Proof. The exceptional fiber is connected, by proper birationality and normality of \(P\). Choose a nonzero nonunit \(t\in\mathcal O_P\) and write \[\operatorname{div}_T(t)=A+\sum_i c_iG_i,\qquad c_i>0,\] where \(A\) is the strict-transform part. It meets the exceptional fiber: the strict transform of a component of the divisor of \(t\) has a point over the closed point by properness. For \(C=\sum_i c_iG_i\) we have \[C\cdot G_i=-A\cdot G_i\leq0.\] Distinct curves have nonnegative intersections. Connectedness therefore shows \(G_i^2<0\): 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 this maximum is attained. Choose \(j\in I\) which either meets \(A\) or has a neighbor outside \(I\). Such a vertex exists by connectedness and the fact that \(A\) meets the fiber. Then \[Z\cdot G_j =aC\cdot G_j+\sum_i(z_i-ac_i)G_i\cdot G_j<0.\] The term with \(i=j\) is zero; all other terms are nonpositive, and the chosen vertex makes at least one term strictly negative. ◻

Proof. Localize at a codimension-two point. This gives an excellent normal local surface \(P\), with residue field \(l\), and a projective regular resolution \(g:T\to P\) obtained by localizing a threefold resolution. The residue field \(l\) need not be perfect. We keep all intersection degrees over \(l\).

For an integral exceptional curve \(G\), divisor adjunction and curve duality give \[ (K_T+G)\cdot G =2\dim_lH^1(G,\mathcal O_G)-2\dim_lH^0(G,\mathcal O_G). \tag{7}\] To specify the duality in this formula, use the dualizing complex on \(P\) localized from the threefold and its exceptional inverse image on \(T\). Transitivity for \(G\to T\to P\) and \(G\to\mathop{\mathrm{Spec}}l\to P\) identifies the adjunction sheaf on \(G\) with its dualizing sheaf over \(l\), up to tensoring with a one-dimensional \(l\)-vector space. This does not change degree. Proper duality and the formula \(\chi(\mathcal L)-\chi(\mathcal O_G)=\deg_l\mathcal L\) prove (7) (Hartshorne 1966, 1977).

Suppose \(K_T\cdot G<0\), and set \(l'=H^0(G,\mathcal O_G)\) and \(d=[l':l]\). Both \(G^2\) and \(K_T\cdot G\) are negative multiples of \(d\). Equation (7) and Lemma 3 force \[H^1(G,\mathcal O_G)=0,\qquad G^2=K_T\cdot G=-d.\] The line bundle \(\mathcal O_G(-G)\) consequently has degree one over \(l'\). Riemann–Roch gives at least two independent sections over \(l'\). Every nonzero section has a zero scheme of length one. Two independent sections have no common zero: otherwise their effective Cartier divisors coincide, so their ratio is a global unit, an element of \(l'^\times\). They define a nonconstant map \(G\to\mathbb P^1_{l'}\) of degree one. It is finite, and a finite birational morphism to a normal curve is an isomorphism. Hence \[G\simeq\mathbb P^1_{l'},\qquad \mathcal O_G(G)\simeq\mathcal O_{\mathbb P^1_{l'}}(-1).\]

For completeness this curve can be contracted while retaining a projective regular resolution over \(P\). Choose a relatively very ample Cartier divisor \(A\), put \(a=\deg_{l'}(A|_G)>0\), and set \(Q=A+aG\). The divisor \(Q-G\) has positive degree on every component of the closed fiber, and is relatively ample by the fiber criterion for ampleness (Grothendieck 1961). The restriction \(Q|_G\) is trivial. For every \(n\geq1\), a trivializing section of \(\mathcal O_G(nQ)\) extends successively to \(nG\), because the obstruction groups are \[H^1\bigl(G,\mathcal O_G(nQ-jG)\bigr) =H^1(\mathbb P^1_{l'},\mathcal O(j))=0\qquad(j\geq1).\] For large \(n\), relative Serre vanishing applied to \(n(Q-G)\) lifts these sections from \(nG\) to \(T\). They generate along \(G\). Away from \(G\), the sections of \(n(Q-G)\), multiplied by the canonical section of \(nG\), generate and separate points and tangent directions. Thus \(nQ\) defines a projective contraction whose only nontrivial fiber is \(G\). Its Stein factorization has normal target and is an isomorphism off \(G\).

Let \(q\) be the image of \(G\). Formal functions identifies the completed local ring at \(q\) with \[\varprojlim_n H^0(nG,\mathcal O_{nG}).\] The \(G\)-adic and inverse-image \(q\)-adic neighborhoods are cofinal, since they have the same support. The successive kernels have graded pieces \[H^0(G,\mathcal O_G(-nG))=H^0(\mathbb P^1_{l'},\mathcal O(n)).\] There is no obstruction in \(H^1\), so the associated graded algebra is the polynomial ring \(l'[x,y]\). More explicitly, let \(F_n\) be the kernel of the map from the completed local ring to \(H^0(nG,\mathcal O_{nG})\). Then \(F_1\) is its maximal ideal, and the \(F_n\)-filtration is cofinal with the maximal-ideal-adic filtration. Choose lifts of the two degree-one generators. Generation of the graded algebra gives \(F_1\subset(x,y)+F_n\) for every \(n\), by successive approximation, using arbitrary lifts of coefficients from \(l'\). Finitely generated ideals in the complete Noetherian local ring are closed, so \(F_1=(x,y)\). No embedding of the residue field \(l'\) into the completed local ring is required. Its dimension is two; hence it, and then the local ring at \(q\), is regular. This proves the claimed contraction using formal functions and Serre vanishing (Hartshorne 1977, III, Sections 5 and 11).

There are only finitely many exceptional curves. Contracting a \(K_T\)-negative one whenever possible therefore yields a regular resolution with \(K_T\) relatively nef. If any exceptional curves remain, terminality gives \[E=K_T-g^*K_P=\sum_i a_iG_i,\qquad a_i>0.\] The discrepancies of surviving curves have not changed, since they are read at their generic points. Lemma 3 gives \(E\cdot G_j<0\) for some \(j\), contradicting relative nefness. There are no exceptional curves left, and the proper birational map is an isomorphism. Thus the codimension-two point was regular. Normality already gives regularity in codimension one, so the remaining singular locus of the projective threefold is finite. ◻

Proof. Replacing \(Z\) by \(Z\cap|L|\), we may assume \(Z\subset|L|\). For \(z\in Z\), put \(R=\mathcal O_{Y,z}\) and let \(\mathfrak m\) be its maximal ideal. Normality gives depth at least two, so \[H^1_{\mathfrak m}(R)=0.\] Every nonmaximal localization of \(R\) is a normal local ring of dimension at most two, and is consequently Cohen–Macaulay. This implies that \(H^2_{\mathfrak m}(R)\) has finite length. Explicitly, present \(R=A/I\), where \(A\) is a regular local ring of dimension \(n\), essentially of finite type over the field. The height of \(I\) is \(n-3\). At a nonmaximal prime in \(\mathop{\mathrm{Spec}}R\), catenarity and Auslander–Buchsbaum give projective dimension \(n-3\) over the corresponding localization of \(A\). Hence \(\mathop{\mathrm{Ext}}_A^{n-2}(R,A)\) is supported only at the closed point. It is a finite module, thus has finite length. Local duality identifies \(H^2_{\mathfrak m}(R)\) with its Matlis dual, up to completion, and proves the assertion (Hartshorne 1967).

Let \(t_z\) be a local equation of \(L\). Choose \(j\) so large that \(t_z^{j-1}\) kills \(H^2_{\mathfrak m}(R)\) for every \(z\in Z\). There are only finitely many points, so the same \(j\) works at all of them. The restriction sequences fit into a commutative diagram \[\begin{tikzcd}[column sep=small] 0 \arrow[r] & \mathcal O_Y((u-j)L) \arrow[r] \arrow[d] & \mathcal O_Y(uL) \arrow[r] \arrow[d,equal] & \mathcal O_{jL}(uL) \arrow[r] \arrow[d] & 0\\ 0 \arrow[r] & \mathcal O_Y((u-1)L) \arrow[r] & \mathcal O_Y(uL) \arrow[r] & \mathcal O_L(uL) \arrow[r] & 0 . \end{tikzcd}\] In local trivializations the left vertical map is multiplication by \(t_z^{j-1}\). The vanishing of the ambient \(H^1_Z\) makes the connecting maps from the \(H^1_Z\) of the rightmost terms to the ambient \(H^2_Z\) injective. The choice of \(j\) therefore gives \[ H^1_Z(Y,\mathcal O_{jL}(uL)) \longrightarrow H^1_Z(Y,\mathcal O_L(uL)) \quad\text{equal to zero for every integer }u . \tag{12}\] The assertion is uniform in \(u\): twisting by \(uL\) only changes local trivializations of these finite-length modules.

