers u<i, the quotient
[
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 O_V(M_eD) when M_e=rp^e and r is the order of O_D(D).

# Birational preparation


The goal of this section is to reduce the maximal-nef-dimension case to a clean geometric situation. The output is the following proposition.

[The prepared cycle]
Suppose k is uncountable and algebraically closed of characteristic p>3, and X is a projective terminal threefold with K_X nef, nu(K_X)=1, and n(X,K_X)=3. Then there exist:
 a smooth projective threefold V;
 a generically finite separable morphism rho:Vto X;
 a very ample divisor H on V;
 a nonzero effective Cartier divisor
[
D=sum_alpha m_alpha D_alpha,

m_alphainZ_{>0},

gcd_alpha m_alpha=1,
]

such that:
[label=(roman*)]
 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
[
D· H^2>0,

D^2· H=0.
]
 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
~= O_U(B_*),

B_*=sum_alpha b_alpha D_alpha.

Consequently K_V· D· A=0 for every Cartier divisor A on V.



The proof has several steps. We give them in a form that highlights the geometry and records the technical inputs.

## Nonvanishing and the maximal nef-dimension branch
By nonvanishing cite[Theorem~1.4]{xu2024}, there is a nonzero effective Cartier divisor
[
0ne N_0sim mK_X,

m>0.
]
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
[
n(X,K_X)=3.
]
The nef reduction exists over our uncountable algebraically closed field cite[Theorem~2.2]{xu2026}. The assumption n(X,K_X)=3 says that no N_0-trivial curve passes through a very general point.

## Isolating a boundary component
Let lambda>0 be the log canonical threshold of N_0 on X. It is finite, rational, and attained on a log resolution. Thus (X,lambda N_0) is lc and not klt. Take a crepant projective Q-factorial dlt modification
[
mu:(W,B)→(X,lambda N_0).
]
All mu-exceptional divisors occur in B with coefficient one. Put N=mu^*N_0 and tau=m^{-1}+lambda>0. Then

K_W+Bsim_{Q}tau N,

Supp B=Supp N.


Choose a coefficient-one component S of B. For small rational ε>0,
[
B-ε Sge0,

tau N-ε Sge0.
]
Run a (K_W+B-ε S)-MMP that is (K_W+B)-trivial. Termination is available because the lowered pair is dlt and its log divisor is pseudoeffective. There is no Mori fiber output, since a pseudoeffective divisor cannot be anti-ample on a covering family.

On a common resolution of each step, the pullbacks of N agree. The difference is exceptional over the new model. A curve contracted over the new model maps to a curve of N-degree zero on the old model, because the extremal ray is N-trivial. Negativity therefore makes the difference numerically trivial, hence zero. Thus N remains nef and numerically trivial on every component of its support.

Let Y,B_Y,S_Y,N_Y be the output. The partial-MMP conclusion gives

K_Y+B_Ysim_{Q}tau N_Y,

K_Y+B_Y-delta S_Y nef

for some rational delta>0. If Tne S_Y is a component of N_Y, then -S_Y|_T is nef by (eq:isolated-nef-pair), while N_Y|_Tequiv0. But S_Y|_T is effective and Q-Cartier. If nonzero, it has positive degree against an ample divisor on the integral projective surface T, contradiction. Hence
[
S_Ycap T=∅

(Tne S_Y).
]
Writing s=mult_{S_Y}N_Y>0, we get

S_Y|_{S_Y}equiv0,

K_Y+S_Ysim_{Q}tau s S_Y

on a neighborhood of S_Y.

## Adjunction on the reduced boundary
Take a new crepant projective Q-factorial dlt modification
[
f:(Y_d,Σ)→(Y,S_Y),
]
where Σ is the strict transform of S_Y together with all exceptional divisors, all reduced. One checks
[
Supp f^*S_Y=SuppΣ.
]
For an extracted divisor E, log canonicity gives
[
0le a_E(Y,B_Y)=a_E(Y,S_Y)-ord_E(B_Y-S_Y)
=-ord_E(B_Y-S_Y).
]
Both terms vanish. Crepancy with (X,lambda N_0) gives a_E(X,lambda N_0)=0. Since X is klt, ord_E N_0>0, hence ord_E N_Y>0 and ord_E S_Y>0. This proves the support assertion.

Near Σ,

K_{Y_d}+Σsim_{Q}tau s f^*S_Y,

f^*S_Y|_Σequiv0.


Let sigma:Σ'toΣ be the finite S_2-ification, an isomorphism away from finitely many closed points. Reduced-boundary adjunction gives

(K_{Y_d}+Σ)|_{Σ'}=K_{Σ'}+Delta',

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 Z· G_j<0.


Proof. 
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,
[
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
[
Z· G_j
=aC· G_j+sum_i(z_i-ac_i)G_i· G_j<0.
]


[Terminal threefolds are regular in codimension two]
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)· G
=2dim_lH^1(G,O_G)-2dim_lH^0(G,O_G).
]
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,O_G)=0,

G^2=K_T· G=-d.
]
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,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,O_{nG}).
]
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 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

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.

## Torsion on the full Cartier cycle
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 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=O_{Y,z}. Normality gives depth at least two, so H^1_{m}m(R)=0. Every nonmaximal localization of R is a normal local ring of dimension at most two, hence Cohen--Macaulay. This implies H^2_{m}m(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_{m}m(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_{m}m(R) for every zin Z. Then the restriction diagram
[
[column sep=small]
0 arrow[r] &
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] &
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,O_{jL}(uL))→ H^1_Z(Y,O_L(uL))
]
is zero for every integer u.

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,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,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 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 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
[
X_{reg}∖Supp N_0.
]
Let rho_0:V_0to X. The pullbacks of N_Y and N_0 agree on V_0.

The section cutting out N_0sim mK_X gives a rational section theta of omega_{V_0}^{ ⊗  m} with
[
div(theta)
=rho_0^*N_0+m(K_{V_0}-rho_0^*K_X).
]
Every exceptional center outside Supp N_0 is an isolated singular point of X, by Lemma~lem:terminal-codimension-two. The divisors over those centers are disjoint from rho_0^*N_0. Thus the support consists of the total pullback support and possibly divisors disjoint from it.

Write m=p^bm' with (m',p)=1. Choose a nonzero rational generator eta of omega_{V_0}^{ ⊗  p^b}, so theta=aeta^{m'} for some ain k(V_0)^times. Adjoin one root v of v^{m'}=a and normalize V_0 in the resulting field. The extension is finite separable, possibly of degree smaller than m'. Away from Suppdiv(theta) it is finite 'etale, since m' is invertible in k.

Resolve the normalization and then the total divisor support, keeping the regular 'etale open set unchanged. Denote the result by V, with map pi:Vto V_0. The root veta is a rational section of pi^*omega_{V_0}^{ ⊗  p^b}. The determinant of the differential of pi is a rational section of omega_V ⊗ pi^*omega_{V_0}^{-1}, nonzero because pi is separable. Multiplying veta by the p^b-th power of this Jacobian gives a rational section of omega_V^{ ⊗  p^b} whose divisor is supported over Suppdiv(theta).

Let L_V be the pullback of L from Y_d, and choose one connected component C of Supp L_V. The isolation of S_Y shows that C is disjoint from the rest of the total pullback support of N_0. Write the part of L_V supported on C as
[
aD,

D=sum_alpha m_alpha D_alpha,

a=gcd_alpha(mult_{D_alpha}L_V).
]
Then D is a nonzero primitive effective Cartier divisor. Its support is connected and snc by the final resolution.

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^2· H=0,

D· H^2>0.
]

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
[
O_D(taD)~=O_D.
]
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_V· D· A
=B_*· D· A
=sum_alpha b_alpha,c_1(O_{D_alpha}(D))· A|_{D_alpha}=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^2· H=N_{tot}^2· H
=D· N_{tot}· H=0,

D· H^2>0,

N_{tot}· H^2>0.
]
The surface Hodge index theorem gives a positive rational c such that

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{D· H^2}{N_{tot}· H^2}.
]
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.

# Slopes, foliations, and polar parts


We now work on the smooth projective threefold V with the cycle D from Proposition~prop:prepared-cycle. Fix a very ample divisor H. We use

H^2D>0,

HD^2=0,

K_VDA=0
quadfor every divisor A.

The divisor D is nef, and no D-trivial curve passes through a very general point. The field is uncountable, algebraically closed, and has characteristic p>3.