Fix this \(j\). The kernel of \(\mathcal O_{jL\setminus Z}\to\mathcal O_{L_{\mathrm{red}}\setminus Z}\) is nilpotent. Lift a chosen generator of the trivial line \(\mathcal O_Y(L)|_{L_{\mathrm{red}}\setminus Z}\) to local generating sections on a finite open cover of \(jL\setminus Z\). On overlaps their differences are 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. The powers therefore give a trivializing section \[s_j\in H^0(jL\setminus Z,\mathcal O_{jL}(uL)).\] Project it to a section \(s\) on \(L\setminus Z\). In the exact sequence of sections with support in \(Z\), its obstruction to extending over \(L\) is the image of the obstruction for \(s_j\). Equation (12) kills that image. Thus \(s\) extends to \(H^0(L,\mathcal O_L(uL))\).

This extension is a generator everywhere. On each integral projective surface component \(T\) of \(L_{\mathrm{red}}\) 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: a nonzero section of an invertible sheaf on an integral scheme is a non-zero-divisor in local trivializations. Such a divisor has positive degree against an ample class, whereas \(\mathcal O_T(uL)\) is numerically trivial. This is impossible. No normality of \(T\) is needed for this intersection argument. Nonvanishing after reduction implies invertibility on \(L\), by the local criterion for a unit modulo a nilpotent ideal. We conclude \(\mathcal O_L(uL)\simeq\mathcal O_L\), as required. ◻

Proof. The induced map is \(\mathcal O_V\)-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 of the ambient generic vector space. In a Grassmannian chart, represent this subspace by a matrix \((I\mid M)\). Horizontality says that every entry of \(M\) has zero differential. Since \(K\) is separably generated over the perfect field \(k\), the kernel of \(d\colon K\to\Omega_{K/k}^1\) is \(K^p\). Taking \(p\)-th roots of the entries of \(M\) therefore defines a subspace before Frobenius pullback.

Saturate this subspace to a subsheaf \(B_0\subset E\). At every codimension-one point the relevant torsion-free quotients are free over the discrete valuation ring. Frobenius is flat on the smooth variety, so \(F_V^*B_0\) is saturated there and agrees with \(B\) there: they have the same generic subspace. Consequently \[\deg(B)=p\deg(B_0),\qquad \mu(B_0)=\frac{\mu(B)}p> \frac{\mu(F_V^*E)}p=\mu(E),\] contradicting semistability of \(E\). ◻

Proof. Put \(Q=T_V/A\) and \(a=\mu(A)\). By Equation (15), \(\mu(T_V)=0\), so \[a>0,\qquad \mu_{\max}(Q)<a.\] The rank of \(A\) is one or two. Work where \(A\) and \(Q\) are bundles. The bracket modulo \(A\) is an \(\mathcal O_V\)-linear map \(\bigwedge^2A\to Q\). It vanishes automatically in rank one. In rank two, \(Q\) has rank one and slope \(-2a\), whereas \(\bigwedge^2A\) has slope \(2a\); Equation (16) again makes the map zero.

Once bracket closure holds, the \(p\)-th power of derivations modulo \(A\) defines an \(\mathcal O_V\)-linear map \[ F_V^*A\longrightarrow Q. \tag{17}\] For completeness, the two restricted-Lie identities needed for this assertion can be seen directly. For local derivations \(u,v\in A\), set \(w=tu+v\) with \(t\) a formal scalar. In characteristic \(p\), \[\frac{d}{dt}w^p =\sum_{j=0}^{p-1}w^juw^{p-1-j} =\operatorname{ad}(w)^{p-1}(u).\] The last equality follows from \(\binom{p-1}{j}\equiv(-1)^j\pmod p\). Every coefficient of the right-hand side lies in the bracket-closed module \(A\). The mixed coefficients of \(w^p-t^pu^p-v^p\), whose degrees in \(t\) lie between \(1\) and \(p-1\), thus belong to \(A\). This proves additivity modulo \(A\).

For a local function \(f\), expanding \((fu)^p\) as a differential operator 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 \(u\ne0\), choose \(x\) with \(u(x)\ne0\). For a largest index \(j\ge2\) with \(c_j\ne0\), the \(j\)-fold commutator with multiplication by \(x\) is \(j!c_j(u(x))^j\ne0\), while it must vanish for a derivation. Hence only the \(j=1\) term can remain. The scalar discrepancy is therefore a multiple of \(u\), proving Frobenius linearity modulo \(A\) and hence Equation (17).

If \(\mathop{\mathrm{rk}}A=1\), its source has slope \(pa>a>\mu_{\max}(Q)\) and the map is zero. Suppose \(\mathop{\mathrm{rk}}A=2\). In this case \[\mu(Q)=-2a,\qquad \mu_{\max}(\Omega_V^1)=2a.\] If \(F_V^*A\) is semistable, its minimum slope is \(pa>-2a\). Otherwise let \(L\) be its first Harder–Narasimhan term and let \(R=F_V^*A/L\); both have rank one. Lemma 6 supplies a nonzero map \[L\longrightarrow R\otimes\Omega_V^1.\] Since \(R\) has rank one, the slope rules give \[\mu(L)-\mu(R)\le2a,\qquad \mu(L)+\mu(R)=2pa.\] It follows that \[\mu_{\min}(F_V^*A)=\mu(R)\ge pa-a>-2a.\] Thus Equation (16) makes Equation (17) zero in either case. ◻

Proof. On an affine open with ring \(R\), let \[P=\{f\in R: u(f)=0\text{ for every local section }u\in A\}.\] The ring \(P\) lies between \(R^p\) and \(R\). Since \(R\) is finite over \(R^p\), \(P\) is finite over \(R^p\) and \(R\) is finite over \(P\). These invariant rings glue to the asserted factorization of Frobenius; in particular both maps are finite and their underlying maps of topological spaces are homeomorphisms. The ring \(P\) is normal. Indeed, an element of \(\mathop{\mathrm{Frac}}(P)\) integral over \(P\) is integral over \(R\), hence belongs to the normal ring \(R\), and all the derivations annihilate it. Thus it belongs to \(P\). The variety \(W\) is projective because it is finite over the projective variety \(V^{(1)}\).

Remove the codimension-at-least-two loci where \(A\) or \(T_V/A\) fails to be locally free. Near any remaining point choose functions \(x_1,\ldots,x_d\) whose differentials restrict to a basis of \(A^\vee\), and choose the dual local derivations \(u_1,\ldots,u_d\). Thus \(u_i(x_j)=\delta_{ij}\). Bracket closure implies \([u_i,u_j]\in A\); this bracket annihilates every \(x_l\) and is therefore zero. Likewise \(u_i^p\in A\) annihilates all the \(x_l\), so \(u_i^p=0\).

These commuting nilpotent derivations give an explicit invariant-ring presentation. For one derivation \(u\) with \(u(x)=1\), the coefficient \((p-1)!^{-1}u^{p-1}(f)\) lies in its kernel. Subtract its product with \(x^{p-1}\) and repeat with successively smaller powers. This writes every \(f\) uniquely as a polynomial of degree less than \(p\) in \(x\), with coefficients in the kernel of \(u\). Applying this procedure successively to the commuting \(u_i\) gives \[ R=\bigoplus_{0\le I_1,\ldots,I_d<p}P\,x^I \simeq P[X_1,\ldots,X_d]/(X_1^p-c_1,\ldots,X_d^p-c_d), \qquad c_i=x_i^p\in P. \tag{19}\] Uniqueness follows by applying the same successive highest derivatives to a putative relation. In particular, \(\pi\) is locally free of degree \(p^d\) on this open set.

We may further remove the singular locus of \(W\) and its inverse image; normality and finiteness ensure that these removed sets also have codimension at least two. The presentation in Equation (19) now yields the exact sequence of vector bundles \[ 0\longrightarrow F_V^*(A^\vee) \longrightarrow\pi^*\Omega_W^1 \longrightarrow\Omega_V^1 \longrightarrow A^\vee\longrightarrow0. \tag{20}\] Here the last map is evaluation on \(A\). To verify local freeness of the first image, put \(N=\langle dc_1,\ldots,dc_d\rangle\) inside \(R\otimes_P\Omega_{P/k}^1\). The presentation gives \[\Omega_{R/k}^1\simeq \bigl((R\otimes_P\Omega_{P/k}^1)/N\bigr) \oplus\bigoplus_{i=1}^dR\,dx_i,\] because the relations have no differential in the \(X_i\) directions. 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 \(R\otimes_P\Omega_{P/k}^1\). The displayed \(d\) generators of \(N\) are a basis: locally their surjection from \(R^d\) onto a free module of rank \(d\) is an isomorphism. This proves the required exactness at every point of the chosen open set.

To identify its transition maps, let \(x'_j\) be another such set of functions and write it in the basis \(x^I\) over \(P\). Taking \(p\)-th powers and differentiating downstairs shows that, after pullback, the coefficient of \(dc_l\) in \(d((x'_j)^p)\) is \(u_l(x'_j)^p\). On the other hand, the coefficient of the \(l\)-th basis vector of \(A^\vee\) in \(dx'_j|_A\) is \(u_l(x'_j)\). The first term of Equation (20) is therefore precisely \(F_V^*(A^\vee)\). Taking determinants in that sequence gives \[\pi^*\omega_W \simeq(\det A)^{-p}\otimes\omega_V\otimes\det A,\] which is Equation (18). ◻

Proof. We first prove semistability by the foliation and rational-curve argument of (Xu and Zhang 2019, Lemma 2.10), with the quantitative bend-and-break statement specified below. Suppose that \(A\) is the first destabilizing term, with \(a=\mu(A)>0\) and \(d=\mathop{\mathrm{rk}}A\). By Lemmas 7 and 8, it has a finite purely inseparable quotient \(\pi\colon V\to W\). On the big smooth open set where Equation (18) holds, put \(\Lambda=-\pi^*K_W\). Its extension as a divisor class on \(V\) is \[\Lambda=(p-1)c_1(A)-K_V, \qquad \Lambda HD=(p-1)da>0.\] No global Cartier property of \(K_W\) is needed: its degree will be taken only on curves contained in this smooth open set.

Let \(H^{(1)}\) and \(D^{(1)}\) denote the corresponding divisor classes on \(V^{(1)}\), and set \[H'=\rho^*H^{(1)},\qquad D'=\rho^*D^{(1)}.\] Then \(H'\) is ample and Cartier, \(D'\) is nef and Cartier, and \[\pi^*H'=pH,\qquad \pi^*D'=pD.\] For a positive integer \(t\), take sufficiently divisible positive multiples of \(H'\) and \(H'+tD'\) which are very ample. Their general complete intersection is a smooth projective integral curve \(C_t\) wholly contained in the open set under consideration. Here is the avoidance point in this assertion. The complement 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 finitely many points. Bertini on the smooth open set, and irreducibility for the successive very ample sections, give the required curve.

Projection formula, computed where \(\pi\) is locally free, cancels both the degree of \(\pi\) and the chosen complete-intersection multiples from the degree ratio. It gives \[\begin{align*} \frac{D'C_t}{-K_WC_t} &=\frac{pD\,H(H+tD)}{\Lambda H(H+tD)} \\ &=\frac{pDH^2}{\Lambda H^2+t\Lambda HD} \longrightarrow0 \qquad(t\longrightarrow\infty). \tag{22}\end{align*}\] The denominator is positive for all sufficiently large \(t\).

We use the following quantitative form of bend-and-break (Jovinelly et al. 2026, Theorem 1.1): if \(Y\) is projective over an algebraically closed field, \(C\subset Y_{\mathrm{sm}}\) 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 \(\Gamma\subset Y\) satisfying \[N\Gamma\le(\dim Y+1)\frac{NC}{-K_YC}.\] This statement is valid in arbitrary characteristic. Choose \(t\) so large that the ratio in Equation (22) is less than \(1/4\). Applied on \(W\) with \(N=D'\), the theorem gives, through every closed point \(x\in C_t\), a rational curve \(\Gamma\) with \[0\le D'\Gamma<1.\] As \(D'\) is Cartier, this degree is an integer and hence is zero.

We choose \(C_t\) to witness the very-general-point contradiction as well. Fix the product of the two chosen very ample linear systems, and let \(\mathcal T\) be its nonempty open locus of smooth integral complete intersections avoiding the bad locus above. This parameter space is irreducible. Its universal curve \(\mathcal C\to\mathcal T\) is smooth: each fiber is a transverse complete intersection inside \(W_{\mathrm{sm}}\), so the relative Jacobian criterion applies. Its evaluation map \(\mathcal C\to W\) is dominant. Indeed the ambient incidence of points and pairs of divisors through them is a product of projective bundles over \(W\), hence is irreducible and surjects onto \(W\); restricting it to the nonempty open parameter locus \(\mathcal T\) leaves a dense open subset, whose evaluation still has dense image.

Let \(Z_1,Z_2,\ldots\) be proper closed subsets of \(V\) outside whose union no \(D\)-trivial curve passes. The finite map \(\pi\) has proper closed images \(\pi(Z_j)\). For each \(j\), the locus in \(\mathcal T\) of curves not contained in \(\pi(Z_j)\) is the image of \(\mathcal C\setminus\operatorname{ev}^{-1}(\pi(Z_j))\) under the smooth map \(\mathcal C\to\mathcal T\). It is open because smooth maps are open, and nonempty because evaluation is dominant. Over the uncountable field, the countably many resulting nonempty opens in \(\mathcal T\) have a common \(k\)-point. On its integral curve \(C_t\), each \(C_t\cap\pi(Z_j)\) is finite. Choose a closed point \(x\) outside their union. Its unique inverse-image point \(y\in V\) lies outside every \(Z_j\).

The reduced inverse image of the rational curve \(\Gamma\) through \(x\) contains a curve \(\widetilde\Gamma\) through \(y\), finite and surjective over \(\Gamma\). Projection formula gives \[pD\widetilde\Gamma =(\pi^*D')\widetilde\Gamma =\deg(\widetilde\Gamma/\Gamma)D'\Gamma=0.\] This contradicts the choice of \(y\). Thus \(T_V\) is semistable.

Suppose next that \(e\ge1\) is the first integer for which \(F_V^{e*}T_V\) is unstable. Let \(A'\) be its first Harder–Narasimhan term and \(Q'=F_V^{e*}T_V/A'\). Applying Lemma 6 to the semistable sheaf \(F_V^{(e-1)*}T_V\) gives a nonzero map \[A'\longrightarrow Q'\otimes\Omega_V^1, \qquad\text{equivalently}\qquad A'\otimes(Q')^\vee\longrightarrow\Omega_V^1\] on a big open set. The cotangent bundle is semistable of slope zero, because \(T_V\) is. Since \(\mathop{\mathrm{rk}}A'+\mathop{\mathrm{rk}}Q'=3\), at least one tensor factor in the second source has rank one. If \(\mathop{\mathrm{rk}}Q'=1\), semistability of \(A'\) and the rank-one twisting rule give the minimum slope of the source as \(\mu(A')-\mu(Q')\). If \(\mathop{\mathrm{rk}}A'=1\), the quotient \(Q'\) need not be semistable; use instead \[\mu_{\min}((Q')^\vee)=-\mu_{\max}(Q')\] and twist by the rank-one factor \(A'\). In both cases, \[\mu_{\min}\bigl(A'\otimes(Q')^\vee\bigr) =\mu(A')-\mu_{\max}(Q')>0.\] This contradicts Equation (16). All Frobenius pullbacks of \(T_V\) are therefore semistable.

Finally apply Langer’s Bogomolov inequality (Langer 2004b, Theorem 3.2); see also the addendum (Langer 2004a). This theorem is formulated for a nef tuple \(D_1,\ldots,D_{n-1}\) with numerically nonzero product, and for a strongly semistable torsion-free sheaf gives \(\Delta(E)D_2\cdots D_{n-1}\ge0\), where \(\Delta(E)=2\mathop{\mathrm{rk}}(E)c_2(E)-(\mathop{\mathrm{rk}}(E)-1)c_1(E)^2\). Here the tuple is \((H,D)\), whose product is nonzero since \(H^2D>0\). For the rank-three tangent bundle this yields \[\bigl(6c_2(V)-2K_V^2\bigr)D\ge0.\] Equation (15) gives \(K_V^2D=0\) and proves Equation (21). ◻

Proof. A nonzero section has an effective zero divisor, so its degree is nonnegative and hence must be zero. Suppose that two independent sections \(s_0,s_1\) exist. Remove the base locus of the resulting pencil. Through every point \(x\) of the remaining open set there is a member of the pencil: take a nonzero linear combination vanishing at \(x\). Its divisor has degree zero. Each prime component \(S\) through \(x\) therefore satisfies \(DHS=0\), because all component degrees are nonnegative.

The restriction of \(D\) to \(S\) is nef and numerically trivial. One way to see the latter assertion, allowing \(S\) to be singular, is to take a projective resolution \(\sigma\colon\widetilde S\to S\). The class \(L_S=\sigma^*(D|_S)\) is nef and is orthogonal to the big and nef class \(A_S=\sigma^*(H|_S)\), with \(A_S^2>0\). The surface Hodge index theorem (Hartshorne 1977, V, Section 1) makes the intersection form negative definite on \(A_S^\perp\), whereas nefness gives \(L_S^2\ge0\). Hence \(L_S\) is numerically zero, and the projection formula gives the assertion on \(S\). A projective integral curve in \(S\) through \(x\), obtained for example as a component of a sufficiently ample section through \(x\), consequently has \(D\)-degree zero. Choosing \(x\) very general contradicts the prepared-cycle property. ◻

Proof. First suppose both sheaves are stable. 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 would make the first inequality strict by stability of \(G_1\). An image of rank smaller than \(\mathop{\mathrm{rk}}G_2\) would make the second inequality strict by stability of \(G_2\). Thus every nonzero map has full rank, and the two sheaves have the same rank, say \(n\).