## Slope calculus
For a torsion-free coherent sheaf E of positive rank, set
[
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 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.
]
For ranks bounded by rk E, slopes form a discrete subset of Q, so a maximal slope is attained. The standard sum-and-intersection argument gives the Harder--Narasimhan filtration.

Write mu_{max}(E) and mu_{min}(E) for the largest and smallest slopes of its factors. Additivity of degree and the image of a morphism give

mu_{min}(E)>mu_{max}(F)
⇒
Hom(E,F)=0.

Dualizing on the locally free open set gives
[
mu_{max}(E^vee)=-mu_{min}(E),

mu_{min}(E^vee)=-mu_{max}(E).
]
Tensoring by a line bundle adds its degree to every slope. In particular, twisting by iD leaves all slopes unchanged.

## Connections and foliations
Let F_V:Vto V be absolute Frobenius. A Frobenius pullback F_V^*E has a canonical connection that differentiates the coefficients in a pulled-back local frame.

[The connection detects destabilization]
Let E be a semistable torsion-free sheaf. If F_V^*E is unstable and B is its first Harder--Narasimhan term, then the connection induces a nonzero morphism on a big open set
[
B→ (F_V^*E/B) ⊗ Ω_V^1.
]


Proof. 
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
[
deg(B)=pdeg(B_0),

mu(B_0)=frac{mu(B)}p>
frac{mu(F_V^*E)}p=mu(E),
]
contradicting semistability of E.


[The first destabilizing term is a foliation]
If T_V is unstable, its first Harder--Narasimhan term A is closed under Lie brackets and under p-th powers of derivations.


Proof. 
Put Q=T_V/A and a=mu(A). By (eq:slope-numerical-input), mu(T_V)=0, so
[
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 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 O_V-linear map

F_V^*A→ Q.

For local derivations u,vin A, set w=tu+v with t a formal scalar. In characteristic p,
[
(d)/(dt)w^p
=sum_{j=0}^{p-1}w^juw^{p-1-j}
=ad(w)^{p-1}(u).
]
Every coefficient lies in the bracket-closed module A. The mixed coefficients of w^p-t^pu^p-v^p thus belong to A. This proves additivity modulo A.

For a local function f, expanding (fu)^p gives
[
(fu)^p-f^pu^p=sum_{j=1}^{p-1}c_ju^j.
]
The left side is a derivation. At the generic point, if une0, choose x with u(x)ne0. For a largest jge2 with c_jne0, the j-fold commutator with multiplication by x is j!c_j(u(x))^jne0, while it must vanish for a derivation. Hence only the j=1 term remains, so the scalar discrepancy is a multiple of u. This proves Frobenius linearity modulo A and hence (eq:p-curvature-obstruction).

If rk A=1, its source has slope pa>a>mu_{max}(Q), so the map is zero. Suppose rk A=2. Then
[
mu(Q)=-2a,

mu_{max}(Ω_V^1)=2a.
]
If F_V^*A is semistable, its minimum slope is pa>-2a. Otherwise let L be its first Harder--Narasimhan term and R=F_V^*A/L. Both have rank one. Lemma~lem:connection-destabilization gives a nonzero map
[
L→ R ⊗ Ω_V^1.
]
Since R has rank one, the slope rules give
[
mu(L)-mu(R)le2a,

mu(L)+mu(R)=2pa.
]
Hence
[
mu_{min}(F_V^*A)=mu(R)ge pa-a>-2a.
]
Equation~(eq:slope-hom-vanishing) makes (eq:p-curvature-obstruction) zero in either case.


## The inseparable quotient
[Quotient by the distribution]
Let Asubset T_V be a saturated bracket-closed and p-closed subsheaf of rank d. There is a finite purely inseparable map pi:Vto W to a normal projective variety, with a factorization
[
Vxrightarrow{pi}Wxrightarrow{rho}V^{(1)}
]
of relative Frobenius. On open sets whose complements have codimension at least two,

omega_V~=pi^*omega_W ⊗ (det A)^{ ⊗ (p-1)}.



Proof. 
On an affine open with ring R, let
[
P={fin R:u(f)=0 for every local section uin A}.
]
The ring P lies between R^p and R. Since R is finite over R^p, it is finite over P and P is finite over R^p. These invariant rings glue. The ring P is normal: an element of Frac(P) integral over P is integral over R, hence belongs to R, and all derivations annihilate it, so it belongs to P. The variety W is projective because it is finite over V^{(1)}.

Remove the codimension-at-least-two loci where A or T_V/A fails to be locally free. Near any remaining point, choose functions x_1,...,x_d whose differentials restrict to a basis of A^vee, and let u_1,...,u_d be the dual local derivations, so u_i(x_j)=delta_{ij}. Bracket closure forces [u_i,u_j]=0: their bracket lies in A but annihilates every x_l. Likewise u_i^pin A annihilates every x_l, so u_i^p=0.

These commuting nilpotent derivations give an explicit invariant-ring description. For one derivation u with u(x)=1, the coefficient (p-1)!^{-1}u^{p-1}(f) belongs to ker u. Subtracting its product with x^{p-1} and iterating writes every f uniquely as
[
f = sum_{k=0}^{p-1} x^k, g_k,
 g_k in ker u.
]
Applying this successively to the commuting u_i gives

R = ⊕_{0le I_1,...,I_d<p} P x^I
~= 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.

Remove further the singular locus of W and its inverse image; normality and finiteness make these also codimension at least two. Then (eq:foliation-local-presentation) produces an exact sequence of vector bundles

0 → F_V^*(A^vee)
  → pi^*Ω_W^1
  → Ω_V^1
  → A^vee → 0,

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 ~= ((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.

The transition maps match those of F_V^*(A^vee). Let x'_j be another such set of functions and write x'_j in the basis x^I over P. Taking p-th powers and differentiating downstairs, the coefficient of dc_l in d((x'_j)^p) is u_l(x'_j)^p; the coefficient of the l-th basis vector of A^vee in dx'_j|_A is u_l(x'_j). Hence the first term of (eq:foliation-cotangent) is indeed F_V^*(A^vee).

Taking determinants in (eq:foliation-cotangent) gives
[
pi^*omega_W ~= (det A)^{-p} ⊗ omega_V ⊗ det A,
]
which is (eq:foliation-canonical).


## Strong semistability of the tangent bundle

[Strong semistability]
The tangent bundle T_V is strongly semistable for the slope c_1(-)HD/rk(-), and

c_2(V)D ge 0.



Proof. 
We first prove semistability by the foliation and rational-curve argument of cite[Lemma~2.10]{xuzhang2019}, upgraded by the quantitative bend-and-break of cite[Theorem~1.1]{jlr2026}. Suppose Asubset T_V is the first destabilizing term, with a=mu(A)>0 and d=rk A. By Lemmas~lem:destabilizing-foliation and lem:foliation-quotient, it gives a finite purely inseparable quotient pi:Vto W. On the big smooth open set where (eq:foliation-canonical) holds, set Λ=-pi^*K_W. As a divisor class on V,
[
Λ = (p-1)c_1(A)-K_V,

Λ HD = (p-1)da>0.
]
No global Cartier property of K_W is required: its degree is only taken on curves inside this smooth open set.

Let H^{(1)},D^{(1)} be the corresponding classes on V^{(1)}, and set H'=rho^*H^{(1)}, D'=rho^*D^{(1)}. Then H' is ample Cartier, D' is nef Cartier, and
[
pi^*H'=pH,

pi^*D'=pD.
]
For a positive integer t, take sufficiently divisible very ample multiples of H' and H'+tD'; their general complete intersection C_t is a smooth projective integral curve.

Why C_t can be chosen in W_{sm}: the bad complement in W has dimension at most one. A general first divisor meets it in finitely many points and contains none of its curve components; a general second divisor avoids those points. Bertini on the smooth open set, plus irreducibility for the successive very ample sections, gives the required curve.

Projection formula, computed over the locally free locus of pi, cancels the degree of pi and the chosen multiples, giving

(D'C_t)/(-K_WC_t)
&=(pD,H(H+tD))/(Λ H(H+tD)) notag
&=(pDH^2)/(Λ H^2+tΛ HD)
→ 0
(tto∞).


The denominator is positive for all large t.