The determinant of each map is a global section of \[L=\det(G_2)\otimes\det(G_1)^{-1},\qquad\deg L=0.\] It is initially defined on the locally free open set and extends across its codimension-at-least-two complement. By Lemma 10, \(H^0(V,L)\) has dimension at most one. If \(f,g\) were linearly independent maps, fix a nonzero section \(\tau\) spanning this space and write \[\det(sf+tg)=P(s,t)\tau\] for a homogeneous polynomial \(P\) of degree \(n\). Since \(f\) has full rank, \(P\) is nonzero; since \(n>0\) and \(k\) is algebraically closed, \(P\) vanishes at some point of \(\mathbb P^1(k)\). The corresponding linear combination is a nonzero map by independence, but has zero determinant, a contradiction. Thus the Hom space between stable factors has dimension at most one.

Every slope-zero semistable sheaf admits a finite filtration by saturated subsheaves with torsion-free stable factors of slope zero. Indeed if it is not stable, a smaller-rank equal-slope subsheaf can be saturated; its degree cannot increase strictly because of semistability. Both it and the quotient are semistable of slope zero, and induction on rank finishes the construction. Apply the left-exact Hom sequences successively to such filtrations of \(G_1\) and \(G_2\). The dimension of the Hom space is at most the sum of the dimensions between all pairs of stable factors. Each summand is at most one, and the numbers of factors are at most the respective ranks. This proves Equation (23). ◻

Proof. All the twists \(G(iD)\) are semistable of slope zero. Apply Lemma 11 first to \(\mathcal O_V,G(iD)\) and then to \(G^\vee,G(iD)\). The identities \[H^0\bigl(V,G(iD)\bigr)=\mathop{\mathrm{Hom}}\bigl(\mathcal O_V,G(iD)\bigr), \qquad H^0\bigl(V,G^{\otimes2}(iD)\bigr) =\mathop{\mathrm{Hom}}\bigl(G^\vee,G(iD)\bigr)\] give Equation (24). In particular the second bound uses the two semistable bundles \(G^\vee\) and \(G(iD)\), without needing semistability of \(G^{\otimes2}\).

For \(i>0\), the cohomology sequence of \(0\to J\to J(iD)\to J(iD)/J\to0\) gives \[h^0\bigl(V,J(iD)/J\bigr) \le h^0\bigl(V,J(iD)\bigr)+h^1(V,J).\] The right side is bounded independently of \(i\). Effectivity of \(D\) gives injections of the quotient sheaves as \(i\) increases, hence injections of their finite-dimensional spaces of global sections. A bounded increasing sequence of integer dimensions is eventually constant, so these spaces stabilize.

More explicitly, choose \(i_0\) after stabilization. Every global polar part, at every later cut, is then the image of an element of \(H^0(V,J(i_0D)/J)\). At the generic point of a prime component \(D_\alpha\) of multiplicity \(m_\alpha\), a local bundle frame measures its pole order coefficientwise. Such an image has pole order at most \(i_0m_\alpha\) there. This bound is independent of the later cut and of the element chosen. ◻

Proof. Apply Proposition 12 to \(G=\mathcal O_V\). Suppose a nonzero scalar polar part \(\eta\) exists. It has a genuine negative valuation at the generic point of some component of \(D\). To justify this detection, take local rational-function representatives. If all their component valuations were nonnegative, they would be regular by normality of \(V\); they already have no poles off \(D\). The quotient section would then be zero.

For every \(e\ge0\), raising the local representatives to their \(p^e\)-th powers defines a global section \[\eta^{p^e}\in H^0\bigl(V,\mathcal O_V(p^eiD)/\mathcal O_V\bigr).\] This operation is well defined on polar parts: two representatives differing by a regular function have \(p^e\)-th powers differing by a regular function. At the selected component the negative valuation is multiplied by \(p^e\). These pole orders are unbounded, contradicting the uniform bound in Proposition 12. ◻

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

If \(d_s[x]=0\), the leading term of \(\partial x\) is represented by \(\partial y\) with \(y\in C_{i-1}^a\) and \(\partial y\in C_{i-s}^{a+1}\). 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 \(\partial z\in C_i^a\) with \(z\in C_{i+s}^{a-1}\). This proves the next-page assertion. Equation 27 identifies page one. Since the quotients are supported on the projective surface scheme \(D\), their cohomology is zero outside degrees \(0,1,2\); the same is true of every subsequent page.

The differentials through a middle-row term are arranged as \[\begin{tikzcd}[column sep=large] E_s^{i+s,0}(E) \arrow[r,"d_s"] & E_s^{i,1}(E) \arrow[r,"d_s"] & E_s^{i-s,2}(E). \end{tikzcd}\] Moving right lowers weight by \(s\) and raises cohomological degree by one; the composite is zero. There is no incoming differential in degree zero. Its representatives are exactly the sections of \(P_i^s(E)\), proving Equation 28.

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

The two-term locally free presentation of \(P_j^s(E)\) gives \[\mathcal E xt^1(P_j^s(E),\omega_V) =\frac{E^!((s-j)D)}{E^!(-jD)}=P_{s-j}^s(E^!),\] and all other sheaf Ext groups vanish. Serre duality on the smooth projective threefold consequently identifies \(H^2(P_j^s(E))^\vee\) with \(H^0(P_{s-j}^s(E^!))\) (Hartshorne 1977, III, Section 7). Under this identification the dual of the displayed inclusion is \[H^0(P_{1-i}^s(E^!))\longrightarrow H^0(P_{1-i}^1(E^!)).\] One can check the map, as well as its index, directly: the map of the Cartier presentations is the identity on \(E((i-1)D)\) and the inclusion on their upper terms, so dualizing gives precisely the indicated projection. This proves Equation 29. The ranks of the two differentials adjacent to the middle term are \(\delta_s^0(i+s)\) and \(\delta_s^2(i-s)\), respectively. Taking its cohomology proves Equation 30.

Triviality of \(\mathcal O(MD)\) on \(sD\) identifies each quotient of width \(t\leq s\) with its translate by \(M\), compatibly with its layer maps. Equations 28 and 29 therefore give periodicity of both outer rows at every such width. Page one is periodic in all degrees; Equation 30 then gives periodicity of the middle row by induction. If \(e_t(i)=g_t^0(i)-g_t^1(i)+g_t^2(i)\), that equation 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 \(M\). Their sums cancel, proving Equation 31 by induction from page one.

For the last assertion, set \(A_s=H^0(P_i^s(E))\) and \[B_s=\bigcap_{t\geq s}\mathop{\mathrm{im}}(A_t\longrightarrow A_s).\] Each \(A_s\) is finite dimensional, so the intersection defining \(B_s\) is attained at a finite stage. Choosing one stage beyond stabilization at both \(s\) and \(s+1\) shows that \(B_{s+1}\longrightarrow B_s\) is surjective. Every element of \(B_1\) therefore admits compatible lifts in all the \(B_s\). Their inverse limit is a global section of \(\widehat{E(iD)}\), since sections commute with inverse limits of sheaves. Conversely a completed section restricts to the image of every \(A_s\). This proves the assertion directly. ◻

Proof. Choose local algebraic lifts \(f_\lambda\) of \(\xi\) on a finite affine cover. Membership in \(\mathcal O(uD)\) is checked by valuations at the prime divisors of the smooth, hence normal, variety \(V\). Thus some component detects the nonzero quotient section. The maximum of the ratios \(-\mathop{\mathrm{ord}}_{D_\alpha}(f_\lambda)/m_\alpha\) that exceed \(u\) is a well-defined rational number \(t\in(u,i]\): changing a lift adds a section of \(\mathcal O(uD)\) and cannot alter such a valuation. All lifts belong to \(\mathcal O(tD)\) in the sense of these rational divisor bounds.

Choose a positive integer \(n\) with \(nt\in\mathbb Z\). Then \(f_\lambda^n\in\mathcal O(ntD)\), and these sections agree on \(D_{\mathrm{red}}\). Indeed \(f_\lambda-f_\mu\in\mathcal O(uD)\), 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 \(\mathcal O(ntD)\) its valuation at every \(D_\alpha\) is at least \((t-u)m_\alpha>0\). They therefore define a section of \(\mathcal O(ntD)|_{D_{\mathrm{red}}}\), 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. It is consequently nowhere zero on that component. At an intersection with another component its common value is nonzero, so its restriction to that component is also nonzero and nowhere zero. Connectedness of \(|D|\) propagates this conclusion to all components.

It follows that \(\mathop{\mathrm{ord}}_{D_\alpha}(f_\lambda)=-tm_\alpha\) for every \(\alpha\). In particular every \(tm_\alpha\) is an integer. Primitivity of the cycle, or an integral linear combination of the \(m_\alpha\) equal to one, gives \(t\in\mathbb Z\). The original lifts are now sections of the honest line bundle \(\mathcal O(tD)\), are units in it along \(D\), and agree modulo \(\mathcal O(uD)\). Since \[\frac{\mathcal O(tD)}{\mathcal O(uD)} =\mathcal O(tD)|_{(t-u)D},\] they give the asserted trivialization.

A jet of width \(s\) at upper weight \(i\) has nonzero image in the leading layer exactly when the uniform ratio just proved is \(t=i\). This proves existence in Equation 32, in both directions. To see that the image dimension is one, take two such jets. Their normalized unit sections on \(D_{\mathrm{red}}\) are proportional: a connected reduced projective scheme over \(k\) has only constant global functions. Subtract the appropriate multiple. If its leading image were still nonzero, the preceding argument would again give a unit on the reduction with ratio \(i\), contradicting the cancellation. Thus the two leading images are proportional. ◻

Proof. Suppose instead that \(g_s^0(l;\omega_V)=0\) for every \(l\) and every \(s\geq S\), for some \(S\). By Equation 29, the top row for \(E=\mathcal O_V\) then vanishes at all these widths. Take \(e\) sufficiently large that \(s_e=d_{e-1}+1\geq S\). At page \(s_e\) the bottom row is one precisely at multiples of \(M_e\), by Equation 35 and \(s_e\leq d_e\). This page is periodic modulo \(M_e\). Equations 31 and 36 give \[\sum_{i\bmod M_e}g_{s_e}^1(i;\mathcal O_V) =1-M_e\gamma\leq1.\] At differential length \(d_e\), the bottom entries at \(jM_e\), \(p\nmid j\), each drop by one. By Equation 30 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 \(-d_e\pmod{M_e}\).

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 \(-d_e\pmod{M_{e+1}}\). Applying the same argument with \(e+1\) says that it must instead lie at \(-d_{e+1}\pmod{M_{e+1}}\). But \(0<d_e<d_{e+1}<M_{e+1}\), so these residues are different. This contradiction proves the lemma. ◻

Proof. Put \(P=p^b\), and choose a sequence of nonzero leading canonical jets of widths \(s\to\infty\), with upper weights \(l\) that may vary. Raising local representatives to the \(P\)th power and using Equation 26 gives a scalar jet with bounds \[A=PlD+B_*,\qquad B=P(l-s)D+B_*.\] The lower 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 \(c_*\geq\max_\alpha\beta_\alpha\). Enlarging both bounds gives a section of \[\frac{\mathcal O((Pl+c_*)D)}{\mathcal O((P(l-s)+c_*)D)}.\] This section is nonzero for all sufficiently large \(s\), uniformly in \(l\). To check this, at a component detecting the original leading layer, a local coefficient \(f\) in an ordinary frame of \(\omega_V\) has \(-\mathop{\mathrm{ord}}_{D_\alpha}(f)/m_\alpha>l-1\). Its powered scalar coefficient has pole ratio \[-P\frac{\mathop{\mathrm{ord}}_{D_\alpha}(f)}{m_\alpha} +\beta_\alpha>P(l-1)+\min_\alpha\beta_\alpha.\] The new lower bound is \(P(l-s)+c_*\), which is smaller for large \(s\).

Lemma 15 now shows that all these powered coefficients have one integral pole ratio \(t\). The preceding lower estimate and the upper bound give a constant \(C\), independent of \(s,l\), such that \[ \begin{gathered} |t-Pl|\leq C,\\ \mathcal O(tD)\text{ is trivial to a width at least }Ps-C. \end{gathered} \tag{38}\] For each fixed \(J\), widths greater than \(d_{J-1}\) can occur only at multiples of \(M_J\), by the scalar classification; zero is allowed as a multiple. Since the widths in Equation 38 tend to infinity, we can assign integers \(J\to\infty\) along this sequence such that \(M_J\mid t\).

Fix a component \(D_\alpha\). If \(f\) is the original local coefficient in a frame of \(\omega_V\), the local generator of \(\mathcal O(B_*)\) has order \(-b_\alpha\). Consequently \[ b_\alpha-tm_\alpha=P\mathop{\mathrm{ord}}_{D_\alpha}(f). \tag{39}\] For large \(J\), the integer \(t\) is divisible by \(P\). Equation 39 therefore shows that \(P\mid b_\alpha\) for every \(\alpha\). This proves \(B_*=PB_0\) with \(B_0\) integral.

For the same sequence put \(h_0=t/P\). The order identity says 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 by this line bundle. 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 \(L_{h_0}^{\otimes P}\simeq\mathcal O(tD)\), and the \(P\)th powers of these sections are units along all of \(D\) by Lemma 15. The original sections are therefore themselves units along \(D\). They agree modulo the original lower lattice \(\omega_V((l-s)D)\). Hence they trivialize \(L_{h_0}\) on the effective thickness \[T=(h_0-l+s)D-B_0.\] By Equation 38, \(|h_0-l|\) is bounded independently of the sequence. Thus \(T\) contains \(nD\) for integers \(n\to\infty\). Also \(M_J\mid t\) implies \(M_{J-b}\mid h_0\) once \(J\geq b\). The line bundle \(\mathcal O(h_0D)\) is therefore trivial on \(d_{J-b}D\), and \(d_{J-b}\to\infty\) by Equation 34. Removing this twist shows that \(L=\omega_V(-B_0)\) is trivial on arbitrarily large finite thickenings \(nD\).

We finally choose these trivializations compatibly. Apply the stabilized-image argument in the proof of Lemma 14 to the spaces \(H^0(nD,L|_{nD})\), also including restriction to \(D_{\mathrm{red}}\). The latter line bundle is trivial, and its section space is one dimensional because \(D_{\mathrm{red}}\) 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 \(D_{\mathrm{red}}\) compatibly to all thickenings. Each lift is a unit, as may be checked after reduction, and their inverse limit trivializes \(\widehat L\). This gives Equation 37. ◻

Proof. For all sufficiently large \(e\) set \[s_e=\max(d_{e-1},d'_{e-1})+1.\] Equations 34 and 45 show that these are strictly increasing and \(s_e\leq\min(d_e,d'_e)\). Work with the three rows for \(E=\mathcal O_V\). 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 Equation 35. For the top entry, Equations 29 and 41 associate to a top weight \(i\) the scalar ratio \(t=1-i-h=q-i\). Its entry is therefore at \(q\pmod{M_e}\).

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 \[-d_e+jM_e,\qquad p\nmid j.\] The top drops, of length \(d'_e\), consume middle-row dimensions at \[q+d'_e+jM_e,\qquad p\nmid j;\] here replacing \(j\) by \(-j\) accounts for the sign in the top ratio. Put \[A_e=-d_e,\qquad B_e=q+d'_e.\] The two lists, including their multiplicities when they meet, are thus \(A_e+jM_e\) and \(B_e+jM_e\), with \(p\nmid j\).

We spell out the rank count also when the two centers coincide modulo \(M_e\). Fix a residue \(a\pmod{M_e}\) 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.

In general, since \(p>2\), some lift receives all \(m(a)\) drops. Nonnegativity of its final rank gives \(n(a)\geq m(a)\). Summing over \(a\), and then using Equations 31 and 36, gives \[2\leq\sum_{a\bmod M_e}n(a) =2-M_e\gamma\leq2.\] 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 \[ \{-d_e,q+d'_e\} =\{-d_{e+1},q+d'_{e+1}\}\pmod{M_{e+1}}. \tag{47}\] 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 Equation 47 must be the swapped matching, even if a multiset initially has repeated entries. In particular \[d_{e+1}+q+d'_e\equiv0\pmod{M_{e+1}}.\] For large \(e\) its left side is positive and, by \(d_{e+1}<M_{e+1}\) and \(d'_e\leq d_e+C_0<M_e+C_0\), is less than \(2M_{e+1}\). Hence \[ d_{e+1}+q+d'_e=M_{e+1}. \tag{48}\] Divide by \(M_{e+1}=pM_e\) and let \(e\to\infty\). Equation 45 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 Equation 46. ◻

Proof. Write \(s=\lfloor c_0M_e\rfloor\). For large \(e\) we have \(s\le d_e\). A unit of \(\mathcal O(M_eD)|_{sD}\) 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 that no uniform bound on \(|v|\) exists. Passing to a sequence, we may assume that \(v\to+\infty\) or \(v\to-\infty\). 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 12.

In the second case choose a real number \(\lambda\) such that \[ \frac{1-c_0}{2}<\lambda<\frac1p, \qquad p\lambda>\frac12. \tag{51}\] The interval is nonempty by (50). Since the periods increase by the factor \(p\), we can choose \(j\le e\) with \[\lambda\le\frac{|v|}{M_j}\le p\lambda.\] Here \(j\to\infty\) as \(|v|\to\infty\). After a subsequence the ratios tend to a number \(y\in[\lambda,p\lambda]\). There is an integer \(m\in\{1,2\}\) for which \[ 1-c_0<my<1. \tag{52}\] Indeed, if \(y>1-c_0\), take \(m=1\); then \(y\le p\lambda<1\). Otherwise take \(m=2\). Its lower bound follows from \(2y\ge2\lambda>1-c_0\), and its upper bound follows from \(2y\le2(1-c_0)<4/p<1\).

Restrict the original jet to width \(\lfloor c_0M_j\rfloor\), take its \(m\)th tensor power, and multiply by a unit section of \(\mathcal O(M_jD)\) 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 \(G^{\otimes m}\) at upper weight \[w=M_j+mv.\] By (52), \(w\to+\infty\) and \(w<\lfloor c_0M_j\rfloor\) with a margin tending to infinity. It therefore gives a global polar part of \(G^{\otimes m}\).