Quantitative bend-and-break cite[Theorem~1.1]{jlr2026}: if Y is projective over an algebraically closed field, Csubset Y_{sm} is an integral curve with K_YC<0, and N is a nef real Cartier divisor, then every closed point of C lies on a rational curve Gammasubset Y with
[
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 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 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
[
pDwidetildeGamma = (pi^*D')widetildeGamma
= deg(widetildeGamma/Gamma),D'Gamma=0,
]
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,
 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
[
mu_{min}(A' ⊗ (Q')^vee)
=mu(A')-mu_{max}(Q')>0,
]
contradicting (eq:slope-hom-vanishing). All iterated Frobenius pullbacks of T_V are therefore semistable.

Finally, apply Langer's Bogomolov ineuality cite[Theorem~3.2]{langer2004},[cite]: for a strongly semistable torsion-free sheaf with a nef tuple whose product is nonzero, Delta(E)Dge0 where Delta(E)=2rk(E)c_2(E)-(rk(E)-1)c_1(E)^2. Here the tuple is (H,D), which has H^2D>0. For T_V this gives
[
(6c_2(V)-2K_V^2)Dge0,
]
and (eq:slope-numerical-input) gives K_V^2D=0, proving (eq:c2-nonnegative).


## Polar-part bounds

[Zero-degree line bundles]
If a line bundle L has deg Lle0, then h^0(V,L)le1.


Proof. 
A nonzero section has an effective zero divisor, so its degree is nonnegative and hence zero. Suppose two independent sections exist. Remove the base locus of their pencil. Through every point x of the remaining open set there is a pencil member; its divisor has degree zero. Each prime component S through x therefore satisfies DHS=0, because all component degrees are nonnegative.

The restriction D|_S is nef and numerically trivial. To see this for possibly singular S, take a projective resolution sigma:widetilde Sto S. The class L_S=sigma^*(D|_S) is nef and orthogonal to the big-and-nef class A_S=sigma^*(H|_S) with A_S^2>0. The surface Hodge index theorem makes the form negative definite on A_S^perp, while nefness gives L_S^2ge0, so L_S is numerically zero. The projection formula gives the assertion on S.

A projective integral curve in S through x (e.g. a component of a sufficiently ample section through x) then has D-degree zero. Choosing x very general contradicts the prepared-cycle property.


[Hom bound]
If G_1,G_2 are torsion-free semistable sheaves of slope zero, then
[
dim_kHom(G_1,G_2)lerk G_1·rk G_2.
]


Proof. 
First suppose G_1,G_2 are stable of slope zero. For a nonzero map, its image has slope at least zero as a quotient of G_1 and at most zero as a subsheaf of G_2. A positive-rank kernel makes the first ineuality strict; an image of smaller rank than rk G_2 makes the second strict. Hence every nonzero map is full rank and rk G_1=rk G_2=n.

Its determinant is a global section of L=det(G_2) ⊗ det(G_1)^{-1} with deg L=0, so h^0(L)le1 by Lemma~lem:degree-zero-line-sections. If f,g were linearly independent, write det(sf+tg)=P(s,t)tau with tau a global generator and P homogeneous of degree n. Since f has full rank, Pne0; over the algebraically closed field, P vanishes at some (s:t)inPP^1(k). Then sf+tg is nonzero (by independence) but has zero determinant, a contradiction. Hence dimHomle1.

Every slope-zero semistable sheaf has a finite filtration by saturated subsheaves whose factors are torsion-free stable of slope zero: if it is not stable, a smaller-rank equal-slope subsheaf can be saturated, its degree not increasing strictly, and induction on rank gives the filtration. Left-exactness of Hom gives the bound by summing one-dimensional contributions over the pairs of stable factors.


[Polar-part bounds]
Let G be a semistable vector bundle of slope zero. For every integer i,
[
h^0(V,G(iD))lerk G,

h^0(V,G^{ ⊗ 2}(iD))le(rk G)^2.
]
For each fixed J=G or J=G^{ ⊗ 2}, the increasing spaces H^0(V,J(iD)/J) for i=1,2,... have bounded dimension, eventually stabilize, and have a uniform pole bound along every component of D.


Proof. 
Apply Lemma~lem:semistable-hom-bound to O_V, G(iD) and to G^vee, G(iD), using
[
H^0(G(iD))=Hom(O_V, G(iD)),

H^0(G^{ ⊗ 2}(iD))=Hom(G^vee, G(iD)).
]

For i>0, the sequence 0to Jto J(iD)to J(iD)/Jto0 gives
[
h^0(V,J(iD)/J)
le h^0(V,J(iD))+h^1(V,J),
]
bounded independently of i. Effectivity of D gives injections of the quotient sheaves as i increases, hence injections of section spaces. A bounded increasing sequence of dimensions stabilizes.

More explicitly, choose i_0 after stabilization. Every global polar part, being the image of an element of H^0(V,J(i_0D)/J), has pole order bounded by i_0mult_{D_alpha}D at the generic point of D_alpha, uniformly in the cut and the chosen section.


[No scalar polar parts]
For every i>0, H^0(V,O_V(iD)/O_V)=0.


Proof. 
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,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 O_D(D) is torsion. Near D there is an isomorphism

omega_V^{ ⊗  p^b}~=O_V(B_*),

B_*=sum_alpha b_alpha D_alpha,

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 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 Ohat_V for the completion along D, regarded as a sheaf on |D|, and set
[
R=Ohat_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.

Two basic completeness facts. First, on an affine chart with local equation f for D, adic completion is flat and f remains a non-zero-divisor, so the completed lattices embed in their localization. Second, a section of the localization on a quasi-compact open lies in one lattice: finitely many local pole bounds have a common upper bound and the resulting lattice sections glue. Consequently every Laurent cochain has a finite upper weight. Each C_i^bullet(E) is complete and separated for the filtration by C_{i-s}^bullet(E) with sge1. Exactness of completion for finite modules gives

C_i^bullet(E)/C_{i-s}^bullet(E)
quadcomputesquad
H^bullet(V,P_i^s(E)),

P_i^s(E)=(E(iD))/(E((i-s)D)).

Thus P_i^s(E) retains the s layers at weights i-s+1,...,i; call i the upper cut. We abbreviate E^!=E^vee ⊗ omega_V.

For sge1, consider a leading cochain: a class in C_i^a(E)/C_{i-1}^a(E), where a is its cohomological degree. The cycle condition asks whether this class has a representative closed modulo C_{i-s}^bullet(E). The boundary condition asks whether it is the leading term of a coboundary with a primitive in C_{i+s-1}^{a-1}(E). Thus s records both the precision of closedness and the allowed upper cut of a primitive. Define

Z_s^{i,a}&={[x]:xin C_i^a(E), ∂ xin C_{i-s}^{a+1}(E)},
U_s^{i,a}&={[∂ y]:yin C_{i+s-1}^{a-1}(E),
                        ∂ yin C_i^a(E)}.

Every exact cochain is closed, so U_s^{i,a}⊆ Z_s^{i,a}. Put
[
E_s^{i,a}(E)=Z_s^{i,a}/U_s^{i,a},

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 x|->∂ x induces maps
[
d_s:E_s^{i,a}(E)→ E_s^{i-s,a+1}(E)
]
whose cohomology is the next page. At page one, E_1^{i,a}(E)=H^a(V,P_i^1(E)). Every page has only rows a=0,1,2. Writing delta_s^a(i)=g_s^a(i;E)-g_{s+1}^a(i;E) for a=0,2, we have

g_s^0(i;E)
&=dimim(H^0(P_i^s(E))to H^0(P_i^1(E))),

g_s^2(i;E)&=g_s^0(1-i;E^!),
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 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)).

Finally, at each fixed weight i, the stable image in (eq:bottom-endpoint) is precisely the image of the global sections of the completed bundle E(iD)hat in H^0(P_i^1(E)).


Proof. 
If two representatives of a leading cochain differ by yin C_{i-1}^a and both have coboundary in C_{i-s}, their coboundaries differ by a target boundary from C_{i-1}=C_{(i-s)+s-1}. Adding an exact representative has no effect either. So d_s is well defined, and d_s^2=0 follows from ∂^2=0.