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 12 applies to both \(G\) and \(G\otimes G\), giving the required contradiction. ◻

Proof. We separate the argument into the endpoint patterns, their drops, the middle row, and convergence.

Endpoint patterns. For \(e\) large, \(u_e\le d_e\), 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 14 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 \(w\). We 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 \(M_e\to\infty\), 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. These quotients give a finite filtration of the space of global sections. The same observation applies to \(E^!\).

Endpoint 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 must control their lengths, with multiplicities, independently of \(j\) whenever \(p\nmid j\). Consider a bottom position \(i=v+jM_e\). If \(p\mid j\), the next period \(M_{e+1}=pM_e\) retains exactly the same multiplicity there. Monotonicity therefore permits no drop between pages \(u_e\) and \(u_{e+1}\). If \(p\nmid j\), all of this multiplicity disappears by page \(u_{e+1}\). We claim that its drops occur at lengths \[ d_e\le s\le M_e+C \tag{53}\] for a fixed \(C\), and that their multiplicities at each length are independent of \(j\) among integers prime to \(p\).

There is no drop before length \(d_e\), since translation by a unit of \(\mathcal O(M_eD)|_{d_eD}\) identifies the leading images with those at the unshifted stable position \(v\). Choose local formal unit lifts \(t_\lambda\) of this trivializing section of \(\mathcal O(M_eD)|_{d_eD}\), and write on overlaps \[\frac{t_\lambda}{t_\mu}=1+b_{\lambda\mu}, \qquad b_{\lambda\mu}\in\mathcal I_D^{d_e}\widehat{\mathcal O}_V, \qquad \mathcal I_D=\mathcal O_V(-D).\] Fix a constant \(C\) for the moment and a width \[d_e<k\le M_e+C.\] For large \(e\), \(k<2d_e\) by (49). 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 \(\widehat{E(vD)}\) and satisfy \[ \alpha_\mu=(1+jb_{\lambda\mu})\alpha_\lambda \pmod{\mathcal I_D^k\widehat{E(vD)}}. \tag{54}\] Indeed \((1+b_{\lambda\mu})^j=1+jb_{\lambda\mu}\) 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 Equation (27): on each affine intersection this is the identity \(\widehat A/(f^s)=A/(f^s)\). These restrictions commute with the Čech 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 \[\alpha-\beta_{<n}\in C_{v-n}^0(E),\qquad \partial(\alpha-\beta_{<n})\in C_{v-d_e}^1(E).\] 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 inequalities \[ d_e-n\ge 2d_e-M_e-O(1)>u_e, \qquad k-d_e<M_e/2-O(1) \tag{55}\] 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 it to a completed global section and subtract. A finite induction constructs the asserted \(\beta\).

If \(p\nmid j,j'\), put \(q=j'/j\) in the prime field and set \[\alpha'=q\alpha+(1-q)\beta.\] Since \(\beta\) is global, Equation (54) gives \[\alpha'_\mu-(1+j'b_{\lambda\mu})\alpha'_\lambda =j'(1-q)b_{\lambda\mu}(\alpha_\lambda-\beta) \pmod{\mathcal I_D^k\widehat{E(vD)}}.\] The right side vanishes: its two factors have depths \(d_e\) and \(k-d_e\). Retwisting by \(t_\lambda^{j'}\) therefore gives a width-\(k\) jet at \(v+j'M_e\). It has the same normalized leading term, since \(k-d_e\ge1\). Also \(\alpha'-\beta=q(\alpha-\beta)\), so the same \(\beta\) reverses the construction with \(q^{-1}\). The normalized leading-image subspaces are consequently equal at every width under consideration. Their ranks, and hence their drops at each length, 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. Indeed 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. The increasing spaces in (ii) stabilize and forbid such polar parts. Since \(u_{e+1}>M_e+C\) for large \(e\), this proves (53) and the complete drop rule. The identical argument for \(E^!\), followed by the duality of Lemma 14, gives the corresponding top-row rule.

The middle row. We use the nonnegativity and Euler comparison from Proposition 18, now with a multiset of drop positions. Let \(T_e\) be the multiset of the numbers \(v-s\) for the bottom drops and \(w+s\) for the top drops. Each is 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 in Lemma 14 gives \[ \begin{split} H_{e+1}(i)=H_e(i) &-\#\{t\in T_e:t\equiv i\pmod{M_e}\}\\ &+\#\{t\in T_e:t\equiv i\pmod{pM_e}\}. \end{split} \tag{56}\] The last term restores the unshifted copies, which do not drop. All members of \(T_e\) lie in the two intervals \[ [-M_e-O(1),-d_e+O(1)] \quad\hbox{and}\quad [d_e-O(1),M_e+O(1)]. \tag{57}\] 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 \(p\ge5\), some lift is missed. Nonnegativity of \(H_{e+1}\) at that lift and periodicity of \(H_e\) show that \[H_e(i)\ge\#\{t\in T_e:t\equiv i\pmod{M_e}\}.\] Summing over a period gives equality of the two sides, by the Euler identity in Lemma 14 and property (i). The inequality is therefore equality at every residue. Equation (56) now says that \(H_{e+1}\) is exactly the distribution of \(T_e\) modulo \(pM_e\). For each fixed integer \(i\), the bands (57) and \(d_e\to\infty\) imply \[ g^1_{u_e}(i;E)=0\qquad\text{for all sufficiently large }e. \tag{58}\]

Convergence of primitives. Fixed-weight page vanishing alone does not bound the poles of successive primitives. Fix an integer upper bound \(l\) for a Laurent cocycle. We will bound the source \(z=x+s\) of every nonzero incoming differential of length \(s\) to a middle-row weight \(x\le l\) by a function \(B_l(x)\) satisfying \[ \sup_{x\le l}B_l(x)<\infty, \qquad B_l(x)\longrightarrow-\infty\quad(x\to-\infty). \tag{59}\]

At a sufficiently late period, the drop rule gives \(z=v+jM_e\), \(p\nmid j\), and \(d_e\le s\le M_e+C\). If \(j\ge2\), then \(x\ge M_e-A-C\), which exceeds \(l\) for all sufficiently 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 \(e\ge e_0\) and every middle rank in this finite set is zero at page \(u_{e_0}\). Equation (58) 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 \(z\le x+S_l\). Every later length belongs to a period \(u_e\le s<u_{e+1}\) with \(e\ge e_0\); the preceding exclusions leave only \(j\le-1\). In that case \(z\le-M_e+A\), and hence \(M_e\le-z+A\). Thus \(x=z-s\ge2z-A-C\), or \[z\le\frac{x+A+C}{2}.\] The real-valued upper bound \[B_l(x)=\max\left\{x+S_l,\frac{x+A+C}{2}\right\} \qquad(x\le l)\] therefore bounds every possible source and satisfies both assertions of Equation (59).

Let \(\gamma\) be an actual Laurent \(1\)-cocycle of upper cut \(x\le l\). Its leading cochain belongs to every cycle space \(Z_s^{x,1}\). By (58), it eventually belongs to the boundary space \(U_s^{x,1}\). More precisely, the page construction identifies \[U_{s+1}^{x,1}/U_s^{x,1} =\mathop{\mathrm{im}}\bigl(d_s:E_s^{x+s,0}\longrightarrow E_s^{x,1}\bigr),\] where \(E_s^{i,a}=Z_s^{i,a}/U_s^{i,a}\). Thus \(U_s^{x,1}\) stops increasing once \(x+s>B_l(x)\). By its definition, each increase from a source cut \(z=x+s\) has a representative primitive in \(C_z^0(E)\); the initial space \(U_1^{x,1}\) 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 (59). Their sum converges in a single bounded-pole completed lattice. The differential is continuous, and the filtration is separated, so its coboundary is \(\gamma\). All arguments use finite covers by affine charts and their affine intersections, and remain valid after refinement. This proves the theorem. ◻

Proof. The algebraic calculation is the \(p\)-basis form of Cartier’s theorem (Cartier 1957, Theorem 1); see also (Katz 1970, Theorem 7.2). We first check that its finite basis survives the completion and localization used here. For an étale algebra the relative Frobenius square is a base change square. One can check this because the relative Frobenius is étale and induces isomorphisms on geometric fibers. The polynomial-coordinate monomial basis over \(p\)th 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 \((\widehat A)^p\). The basis thus persists after completion and then after inverting \(f\); in the latter step, \(1/f=f^{p-1}/f^p\).

Every \(k\)-derivation kills \(p\)th 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 \(\Omega^1_{R/k}\) is free on \(dx_j\). There is no additional continuity assumption on differentials.

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 \[f\longmapsto f^p,\qquad df\longmapsto[f^{p-1}df].\] The rule is intrinsic. Leibniz holds directly, while 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 \[R\otimes_A F_*\Omega_A^\bullet\longrightarrow F_*\Omega_R^\bullet, \qquad a\otimes\eta\longmapsto a^p\eta.\] The monomial and coordinate-wedge bases make this an isomorphism. This proves the assertions about (60). Finally, the pairing obtained by Cartier on top forms is perfect: in coordinates it extracts the coefficient of \(x_1^{p-1}x_2^{p-1}x_3^{p-1}dx_1\wedge dx_2\wedge dx_3\). It gives (61) and is intrinsic by the same Cartier rule. ◻

Proof. For \(E=\Omega\), both \(E\) and \(E^!=T_V\otimes\omega_V\) are semistable of slope zero by Proposition 9 and the canonical intersection identities. Proposition 12 and Lemma 19 give conditions (ii) and (iii) of Theorem 20; passing from width \(\lfloor c_0M_e\rfloor\) to \(u_e\) can only decrease the leading images. Hirzebruch–Riemann–Roch gives condition (i). Indeed the degree-two terms in \(\mathop{\mathrm{ch}}(\Omega)\mathop{\mathrm{td}}(T_V)\) 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 (60) gives condition (i). By projection formula, a single cell for \(F_*\mathcal O_V\) has the Euler characteristic of \(p\) successive scalar cells; each is zero by Proposition 18. Frobenius changes the \(k\)-module action but not these finite dimensions.

There are subbundle inclusions with locally free quotients \[ B^1% \mathrel{% \BeginAccSupp{method=hex,unicode,space=false,pdfliteral=direct,ActualText=21AA}% \lhook\joinrel\longrightarrow% \EndAccSupp{pdfliteral=direct}% }% F_*\Omega, \qquad (B^1)^!% \mathrel{% \BeginAccSupp{method=hex,unicode,space=false,pdfliteral=direct,ActualText=21AA}% \lhook\joinrel\longrightarrow% \EndAccSupp{pdfliteral=direct}% }% F_*\omega_V. \tag{62}\] The first follows from Lemma 21; the second follows by dualizing the first sequence in (60) and using (61). For a subbundle \(E\subset F_*G\), the injection remains injective on every quotient \(E(nD)/E\). Projection formula embeds that quotient in \(F_*(G(pnD)/G)\). The polar bounds for \(G=\Omega\) and \(G=\omega_V\) therefore give condition (ii) for both bundles in (62). 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 that is useful to make explicit. A width-\(u_e\) jet of \(E\subset F_*G\) with a nonzero leading term maps to a width-\(pu_e\) jet of \(G\) at upper weight \(pi\). Its leading term is nonzero 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 all sufficiently large \(e\) we have \[pu_e-p\ge\lfloor c_0M_{e+1}\rfloor.\] Lemma 19 at period \(M_{e+1}\) implies that \(a\), and hence \(pi\), is at bounded distance from \(M_{e+1}\mathbb Z\). Dividing by \(p\) gives the required bounded distance from \(M_e\mathbb Z\).

Theorem 20 now applies to \(\Omega\) and \(B^1\), including finite-dimensionality for the former. For a \(Z^1\) cocycle, first solve its image in \(\Omega\) 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 21; refining the finite cover if necessary preserves every preceding argument. This proves the assertion for \(Z^1\). ◻

Proof. Corollary 22 gives finite-dimensionality of \(W\) and actual Laurent one-cocycle vanishing for \(\mathcal B^1\) and \(\mathcal Z^1\), proving the last assertion. To lift a global form through \(C\), choose local lifts in \(\mathcal Z^1\) using Equation (63). Their differences form a \(\mathcal B^1\)-cocycle. Subtracting a coboundary makes the lifts agree, so \[C:H^0(|D|,\mathcal Z^1)\longrightarrow W\] is surjective. Its source is a vector subspace of \(W\). Since the ground field is perfect, a semilinear map has the same dimension properties as a linear map. Finite-dimensionality therefore implies \(H^0(\mathcal Z^1)=W\) and bijectivity of \(C\).

For the last assertion about \(C\), set \(F=C^{-1}\), a bijective \(p\)-semilinear map. Choose coordinates on \(W\simeq k^n\). Then \(F(x)=A x^{[p]}\) with \(A\in\operatorname{GL}_n(k)\), where \(x^{[p]}\) means componentwise \(p\)-th powers. The polynomial map \(F-\mathop{\mathrm{id}}\) extends to \[ \Psi:\mathbb P^n\longrightarrow\mathbb P^n, \qquad [x:t]\longmapsto [A x^{[p]}-t^{p-1}x:t^p]. \tag{64}\] There is no base point: if \(t=0\), the first coordinates vanish only when \(x=0\). Also \(\Psi^*\mathcal O_{\mathbb P^n}(1)= \mathcal O_{\mathbb P^n}(p)\). A positive-dimensional fiber would contain a projective curve on which this line has both degree zero and positive degree, a contradiction. Hence \(\Psi\) is finite. Its image is closed and has dimension \(n\), so it is all of \(\mathbb P^n\). The inverse image of the standard affine chart is exactly that chart, proving surjectivity of \(F-\mathop{\mathrm{id}}\) over the algebraically closed field. The case \(n=0\) is immediate. Applying \(F\) to \((C-\mathop{\mathrm{id}})\beta=\eta\) gives the equivalent equation \[(F-\mathop{\mathrm{id}})\beta=-F\eta,\] so \(C-\mathop{\mathrm{id}}\) is surjective as well. ◻

Proof. Let \(\mathcal L\) be such a line bundle. Its frame cover can be refined by opens \(U_i\cap|D|\), where \(U_i\subset V\) is affine, has étale coordinates, makes \(D\) principal, and trivializes the finitely many algebraic bundles in use. Quasi-compactness of \(|D|\) permits a finite such cover. 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 4. Taking the maximum over the finitely many coefficients gives a common upper cut for a cochain. These observations also apply after finite refinements for local Cartier lifts. 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 \[d\log g_{ij}=g_{ij}^{-1}dg_{ij}\] is a closed additive one-cocycle. It is Cartier-fixed. Indeed, the intrinsic inverse Cartier formula gives \[C^{-1}(g^{-1}dg) =\bigl[g^{-p}g^{p-1}dg\bigr]=[g^{-1}dg]\] for every Laurent unit \(g\). By Lemma 23, after refining the cover there are closed one-forms \(\alpha_i\) such that \[\alpha_j-\alpha_i=d\log g_{ij}.\] These are the connection forms of a connection defined by \(\nabla e_i=\alpha_i\otimes e_i\). Their closedness is precisely its integrability, since the bundle has rank one.

The forms \(C\alpha_i-\alpha_i\) agree on overlaps and hence define a global form \(\eta\in W\). Choose \(\beta\in W\) with \((C-\mathop{\mathrm{id}})\beta=\eta\), using Lemma 23. Replace every \(\alpha_i\) by \(\alpha_i-\beta\). This preserves their differences and their closedness, and now \(C\alpha_i=\alpha_i\) for all \(i\). ◻

Proof. If \(C\alpha=\sum_i b_i\,dx_i\), the \(p\)-basis Cartier calculation represents \(\alpha\) in the form \[\alpha=dc+\sum_i b_i^p x_i^{p-1}\,dx_i \qquad(c\in R).\] Consequently \[ \partial_i^{p-1}a_i =(p-1)!\,b_i^p=-b_i^p. \tag{67}\] Exact terms disappear because \(\partial_i^p c=0\), and \(b_i^p\) is constant for every coordinate derivation. The Cartier-fixed condition therefore says \(a_i^p+\partial_i^{p-1}a_i=0\).

We recall the operator identity giving its significance. For a derivation \(\partial\) with \(\partial^p=0\) and an element \(a\in R\), \[ (\partial+a)^p=a^p+\partial^{p-1}a. \tag{68}\] First, the left side commutes with multiplication by any \(b\in R\): in characteristic \(p\), its commutator is the \(p\)-fold iterated commutator, namely multiplication by \(\partial^p b=0\). It is therefore multiplication by its value on \(1\). Over the integers, successive use of the product rule expands \((\partial+a)^p(1)\) as a sum indexed by set partitions of \(\{1,\ldots,p\}\), a block of size \(j\) contributing \(\partial^{j-1}a\). This formula follows by adjoining the last label either as a singleton or to one existing block. 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. To check the exceptions, in an invariant partition a block is either the whole set or has an orbit of \(p\) members, in which case all blocks are singletons. The summands have the same value throughout an orbit, so only \(a^p\) and \(\partial^{p-1}a\) survive in characteristic \(p\). This proves Equation (68) and hence \(D_i^p=0\). Finally, \[[D_i,D_j]=\partial_i a_j-\partial_j a_i=0\] by closedness, and the commutator with \(x_j\) follows directly from the Leibniz rule. ◻

Proof. Define \(S\)-linear operators \[ P_i=\sum_{n=0}^{p-1}\frac{(-1)^n}{n!}x_i^nD_i^n, \qquad P=P_1P_2P_3. \tag{70}\] The multiplication operator \(x_i^n\) is placed on the left. All factorials here 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 \(j\ne i\). Hence \(P\) is an idempotent with image \(H_M\).