If d_s[x]=0, the leading term of ∂ x is represented by ∂ y with yin C_{i-1}^a and ∂ yin C_{i-s}^{a+1}. Replacing x by x-y makes its coboundary lie in C_{i-s-1}^{a+1}. Thus the kernel is represented by Z_{s+1}^{i,a}. The incoming image enlarges U_s^{i,a} to U_{s+1}^{i,a}: its representatives are exactly the leading terms of ∂ zin C_i^a with zin C_{i+s}^{a-1}. This proves the next-page assertion. Equation~(eq:jet-quotient) identifies page one. Since these quotients are supported on the projective surface scheme D, their cohomology is zero outside degrees 0,1,2, and so is every subsequent page.

The differentials through a middle-row term are
[
[column sep=large]
E_s^{i+s,0}(E) arrow[r,"d_s"] &
E_s^{i,1}(E) arrow[r,"d_s"] & E_s^{i-s,2}(E).

]
Moving right lowers weight by s and raises cohomological degree by one; the composite is zero. There is no incoming differential in degree zero, and its representatives are exactly the sections of P_i^s(E), giving (eq:bottom-endpoint).

For the other endpoint, consider the inclusion of the bottom layer
[
H^2(P_i^1(E))→ H^2(P_{i+s-1}^s(E)).
]
There is no outgoing differential in degree two. The kernel calculation above and the vanishing of the degree-three page show that Z_s^{i,2}=Z_1^{i,2}. A class represented by xin C_i^2 with ∂ xin C_{i-1}^3 dies under this inclusion exactly when x-∂ yin C_{i-1}^2 for some yin C_{i+s-1}^1. Its kernel is U_s^{i,2}/U_1^{i,2}, so the image has dimension g_s^2(i;E).

The two-term locally free presentation of P_j^s(E) gives
[
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
[
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 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
[
e_{t+1}(i)-e_t(i)=-delta_t^0(i)-delta_t^2(i)+delta_t^0(i+t)+delta_t^2(i-t).
]
For t<s all four functions are periodic modulo M, and their sums cancel, proving the Euler identity by induction from page one.

For the last assertion, set A_s=H^0(P_i^s(E)) and B_s=∩_{tge s}im(A_tto A_s). Each A_s is finite dimensional, so the intersection is attained at a finite stage. Choosing one stage beyond stabilization at both s and s+1 shows that B_{s+1}to B_s is surjective. Every element of B_1 therefore admits compatible lifts in all B_s, and their inverse limit is a global section of E(iD)hat (sections commute with inverse limits of sheaves). Conversely a completed section restricts to the image of every A_s.


## Scalar jets and their last widths

[The scalar jet rule]
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;O_V)=

1,&O(iD)|_{sD}~=O_{sD},
0,&otherwise.




Proof. 
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^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 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 O_D(D). For ege0 put

M_e=rp^e,

d_e=max{sge1:O(M_eD)|_{sD}~=O_{sD}}.

These maxima exist and satisfy

1le d_e<M_e,

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 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 ineuality 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

the last width at i=jM_e is d_e
quadif pnmid j.

This includes negative j. If rnmid i, no scalar leading jet occurs even at width one. Every nonzero multiple of r has a unique expression jM_e with pnmid j; at i=0 the scalar entry persists at every width. These observations classify all scalar outer-row entries and drops.

Finally, Hirzebruch--Riemann--Roch on the smooth threefold gives

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=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=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;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}.

Between pages s_e and s_{e+1}=d_e+1 these are the only drops: there is no top row, and all scalar entries with a different last-width index either disappeared earlier or have last width at least d_{e+1}. Modulo M_{e+1}=pM_e, the drops remove all the p lifts of the middle entry except its unshifted lift. Thus at page s_{e+1} the middle row consists of one entry at -d_epmod{M_{e+1}}. Applying the same argument with e+1 says it must lie at -d_{e+1}pmod{M_{e+1}}. But 0<d_e<d_{e+1}<M_{e+1}, so these residues are different.


[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,

omegahat_V~=Ohat_V(B_0).



Proof. 
Put P=p^b, and take a sequence of nonzero leading canonical jets of widths sto∞, with upper weights l that may vary. Raising local representatives to the Pth power and using (eq:canonical-p-power) gives a scalar jet with bounds
[
A=PlD+B_*,

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
[
(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
[
-Pfrac{ord_{D_alpha}(f)}{m_alpha}+beta_alpha
>P(l-1)+min_alphabeta_alpha,
]
while the new lower bound is P(l-s)+c_*, smaller for large s.

Lemma~lem:scalar-jets shows that all these powered coefficients have one integral pole ratio t. The preceding ineuality and the upper bound give a constant C, independent of s,l, such that

|t-Pl|le 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 O(B_*) has order -b_alpha, so

b_alpha-tm_alpha=Pord_{D_alpha}(f).

For large J, the integer t is divisible by P, so (eq:canonical-order-identity) shows Pmid b_alpha for every alpha. This proves B_*=PB_0 with B_0 integral.

For the same sequence put h_0=t/P. The order identity shows that the original local representatives are sections of
[
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}~=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 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 Lhat.L. This gives (eq:canonical-formal-root).


## The two endpoint residues and the limiting width

Write B_0=sum_alpha b_{0,alpha}D_alpha, and define

h=-Big⌊(min_alphafrac{b_{0,alpha}}{m_alpha}),

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 Ohat_V-linear isomorphism respects every nested divisor lattice, and each width-s quotient is killed by cI_D^s. Since Ohat_V/cI_D^sOhat_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

t=l-h.

To prove this, enlarge the two bounds to uniform multiples of D as in the preceding proof. A component detecting the leading image has ratio within a fixed distance of l, so for large s the enlarged quotient still detects it. Lemma~lem:scalar-jets gives a uniform integral ratio t. Set beta_0=min_alpha b_{0,alpha}/m_alpha. Fitting under every upper bound but not under the preceding layer is exactly the pair of ineualities
[
l-1+beta_0<tle l+beta_0.
]
Its unique integral solution is t=l+floor(beta_0)=l-h.

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 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,&O((l-h)D)|_{T_s}~=O_{T_s},
0,&otherwise,

(s sufficiently large).


The comparison with ordinary widths is uniform and quite explicit. Put
[
C_0=h+Bigllceilmax_alphafrac{b_{0,alpha}}{m_alpha}Bigrrceilge0.
]
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 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.

In particular the last width is finite and exists in the range of (eq:canonical-endpoint-rule). Once T_s contains D, the nilpotent-kernel argument used above shows that the last width is the same for t=jM_e with pnmid j. Also such a trivialization requires rmid t; for tne0 the unique expression t=jM_e with pnmid j classifies its last width. The value t=0 persists forever. To see that the large-e rules exhaust the entries at large width, observe from (eq:thickness-comparison) that an entry at width s gives an ordinary scalar trivialization to width s-C_0. For each fixed J, taking s-C_0>d_{J-1} forces M_Jmid t.

[The limiting scalar widths]
The scalar widths satisfy

a_*=lim_{eto∞}(d_e)/(M_e)
=(p)/(p+1),

Dc_2(V)=0.



Proof. 
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=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
[
-d_e+jM_e, pnmid j.
]
The top drops, of length d'_e, consume middle-row dimensions at
[
q+d'_e+jM_e, pnmid j;
]
here replacing j by -j accounts for the sign in the top ratio. Put
[
A_e=-d_e,

B_e=q+d'_e.
]
The two lists, including their multiplicities when they meet, are thus A_e+jM_e and B_e+jM_e with pnmid j.

We spell out the rank count also when the two centers coincide modulo M_e. Fix a residue apmod{M_e} and let m(a)in{0,1,2} be its multiplicity in the multiset {A_e,B_e}. Its initial middle-row rank n(a) is constant on all p lifts modulo pM_e. Each of the m(a) drop lists omits exactly one lift, namely its unshifted center modulo pM_e; at most two lifts are omitted. For example, when p=5 and m(a)=2, two distinct omitted lifts can be pictured as follows:
[
{c|ccccc}
 & a&a+M_e&a+2M_e&a+3M_e&a+4M_e
first list&circ&bullet&bullet&bullet&bullet
second list&bullet&bullet&circ&bullet&bullet

]
A filled dot consumes one rank; a circle marks that list's unshifted lift. If the omitted lifts coincide, the column is omitted twice and every other column receives both drops. Multiplicities are always added.

Since p>2, some lift receives all m(a) drops. Nonnegativity of its final rank gives n(a)ge m(a). Summing over a and using (eq:periodic-euler) and (eq:scalar-euler) gives
[
2lesum_{abmod M_e}n(a)
=2-M_egammale2.
]
Consequently gamma=0, so Dc_2(V)=0, and n(a)=m(a) for every residue.

After subtracting the drop lists, the final rank at each lift is exactly the number of lists omitting that lift. Thus at page s_{e+1} the middle row is described by the same two-element multiset {A_e,B_e}, now modulo M_{e+1}=pM_e. Applying the just-proved description at index e+1 gives the multiset equality

{-d_e,q+d'_e}
={-d_{e+1},q+d'_{e+1}}pmod{M_{e+1}}.

The entries -d_e and -d_{e+1} are distinct modulo M_{e+1} because 0<d_e<d_{e+1}<M_{e+1}. Therefore the matching in (eq:residue-swap) must be the swapped matching, even if a multiset initially has repeated entries. In particular
[
d_{e+1}+q+d'_eequiv0pmod{M_{e+1}}.
]
For large e its left side is positive and, by d_{e+1}<M_{e+1} and d'_ele d_e+C_0<M_e+C_0, is less than 2M_{e+1}. Hence

d_{e+1}+q+d'_e=M_{e+1}.

Divide by M_{e+1}=pM_e and let eto∞. Equation~(eq:canonical-width-comparison) gives d'_e/M_{e+1}to a_*/p, while d_{e+1}/M_{e+1}to a_*. Thus a_*+a_*/p=1, which proves (eq:limiting-width).


# Laurent cohomology of vector bundles


We work with the smooth threefold V, the cycle D, and the notation of the previous sections. The jet analysis produced

M_e=rp^e,

(d_e)/(M_e)→ a_*:=(p)/(p+1),

c_2(V)D=0,

with O_V(M_eD) trivial through width d_e. Choose real numbers

1-frac2p<c_0<c<2a_*-1.

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=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(-)· H· D. There is a constant A_G such that, for every sufficiently large e,
[
g^0_{floor( c_0M_e)}(i;G)>0
⇒
dist(i,M_eZ)le A_G.
]


Proof. 
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-∞.

In the first case s>v for large e. The jet maps to a global section of G(vD)/G and retains its nonzero leading term. Its pole order tends to infinity, contrary to Proposition~prop:polar-bounds.

In the second case choose a real number lambda such that

(1-c_0)/(2)<lambda<frac1p,

plambda>frac12.

The interval is nonempty by (eq:phase-constants). Since the periods increase by the factor p, we can choose jle e with
[
lambdale(|v|)/(M_j)le plambda.
]
Here jto∞ as |v|to∞. After a subsequence the ratios tend to a number yin[lambda,plambda]. There is an integer min{1,2} for which

1-c_0<my<1.

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 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<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 EEhat for the completion of a bundle E along D, and E_{R}=EEhat ⊗ _{cOhat}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:
[label=(roman*)]
 chi(P_i^1(E))=0 for every integer i.
 For J=E,E^! the spaces H^0(V,J(nD)/J) have dimensions bounded independently of nge1.
 For J=E,E^! there is a constant A_J such that
[
g^0_{u_e}(i;J)>0quad⇒
dist(i,M_eZ)le A_J
]
for all sufficiently large e.

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. 
We split the argument into endpoint patterns, drops, the middle row, and convergence.

smallskipnoindentEndpoint patterns.
For e large, u_ele d_e, so the page at width u_e is periodic modulo M_e. By (iii), the bottom row can be supported only at a fixed bounded set of centered positions. At each such position the descending sequence of leading images stabilizes as the width tends to infinity. The stable-image statement of Lemma~lem:page-calculus realizes every vector in the stabilized image by a global completed section at that cut. Consequently, for all sufficiently large e, there is one fixed bottom pattern, at positions v with fixed multiplicities, repeated by translations by M_e. Duality gives a fixed top pattern at positions w. Enlarge a constant A so that all these v,w lie in [-A,A].

This already proves finite-dimensionality of global Laurent sections. A nonzero section has a least upper cut i: its poles are bounded, and the filtration is separated toward negative infinity. Its nonzero leading class belongs to g^0_{u_e}(i;E) for every large e. With i fixed and M_eto∞, condition (iii) forces i to be an unshifted position of the fixed pattern. There are only finitely many such cuts, and each leading quotient is finite-dimensional, giving a finite filtration of the space of global sections. The same observation applies to E^!.

smallskipnoindentEndpoint drops.
By the middle-row recurrence, a bottom drop of length s at v+jM_e removes middle-row rank at v-s+jM_e; a top drop at w+jM_e removes rank at w+s+jM_e. To determine these losses over the next period we control their lengths, with multiplicities, uniformly in j whenever pnmid j.

Consider a bottom position i=v+jM_e. If pmid j, the next period M_{e+1}=pM_e retains exactly the same multiplicity there; monotonicity therefore forbids any drop between pages u_e and u_{e+1}. If pnmid j, all of this multiplicity disappears by page u_{e+1}. We claim the drops occur at lengths

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 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}Ohat_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

alpha_mu=(1+jb_{lambdamu})alpha_lambda
pmod{cI_D^kE(vD)hat}.

Indeed (1+b_{lambdamu})^j=1+jb_{lambdamu} to this precision, for positive or negative j alike, because k<2d_e. In particular, the cochain alpha glues to width d_e.

We next match alpha to a single global completed section beta in that untwisted lattice through width k-d_e. Every completed section and every formal lattice map restricts to the actual algebraic finite jets of (eq:jet-quotient): on each affine intersection this is the identity Ahat/(f^s)=A/(f^s).A/(f^s)=A/(f^s). These restrictions commute with the v Cech maps and with projection to leading layers. Start with beta_{<0}=0. After matching the first n depths, the sum beta_{<n} of the completed global sections already chosen satisfies
[
alpha-beta_{<n}in C_{v-n}^0(E),

∂(alpha-beta_{<n})in C_{v-d_e}^1(E).
]
The second inclusion persists because each subtracted section is global. Thus, for n<k-d_e, the residual is a finite jet at upper weight v-n of width at least d_e-n. The ineualities

d_e-nge 2d_e-M_e-O(1)>u_e,
 k-d_e<M_e/2-O(1)

hold with margins growing linearly in M_e, by c<2a_*-1 and a_*>1/2. Thus v-n is centered, and condition (iii) says that its leading image is either zero or one of the unshifted stabilized images. In the latter case lift and subtract a completed global section. A finite induction constructs beta.

If pnmid j,j', put q=j'/j in the prime field and set alpha'=qalpha+(1-q)beta. Since beta is global, (eq:normalized-jet-overlap) gives
[
alpha'_mu-(1+j'b_{lambdamu})alpha'_lambda
=j'(1-q)b_{lambdamu}(alpha_lambda-beta)
pmod{cI_D^kE(vD)hat}.
]
The right side vanishes: its two factors have depths d_e and k-d_e. Retwisting by t_lambda^{j'} gives a width-k jet at v+j'M_e with the same normalized leading term (since k-d_ege1). Also alpha'-beta=q(alpha-beta), so the same beta reverses the construction with q^{-1}. The normalized leading-image subspaces are therefore equal at every width under consideration, and their ranks and drops are independent of j prime to p.

For j=1, property (ii) kills the entire leading image by width M_e+C once C>A is sufficiently large: this width is at least the upper weight M_e+v, so a surviving leading jet would give a global polar part of order tending to infinity, and the increasing spaces in (ii) stabilize and forbid such polar parts. Since u_{e+1}>M_e+C for large e, this proves (eq:drop-range) and the complete drop rule. The identical argument for E^!, followed by the duality of Lemma~lem:page-calculus, gives the top-row rule.

smallskipnoindentThe middle row.
Let T_e be the multiset of the numbers v-s for bottom drops and w+s for top drops, each counted with its drop multiplicity for one translating integer prime to p. Its total size is the sum of the ranks of the two fixed endpoint patterns. Put H_e(i)=g^1_{u_e}(i;E). The middle-row recurrence of Lemma~lem:page-calculus gives


H_{e+1}(i)=H_e(i)
&-#{tin T_e:tequiv ipmod{M_e}}
&+#{tin T_e:tequiv ipmod{pM_e}}.


The last term restores the unshifted copies. All members of T_e lie in the two intervals

[-M_e-O(1),-d_e+O(1)]
quadhbox{and}
[d_e-O(1),M_e+O(1)].