For one coordinate, the binomial identity gives \[ \sum_{n=0}^{p-1}\frac{x_i^n}{n!}P_iD_i^n=\mathop{\mathrm{id}}_M. \tag{71}\] Indeed, after substituting Equation (70), the coefficient of \(x_i^rD_i^r\), for \(0\le r<p\), is \[\sum_{n+j=r}\frac{(-1)^j}{n!j!}=\begin{cases}1&r=0,\\0&r>0. \end{cases}\] All terms with \(r\ge p\) vanish by \(D_i^p=0\). Multiplying Equation (71) over the three coordinates gives the explicit expansion \[ m=\sum_{0\le I_j<p}x^Ih_I, \qquad h_I=\frac1{I!}P(D^I m)\in H_M. \tag{72}\] Here \(I!=I_1!I_2!I_3!\) and \(D^I=D_1^{I_1}D_2^{I_2}D_3^{I_3}\). The coefficients are unique: for \(h\in H_M\), \[P D^J(x^Ih)=\begin{cases}I!h&I=J,\\0&I\ne J, \end{cases} \qquad 0\le I_i,J_i<p.\] This follows by differentiation and the same binomial sum, which makes \(P(x^Kh)=0\) for every nonzero \(K\) in that range. Together with Equation (65), this proves Equation (69).

As an \(S\)-module, \(M\) is free of rank \(p^3\). The image of the idempotent \(P\) is therefore finite projective. The extension \(S\subset R\) is finite free and faithfully flat; Equation (69) shows that this projective module has rank one.

For restriction, let \(R\to R'\) be the homomorphism of Laurent chart rings and set \(S'=(R')^p\). The shared monomial bases give an isomorphism \[S'\otimes_S R\xrightarrow{\ \sim\ }R'.\] The restricted module is consequently \(S'\otimes_S M\), its operators are \(1\otimes D_i\), and its projector is \(1\otimes P\). Taking the image of a split idempotent commutes with every scalar extension, so its horizontal module is \(S'\otimes_S H_M\). This argument does not require a separate flatness assertion for the restriction homomorphism. Since horizontality means \(\nabla m=0\), the construction is also independent of the coordinate system and the chosen line frame. ◻

Proof. The Noetherian completion theorem gives that \(B\) is Noetherian and flat over \(A\), and \(B/(f^n)=A/(f^n)\) (The Stacks Project Authors 2026, Lemmas 10.97.1–10.97.5). 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. More generally, an element invertible modulo \(f\) is invertible in \(B\) (The Stacks Project Authors 2026, Lemma 10.96.6).

Every maximal ideal \(\mathfrak n\subset B\) thus corresponds to a maximal ideal \(\mathfrak m\subset A\) containing \(f\), and \(\mathfrak n=\mathfrak mB\). For each \(s\), completion modulo \(\mathfrak m^s\) does nothing, since \(f^s\) already vanishes there. Hence \[B/\mathfrak n^s\simeq A/\mathfrak m^s, \qquad \widehat{B_{\mathfrak n}}\simeq\widehat{A_{\mathfrak m}}.\] The latter ring is regular, as \(A_{\mathfrak m}\) is a smooth local ring. A Noetherian local ring is regular if and only if its maximal-ideal completion is regular (The Stacks Project Authors 2026, Lemma 15.44.4), so every \(B_{\mathfrak n}\) is regular. Localization now gives regularity at every prime of \(B\).

A regular Noetherian ring is locally factorial (The Stacks Project Authors 2026, Lemma 15.123.2). 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 \(\mathop{\mathrm{Spec}}R\) and take its Weil divisor. The closure of this finite Weil divisor on \(\mathop{\mathrm{Spec}}B\) is Cartier by local factoriality. Its line bundle restricts to the original one. Doing this on all components gives the asserted extension. This is the Cartier–Weil correspondence on a regular scheme (Hartshorne 1977, II, §6); global unique factorization of \(B\) is not required.

Let \(z\in U\cap|D|\). The extended line is free on a basic open \(\mathop{\mathrm{Spec}}B_g\) containing the prime corresponding to \(z\). Choose \(g_0\in A\) whose image modulo \(f\) equals that of \(g\). Then \(g_0(z)\ne0\). On the formal basic neighborhood coming from \(\mathop{\mathrm{Spec}}A_{g_0}\), its ring is \(B'=\widehat{A_{g_0}}\), and the image of \(g\) is invertible: modulo \(f\) it equals the invertible element \(g_0\). The line is therefore free on this formal neighborhood, and remains free 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. ◻

Proof. Give an arbitrary \(\mathcal R\)-line bundle \(\mathcal L\) the connection from Lemma 24. On a chart with a line frame, Lemma 25 gives commuting nilpotent connection operators, and Lemma 26 makes the horizontal module finite projective of rank one over \(R^p\). Its restriction compatibility and intrinsic definition glue these modules into a sheaf \(\mathcal H=\ker\nabla\) over \(\mathcal R^p\). Lemma 27 shows that \(\mathcal H\) is locally free of rank one in the required formal Zariski topology. The local multiplication isomorphisms glue to \[\mathcal R\otimes_{\mathcal R^p}\mathcal H\simeq\mathcal L.\]

Transport \(\mathcal H\) through the sheaf isomorphism \(\mathcal R\to\mathcal R^p\), \(a\mapsto a^p\), to obtain a line bundle \(\mathcal N\) over \(\mathcal R\). Extension from \(\mathcal R^p\) to \(\mathcal R\) is its absolute Frobenius pullback. In local frames, that pullback raises the transition units of \(\mathcal N\) to their \(p\)-th powers, which are also the transition units of \(\mathcal N^{\otimes p}\). Thus \[\mathcal L\simeq\mathcal N^{\otimes p}.\] This is a global root on the stated ringed space, proving the surjectivity assertion. ◻

Proof. Work near a point on an affine chart as above, small enough that each component through it has a prime equation \(f_\alpha\), and write \(f\) as a unit times \(\prod_\alpha f_\alpha^{m_\alpha}\). Every \(f_\alpha\) remains a non-zero-divisor in \(B\), by flatness of completion. It also remains prime, because \[ B/(f_\alpha)\simeq A/(f_\alpha). \tag{74}\] Indeed, the induced \(f\)-adic topology on \(A/(f_\alpha)\) is the zero-ideal topology, so completion leaves this integral ring unchanged.

If \(u\in B[1/f]^\times\), write \(u=a/f^r\) and \(u^{-1}=b/f^s\), with \(a,b\in B\). 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 \[ u=v\prod_\alpha f_\alpha^{n_\alpha}, \qquad v\in B^\times,\quad n_\alpha\in\mathbb Z. \tag{75}\] 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 Equation (73).

On an irreducible space with its Zariski topology, the constant sheaf \(\mathbb Z\) 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 \((\iota_\alpha)_*\mathbb Z\), 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 Equation (73) consequently proves surjectivity of the Picard restriction map. The image of the connecting map from the global integer sections is the subgroup generated by \(\mathcal O_V(D_\alpha)\), as is seen from their local equations. Exactness proves the kernel assertion. ◻

Proof. For a line bundle \(\mathcal L\) on the completion, its restriction to each integral projective surface \(D_\alpha\) is an ordinary line bundle. Define the integer \[ \Phi(\mathcal L)= \sum_\alpha m_\alpha \bigl(c_1(\mathcal L|_{D_\alpha})\cdot H|_{D_\alpha}\bigr). \tag{76}\] Intersection with the Cartier class \(H\) gives an integral degree on each surface, so \(\Phi\) is an additive homomorphism to \(\mathbb Z\). The numerical properties of the prepared cycle give \[\Phi(\mathcal O_V(D_\beta)) =\sum_\alpha m_\alpha D_\alpha D_\beta H =DD_\beta H=0.\] Lemma 29 therefore makes \(\Phi\) descend to a homomorphism \[\overline\Phi:\mathop{\mathrm{Pic}}(|D|,\mathcal R)\longrightarrow\mathbb Z.\] It is nonzero, since \[ \overline\Phi(\mathcal O_V(H)|_{\mathcal R})=DH^2>0. \tag{77}\]

On the other hand, Proposition 28 implies that every class in this Picard group is divisible by \(p^n\) for every positive integer \(n\). Its image under \(\overline\Phi\) must therefore be an integer divisible by every \(p^n\), hence zero. This contradicts Equation (77). ◻

Completion of the proof of Theorem 1. The assumption of maximal nef dimension supplied the prepared cycle and all the conclusions of Corollary 22. Proposition 30 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 (Xu 2024, Theorem 1.8) applies to the projective log canonical threefold pair \((X,0)\) in characteristic \(p>3\) with nef canonical divisor. It makes \(K_X\) semiample there.

To finish over the original field, denote it by \(k_0\) and the extension by \(k\). Restore \(X\) for the original threefold and write \(X_k=X\times_{k_0}k\). 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 \(\mathcal O_{X_k}(mK_{X_k})\) 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 \[H^0\bigl(X,\mathcal O_X(mK_X)\bigr)\otimes_{k_0}k \simeq H^0\bigl(X_k,\mathcal O_{X_k}(mK_{X_k})\bigr).\] Their dimensions are equal, proving the required inequality over \(k_0\). In particular \(\kappa(X,K_X)\ge1\); nefness and \(\nu(K_X)=1\) give \(\kappa(X,K_X)=1\), and the established positive-Kodaira-dimension abundance theorem (Xu 2024, Theorem 1.6) also gives semiampleness over the original field. ◻