Each interval has length less than M_e for large e. For any fixed residue modulo M_e, these intervals can occupy at most two of its lifts modulo pM_e. Since pge5, some lift is missed. Nonnegativity of H_{e+1} at that lift and periodicity of H_e show that
[
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 ineuality 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)=0 for all sufficiently large e.


smallskipnoindentConvergence of primitives.
Fix an integer upper bound l for a Laurent cocycle. We bound the source z=x+s of every nonzero incoming differential of length s to a middle-row weight xle l by a function B_l(x) satisfying

sup_{xle l}B_l(x)<∞,
 B_l(x)→-∞(xto-∞).


At a sufficiently late period, the drop rule gives z=v+jM_e, pnmid j, and d_ele sle M_e+C. If jge2, then xge M_e-A-C, exceeding l for large e. If j=1, the target must lie in the fixed finite set of integers in [-A-C,l]. Choose one index e_0, depending on l, beyond the thresholds for the endpoint and drop rules, so that M_e-A-C>l for ege e_0 and every middle rank in this finite set is zero at page u_{e_0}. Equation~(eq:fixed-weight-vanishing) permits this simultaneous choice; middle ranks only decrease, so no nonzero incoming image can have j=1 at any later page.

Put S_l=u_{e_0}ge1. Earlier lengths s<S_l give zle x+S_l. Every later length belongs to a period u_ele s<u_{e+1} with ege e_0, and the preceding exclusions leave only jle-1. In that case zle-M_e+A, so M_ele-z+A, giving x=z-sge2z-A-C, i.e.
[
zle(x+A+C)/(2).
]
The real-valued upper bound
[
B_l(x)=maxBigl{x+S_l,(x+A+C)/(2)Bigr}
(xle l)
]
therefore bounds every possible source and satisfies both assertions of (eq:primitive-cut-bound).

Let gamma be an actual Laurent 1-cocycle of upper cut xle l. Its leading cochain belongs to every cycle space Z_s^{x,1}. By (eq:fixed-weight-vanishing), it eventually belongs to the boundary space U_s^{x,1}. More precisely, the page construction identifies
[
U_{s+1}^{x,1}/U_s^{x,1}
=im(d_s:E_s^{x+s,0}→ E_s^{x,1}),
]
so U_s^{x,1} stops increasing once x+s>B_l(x). Each increase from a source cut z=x+s has a representative primitive in C_z^0(E), and the initial U_1^{x,1} uses C_x^0(E). Consequently the leading cochain of gamma has a primitive of cut at most max{x,B_l(x)}. Subtract that primitive's coboundary. The new cocycle has strictly smaller upper cut. Repeating at successive descending cuts gives primitives whose upper cuts tend to negative infinity by (eq:primitive-cut-bound). Their sum converges in a single bounded-pole completed lattice, and the differential is continuous, so the resulting cochain has coboundary gamma. All arguments use finite covers by affine charts and their affine intersections, and remain valid after refinement.


## Cartier algebra on the Laurent charts

Write F for absolute Frobenius and Ω=Ω_V^1. Let B^1,Z^1subset F_*Ω be the sheaves of exact and closed algebraic one-forms, with the Frobenius module structure.


On a smooth affine chart with 'etale coordinates x_1,x_2,x_3 and a local equation f for D, put R=Ahat[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→O_V→ F_*O_V
→ B^1→0,

0→ B^1→ Z^1
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_*O_V)^vee ⊗ omega_V~= F_*omega_V.



Proof. 
The algebraic calculation is the p-basis form of Cartier's theorem cite[Theorem~1]{cartier1957}; see also cite[Theorem~7.2]{katz1970}. We check that its finite basis survives the completion and localization.

For an 'etale algebra, the relative Frobenius square is a base change square: one checks this because the relative Frobenius is 'etale and induces isomorphisms on geometric fibers. The polynomial-coordinate monomial basis over pth powers therefore gives the asserted basis for A over A^p. The field k is perfect.

Complete the p-power subring along f^p and use this finite basis. The f-adic and f^p-adic topologies on A coincide, and Frobenius identifies the completion of A^p with (Ahat)^p.A)^p. The basis thus persists after completion and after inverting f; in the latter step, 1/f=f^{p-1}/f^p.

Every k-derivation kills pth powers. Since each element has a unique finite expansion in the displayed basis with coefficients in R^p, its differential is determined by ordinary differentiation of those monomials. Conversely the coordinate derivations are defined by that rule. Hence Ω^1_{R/k} is free on dx_j.

In one variable, the de Rham complex decomposes into the monomials of degrees less than p; its cohomology has representatives 1 in degree zero and x^{p-1}dx in degree one. Tensoring the three one-variable complexes proves local freeness of images, kernels, and quotients, and proves the inverse Cartier rule
[
f⟼ f^p,

df⟼[f^{p-1}df].
]
The rule is intrinsic. Leibniz holds directly; additivity for differentials follows modulo exact forms by differentiating the mixed terms of the integral polynomial ((X+Y)^p-X^p-Y^p)/p and reducing modulo p. Thus the local isomorphisms glue.

More explicitly, the scalar extension of the algebraic Frobenius complex to a Laurent chart is the actual Laurent de Rham complex with Frobenius action, through
[
R ⊗ _A F_*Ω_A^bullet→ F_*Ω_R^bullet,
 a ⊗ eta⟼ a^peta.
]
The monomial and coordinate-wedge bases make this an isomorphism. This proves the assertions about (eq:cartier-sequences). Finally, the pairing obtained by Cartier on top forms is perfect: in coordinates it extracts the coefficient of x_1^{p-1}x_2^{p-1}x_3^{p-1}dx_1wedge dx_2wedge dx_3. It gives (eq:frobenius-duality) and is intrinsic by the same Cartier rule.


## Application to one-forms and exact forms


Every Laurent v Cech 1-cocycle with values in Ω, B^1, or Z^1 has a Laurent primitive. Global Laurent one-forms form a finite-dimensional k-vector space.


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 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_*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

B^1longhookrightarrow F_*Ω,

(B^1)^!longhookrightarrow F_*omega_V.

The first follows from Lemma~lem:laurent-cartier-algebra; the second follows by dualizing the first sequence in (eq:cartier-sequences) and using (eq:frobenius-duality). For a subbundle Esubset F_*G, the injection remains injective on every quotient E(nD)/E. Projection formula embeds that quotient in F_*(G(pnD)/G), so the polar bounds for G=Ω and G=omega_V give condition (ii) for both bundles in (eq:exact-form-inclusions). All these bounds concern fixed bundles and their first Frobenius pushforwards; the growing index e below changes the jet period, not the bundle in the polar-part estimate.

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-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.

Theorem~thm:laurent-vanishing now applies to Ω and B^1, including finite-dimensionality for the former. For a Z^1 cocycle, first solve its image in Ω and lift the resulting local primitives through Cartier. After subtraction the remaining cocycle lies in B^1 and can also be solved. These lifts exist on affine Laurent charts by Lemma~lem:laurent-cartier-algebra, and refining the finite cover if necessary preserves every preceding argument.


# Cartier descent and Picard divisibility


We work on the Zariski topological space |D|, with
[
cOhat=Ohat_V,

R=Ohat_V(*D).
]
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_{R}^1→0,

where C is inverse-Frobenius-semilinear for the ordinary scalar structures:
[
C(a^palpha)=aC(alpha).
]
The distinction between these scalar structures and the Frobenius-module structures is essential in using the preceding cohomology calculation.

Corollary~cor:form-vanishing supplies the Laurent cocycle vanishings and the finite-dimensionality of global one-forms that we use to make Pic(|D|,cR) p-divisible. Orders along the components of D then let intersection with H descend to a nonzero integer degree on that same group. These two properties are incompatible.

## Global Cartier adjustment


Every global Laurent one-form is closed. On the finite-dimensional k-vector space
[
W=H^0(|D|,Ω^1_{R}),
]
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. 
Corollary~cor:form-vanishing gives finite-dimensionality of W and actual Laurent one-cocycle vanishing for both B^1 and Z^1, proving the last assertion.

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
[
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 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^*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.



Every cR-line bundle admits an integrable connection whose local connection one-forms are fixed by Cartier.


Proof. 
Let cL be such a line bundle. Its frame cover can be refined by opens U_icap|D| where U_isubset V is affine, has 'etale coordinates, makes D principal, and trivializes the finitely many algebraic bundles in use. Quasi-compactness of |D| permits a finite such cover, and its finite intersections are restrictions of affine opens since V is separated. On each intersection, a Laurent coefficient has a finite pole bound: take a finite subcover of local bounds and glue in their common completed lattice, as in Section~sec:jets. Taking the maximum over the finitely many coefficients gives a common upper cut for a cochain. Thus the cocycles below lie in the actual Laurent complexes to which the vanishing results apply.

Choose frames e_i on this cover, and write e_j=g_{ij}e_i on overlaps. Then g_{ij}g_{jk}=g_{ik}, so
[
dlog g_{ij}=g_{ij}^{-1}dg_{ij}
]
is a closed additive one-cocycle. It is Cartier-fixed. Indeed, the intrinsic inverse Cartier formula gives
[
C^{-1}(g^{-1}dg)
=[g^{-p}g^{p-1}dg]=[g^{-1}dg]
]
for every Laurent unit g.

By Lemma~lem:cartier-adjustment, after refining the cover there are closed one-forms alpha_i such that
[
alpha_j-alpha_i=dlog g_{ij}.
]
These are the connection forms of a connection defined by nabla e_i=alpha_i ⊗  e_i. Their closedness is precisely its integrability, since the bundle has rank one.

The forms Calpha_i-alpha_i agree on overlaps, hence define a global form etain W. Choose betain W with (C-id)beta=eta using Lemma~lem:cartier-adjustment. Replace every alpha_i by alpha_i-beta. This preserves their differences and their closedness, and now Calpha_i=alpha_i for all i.


## Horizontal sections over a Laurent chart

We give the local descent calculation, including compatibility with restriction. This is the algebra underlying Cartier descent cite[Theorem~5.1 and formulas (5.1.2)--(5.1.7)]{katz1970}; the formulas below apply directly to our Laurent rings because they retain a (p)-basis, without a finite-type hypothesis on those rings.

Take a smooth affine chart U=Spec Asubset V with 'etale coordinates x_1,x_2,x_3, and let f be an equation of D on it. Put
[
B=varprojlim_n A/(f^n),

R=B[1/f],

S=R^p.
]
Lemma~lem:laurent-cartier-algebra gives

R=⊕_{0le I_1,I_2,I_3<p}Sx^I,

Ω^1_{R/k}=⊕_{i=1}^3 R,dx_i.

Write ∂_i for the coordinate derivations. They commute, kill S, and satisfy ∂_i^p=0.

[The rank-one curvature calculation]
If alpha=sum_i a_i,dx_i is closed and Calpha=alpha, then the connection operators
[
D_i=∂_i+a_i:R→ R
]
commute and satisfy

D_i^p=0,

[D_i,x_j]=delta_{ij},

where a ring element in an operator formula denotes multiplication by that element.


Proof. 
If Calpha=sum_i b_i,dx_i, the p-basis Cartier calculation represents alpha in the form
[
alpha=dc+sum_i b_i^p x_i^{p-1},dx_i
(cin R).
]
Consequently

∂_i^{p-1}a_i=(p-1)!,b_i^p=-b_i^p.

Exact terms disappear because ∂_i^p c=0, and b_i^p is constant for every coordinate derivation. The Cartier-fixed condition therefore says
[
a_i^p+∂_i^{p-1}a_i=0.
]

We recall the operator identity giving its significance. For a derivation ∂ with ∂^p=0 and an element ain R,

(∂+a)^p=a^p+∂^{p-1}a.

First, the left side commutes with multiplication by any bin R: in characteristic p, its commutator is the p-fold iterated commutator, namely multiplication by ∂^pb=0. It is therefore multiplication by its value on 1. Over the integers, successive use of the product rule expands (∂+a)^p(1) as a sum indexed by set partitions of {1,...,p}, a block of size j contributing ∂^{j-1}a. A cyclic permutation of the p labels has orbits of size p on these partitions, except for the discrete partition and the partition with one block; the summands have the same value throughout an orbit, so only a^p and ∂^{p-1}a survive in characteristic p. This proves (eq:rank-one-pcurvature), hence D_i^p=0. Finally,
[
[D_i,D_j]=∂_i a_j-∂_j a_i=0
]
by closedness, and the commutator with x_j follows directly from the Leibniz rule.


[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_i: Mto M)
]
is a finite projective rank-one S-module, and multiplication gives an isomorphism

R ⊗ _S H_Mxrightarrow{ sim }M.

Formation of H_M commutes with restriction to a smaller Laurent chart retaining the coordinates.


Proof. 
Define S-linear operators

P_i=sum_{n=0}^{p-1}((-1)^n)/(n!)x_i^n D_i^n,

P=P_1P_2P_3,

with the multiplication operator x_i^n on the left. All factorials are invertible in k. The commutator relation gives a telescoping sum
[
D_iP_i=frac{(-1)^{p-1}}{(p-1)!}x_i^{p-1}D_i^p=0.
]
Also P_i is the identity on ker D_i, and distinct P_i's commute with each other and with D_j for jne i. Hence P is an idempotent with image H_M.

For one coordinate, the binomial identity gives

sum_{n=0}^{p-1}(x_i^n)/(n!)P_iD_i^n=id_M.

Indeed, after substituting (eq:horizontal-projector), the coefficient of x_i^rD_i^r for 0le r<p is
[
sum_{n+j=r}((-1)^j)/(n!j!)
=1&r=0,0&r>0,
]
and all terms with rge p vanish by D_i^p=0. Multiplying (eq:horizontal-taylor-one) over the three coordinates gives the explicit expansion

m=sum_{0le I_j<p}x^Ih_I,

h_I=frac1{I!}P(D^I m)in H_M,

with I!=I_1!I_2!I_3! and D^I=D_1^{I_1}D_2^{I_2}D_3^{I_3}. The coefficients are unique: for hin H_M,
[
PD^J(x^Ih)=I!h&I=J,0&Ine J,
 0le I_i,J_i<p.
]
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 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

The projective module in Lemma~lem:horizontal-expansion must be locally trivial on the topology of |D|. The next elementary completion fact supplies this step.


For the smooth affine chart and rings A,B,R above, B is a regular Noetherian ring. Every finite projective rank-one R-module extends to a finite projective rank-one B-module. Its associated line is locally trivial after restriction to formal Zariski neighborhoods of every point of Ucap|D|. The same assertions hold for B^psubset R^p.


Proof. 
The Noetherian completion theorem gives that B is Noetherian and flat over A, and B/(f^n)=A/(f^n) cite[Lemmas~10.97.1--10.97.5]{stacks}. The element f belongs to the Jacobson radical of B: an element congruent to 1 modulo f has an inverse by the convergent geometric series, and more generally an element invertible modulo f is invertible in B cite[Lemma~10.96.6]{stacks}.

Every maximal ideal nsubset B corresponds to a maximal ideal msubset A containing f, with n=n=mB.mB. For each s, completion modulo m^sfrakm^s does nothing, since f^s already vanishes there. Hence
[
B/n^sfrakn^s~= A/m^s,m^s,

hathat{B_B_{n}n}~=hathat{A_A_{m}m}.
]
The latter ring is regular, as A_{m}m 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_{n}n is regular, and localization gives regularity at every prime of B.

A regular Noetherian ring is locally factorial cite[Lemma~15.123.2]{stacks}. Its finitely many irreducible components are disjoint, since each local ring is a domain. On each component choose a rational section of the given line on Spec R and take its Weil divisor. The closure of this finite Weil divisor on Spec B is Cartier by local factoriality. Doing this on all components gives the required extension. This is the Cartier--Weil correspondence on a regular scheme cite[II, S6]{hartshorne1977}; global unique factorization of B is not required.

Let zin Ucap|D|. The extended line is free on a basic open Spec B_g containing the prime corresponding to z. Choose g_0in A whose image modulo f equals that of g. Then g_0(z)ne0. On the formal basic neighborhood coming from Spec A_{g_0}, its ring is B'=hathat{A_A_{g_0}}, and the image of g is invertible: modulo f it equals the invertible element g_0. The line is therefore free there and after inverting f. Finally, Frobenius is an abstract ring isomorphism from B onto B^p and from R onto R^p, because these rings are reduced. It identifies the corresponding formal basic neighborhoods. Thus the same proof applies to the power subrings.


[Divisibility of the Laurent Picard group]
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}cH~=cL.
]

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
[
cL~=cN^{ ⊗  p}.
]
This is a global root on the stated ringed space, proving the surjectivity assertion.


# An integral degree and the contradiction


Infinite Frobenius divisibility forces integral numerical invariants to vanish; this is the numerical mechanism for stratified bundles in cite[Lemma~2.1 and Corollary~2.2]{esnaultmehta2010}. Here the required degree is defined first on the completion. We must prove that it descends to the Laurent Picard group before applying that elementary divisibility obstruction.

## Orders of Laurent units


Let iota_alpha: D_alpha↪|D| be the inclusion of a prime component. There is an exact sequence of abelian sheaves

1→Ohat^{,times}
→cR^times
xrightarrow{ ord }
⊕_alpha(iota_alpha)_*Z
→0.

Consequently restriction gives a surjection
[
Pic(|D|,Ohat)→Pic(|D|,cR),
]
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)~= 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.

If uin B[1/f]^times, write u=a/f^r and u^{-1}=b/f^s, with a,bin B. Since f is a non-zero-divisor, ab=f^{r+s} in B. Successively use primality to cancel each of the finitely many prime factors of the right side from either a or b. At the end the remaining factors multiply to a unit. This expresses u uniquely as

u=vprod_alpha f_alpha^{n_alpha},
 vin B^times, n_alphainZ.

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 O_V(D_alpha), as is seen from their local equations. Exactness proves the kernel assertion.


## The degree contradiction


The prepared cycle of Proposition~prop:prepared-cycle cannot satisfy the conclusions of Corollary~cor:form-vanishing.


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})· 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(O_V(D_beta))
=sum_alpha m_alpha D_alpha D_beta H
=DD_beta H=0.
]
Lemma~lem:laurent-unit-orders therefore makes Phi descend to a homomorphism
[
overlinePhi:Pic(|D|,cR)→Z.
]
It is nonzero, since

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).


## Conclusion in the maximal-nef-dimension case

The assumption of maximal nef dimension in Proposition~prop:prepared-cycle supplied the prepared cycle and all the conclusions of Corollary~cor:form-vanishing. Proposition~prop:laurent-degree-contradiction rules out that assumption. Thus n(X,K_X)le2 over the uncountable algebraically closed extension used in the preparation. The nef-dimension-at-most-two abundance theorem cite[Theorem~1.8]{xu2024} applies to the projective log canonical threefold pair (X,0) in characteristic p>3 with nef canonical divisor. It makes K_X semiample there.

# 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 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,O_X(mK_X)) ⊗ _{k_0}k
~=
H^0(X_k,O_{X_k}(mK_{X_k})).
]
Their dimensions are equal, proving the required ineuality over k_0. In particular kappa(X,K_X)ge1.

Nefness and nu(K_X)=1 give kappa(X,K_X)=1. The positive-Kodaira-dimension abundance theorem cite[Theorem~1.6]{xu2024} also gives semiampleness over the original field. This completes the proof of Theorem~thm:main.

{99}


T.~Bauer, F.~Campana, T.~Eckl, S.~Kebekus, T.~Peternell, S.~Rams, T.~Szemberg, L.~Wotzlaw:
A reduction map for nef line bundles. In: Complex Geometry, Springer, 2002, 27--36.


C.~Birkar: Existence of flips and minimal models for 3-folds in characteristic p. Ann. Sci. 'ENS (4) 49 (2016), 169--212.


P.~Cartier: Une nouvelle op'eration sur les formes diff'erentielles. C.~R. Acad. Sci. Paris 244 (1957), 426--428.


V.~Cossart, U.~Jannsen, S.~Saito: Canonical embedded and non-embedded resolution of singularities for excellent two-dimensional schemes. arXiv:0905.2191 (2009).


V.~Cossart, U.~Jannsen, S.~Saito: Desingularization: Invariants and Strategy, LNM 2270, Springer, 2020.


V.~Cossart, O.~Piltant: Resolution of singularities of threefolds in positive characteristic I. J. Algebra 320 (2008), 1051--1082.


V.~Cossart: Resolution of singularities of threefolds in positive characteristic, lecture notes, RIMS Kyoto, 2008.


V.~Cossart, O.~Piltant: Resolution of singularities of threefolds in positive characteristic II. J. Algebra 321 (2009), 1836--1976.


V.~Cossart, O.~Piltant: Resolution of singularities of arithmetical threefolds. J. Algebra 529 (2019), 268--535.


A.~Grothendieck: 'El'ements de g'eom'etrie alg'ebrique III. Publ. Math. IH'ES 11 (1961), 5--167.


A.~Grothendieck: 'El'ements de g'eom'etrie alg'ebrique III. Publ. Math. IH'ES 17 (1963), 5--91.


H.~Esnault, V.~Mehta: Simply connected projective manifolds in characteristic p>0 have no nontrivial stratified bundles. Invent. Math. 181 (2010), 449--465.


W.~Fulton: Intersection Theory, 2nd ed., Ergebnisse 2, Springer, 1998.


C.~Hacon, C.~Xu: On the three dimensional minimal model program in positive characteristic. J. Amer. Math. Soc. 28 (2015), 711--744.


K.~Hashizume, Y.~Nakamura, H.~Tanaka: Minimal model program for log canonical threefolds in positive characteristic. Math. Res. Lett. 27 (2020), 1003--1054.


R.~Hartshorne: Residues and Duality, LNM 20, Springer, 1966.


R.~Hartshorne: Local Cohomology, LNM 41, Springer, 1967.


R.~Hartshorne: Algebraic Geometry, GTM 52, Springer, 1977.


C.~Hacon, J.~Witaszek: The minimal model program for threefolds in characteristic 5. Duke Math. J. 171 (2022), 2193--2231.


E.~Jovinelly, B.~Lehmann, E.~Riedl: Optimal bounds in bend-and-break. Forum of Math. Pi 14 (2026), e16.


J.-P.~Jouanolou: Th'eor`emes de Bertini et applications, Progress in Math. 42, Birkh"auser, 1983.


N.~M.~Katz: Nilpotent connections and the monodromy theorem. Publ. Math. IH'ES 39 (1970), 175--232.


Y.~Kawamata: Abundance theorem for minimal threefolds. Invent. Math. 108 (1992), 229--246.


S.~L.~Kleiman: Toward a numerical theory of ampleness. Ann. Math. 84 (1966), 293--344.


J.~Koll'ar: Rational Curves on Algebraic Varieties, Ergebnisse 32, Springer, 1996.


J.~Koll'ar, S.~Mori: Birational Geometry of Algebraic Varieties, Cambridge Tracts 134, CUP, 1998.


A.~Langer: Semistable sheaves in positive characteristic. Ann. Math. 159 (2004), 251--276.


A.~Langer: Addendum to ``Semistable sheaves in positive characteristic''. Ann. Math. 160 (2004), 1211--1213.


A.~Langer: Generic positivity and foliations in positive characteristic. Adv. Math. 277 (2015), 1--23.


J.~Liu, Z.~Xu: Non-vanishing implies numerical dimension one abundance. arXiv:2505.05250 (2025).


Y.~Miyaoka: Abundance conjecture for 3-folds: case nu=1. Compositio Math. 68 (1988), 203--220.


Q.~Posva: Abundance for slc surfaces over arbitrary fields. 'Epijournal G'eom. Alg'ebrique 7 (2023), art.~5.


The Stacks Project Authors: The Stacks Project, https://stacks.math.columbia.edu (accessed 2026).


H.~Tanaka: Abundance theorem for semi log canonical surfaces in positive characteristic. Osaka J. Math. 53 (2016), 535--566.


B.~Totaro: Moving codimension-one subvarieties over finite fields. arXiv:0712.2049 (2007).


J.~Witaszek: On the canonical bundle formula and log abundance in positive characteristic. Math. Ann. 381 (2021), 1309--1344.


Z.~Xu: Note on the three-dimensional log canonical abundance in characteristic >3. Nagoya Math. J. 255 (2024), 694--723.


Z.~Xu: Abundance for threefolds in positive characteristic when nu=2. arXiv:2307.03938v3 (2026).


C.~Xu, L.~Zhang: Nonvanishing for threefolds in characteristic p>5. Duke Math. J. 168 (2019), 1269--1301.


L.~Zhang: Abundance for 3-folds with non-trivial Albanese maps in positive characteristic. J. Eur. Math. Soc. 22 (2020), 2777--2820.