Log abundance in numerical dimension one for threefolds in positive characteristic

Proof. For any integral curve C\subset T, a sufficiently large multiple of H has an effective Cartier member E containing C. The effective one-cycle of E has positive coefficient along C. Every component has nonnegative P-degree, whereas PE=0, so PC=0. This proves the first assertion. The second is positivity of the ample degree of each component of the effective one-cycle of a nonzero Cartier divisor.□

Proof. First extend the given field to an uncountable algebraically closed field. Integrality and normality persist, and a log resolution with its discrepancy calculation proves that the base-changed pair remains lc. The canonical divisor can be defined by a rational top form on the smooth locus, so the log canonical class also base changes. Nefness persists by ample approximation, and the prescribed intersection numbers are unchanged.

Semiampleness descends under this extension. After making a generated multiple also divisible by the original Cartier index, it is the base change of a line bundle on the original variety. Global sections commute with field extension, and faithful flatness detects surjectivity of its evaluation map. We may therefore work over the enlarged field.

Write L=K_X+B for the original log canonical class. If n(L)\leq2, Xu’s abundance theorem applies. Suppose n(L)=3. Take a crepant projective \Q-factorial dlt modification (X_d,B_d), and resolve X_d projectively. Run a relative canonical MMP of this smooth resolution over X_d. The MMP with scaling applies by [HW22, Theorem 1.2]. Its output cannot be of fibre type over the three-dimensional birational base, and it gives a projective birational morphism

with X_t terminal and \Q-factorial and K_{X_t} nef over X_d. Here terminality is preserved by a canonical MMP: discrepancies do not decrease, and a divisor contracted in a negative divisorial step has strictly positive ordinary discrepancy over the output. Relative pseudoeffectivity causes no restriction for this birational morphism; adding a sufficiently large pullback of an ample divisor makes the canonical class big.

The exceptional divisor F=K_{X_t}-g^*K_{X_d} is g-nef. The negativity lemma gives F\leq0. Since X_d is \Q-factorial, the divisor

is effective and satisfies K_{X_t}+B_t=g^*(K_{X_d}+B_d). Crepancy makes (X_t,B_t) lc.

The new log canonical class is nef of numerical dimension one. To check the square after pullback, observe first that L^2T=0 for every prime surface T on the original variety: place T in an effective member of a multiple of the ample divisor in the hypothesis, and use nonnegativity of L^2 on all its components. Projection formula proves the vanishing upstairs. Numerical nontriviality persists because a curve downstairs can be dominated by a curve upstairs. Finally, the no-zero-degree-curve property persists over the birational isomorphism locus. Thus the new class still has nef dimension three.

Relabel this pair as (X,B). By nonvanishing [Xu24, Theorem 1.4], a positive multiple of K_X+B has an effective representative N_0. It is nonzero because K_X+B is not numerically trivial. Increasing the multiple makes N_0, mK_X, and mB Cartier, as required.□

Proof. We may replace H by a very ample multiple. The number BNH is nonnegative. If it were positive, the identity (K_X+B)NH=N^2H/a=0 would give K_XNH<0.

For each sufficiently large integer l, choose positive integers u_l,v_l such that u_lH and v_l(H+lN) are very ample. Their general complete intersection is an integral curve C_l contained in the smooth locus of X. Indeed, normality gives a singular locus of dimension at most one; the first general member meets this locus in finitely many points, and the second avoids them. Bertini’s theorem on the smooth open gives the desired integral curve. Its intersections satisfy

and -K_XC_l>0 for large l.

The quantitative bend-and-break theorem of Jovinelly, Lehmann, and Riedl [JLR26, Theorem 1.1] applies to an integral curve in the smooth locus of a projective variety over an algebraically closed field, with negative canonical degree, and any nef Cartier test divisor. In dimension three it supplies, through every closed point of C_l, a rational curve \Gamma satisfying

Fix l so large that the right-hand side is strictly less than one. Since N is Cartier, it follows that N\Gamma=0.

For completeness, these complete intersections pass through points to which the very-general-point condition applies. For the two fixed very ample systems, the incidence of a point of X and a pair of members through it is a product of projective bundles over X, hence is irreducible and dominates X. Restricting to the nonempty open set of pairs giving the curves just considered leaves a dense open incidence and a dominant evaluation map. Its image is constructible and contains a nonempty open subset of X. Over the uncountable field, this open contains a closed point outside the countable union excluded by n(N)=3. The curve \Gamma through that point is a contradiction. This proves (5).

Write B=\sum b_TT with b_T>0. All numbers NHT are nonnegative, so (5) gives NHT=0 for each T. Lemma 2.2 yields N|_T\equiv0. If T is not a component of N, the canonical section of \mathcal O_X(N) restricts to a nonzero section on the integral surface T. Were T\cap\Supp N nonempty, its zero scheme would be a nonzero effective Cartier divisor on T and would have positive ample degree. This contradicts N|_T\equiv0, so the two supports are disjoint.□

Proof. We first find a coefficient-one boundary component lying over \operatorname{Supp}N_0. On a log resolution of (X,B+N_0), the log canonical threshold is the minimum of the finitely many ratios

where a(E,X,B) denotes log discrepancy and the divisors considered include the strict transforms of the components of N_0. Thus \lambda is finite and rational, possibly zero, and (X,B+\lambda N_0) is log canonical. There is a log canonical place E of this pair for which \operatorname{ord}_E(N_0)>0.

Take a projective crepant \mathbb Q-factorial dlt modification

with every extracted divisor having coefficient one; see [Xu26, Theorem 2.27]. Set

Then

The divisor N is nef and numerically trivial on every component of its support: a curve in that support maps to a point or to a curve in \operatorname{Supp}N_0, and the assertion follows by the projection formula.

Every component of G is either contained in or disjoint from \operatorname{Supp}N. For strict transforms of components of B this follows from the hypothesis, while those coming from \lambda N_0 are contained in the support. An extracted divisor has center in \operatorname{Supp}N_0\cup P_{\rm out}, since X is klt and the boundary vanishes outside this union. Its irreducible center lies in one of these two disjoint closed sets. In the first case its order on N_0 is positive, and in the second case the divisor is disjoint from \operatorname{Supp}\mu^*N_0.

The center on W of the place E lies in \operatorname{Supp}N and, by crepancy and the dlt property, in \lfloor G\rfloor. Choose a component S of \lfloor G\rfloor containing this center. The preceding separation then implies S\subset\operatorname{Supp}N. This argument also covers \lambda=0; it does not require all components of N to occur in G.

The modification \mu is an isomorphism outside \operatorname{Supp}N_0\cup P_{\rm out}. Indeed it has no exceptional divisors there, and a projective small birational morphism to a \mathbb Q-factorial variety is an isomorphism. To recall the latter fact, push down a relatively ample Cartier divisor. A Cartier multiple of its pushdown pulls back to the same multiple upstairs, because there are no exceptional divisors. It has degree zero on every contracted curve, contradicting relative ampleness if such a curve exists.

Choose a rational \epsilon>0 sufficiently small that

Apply the partial MMP with lowered boundary G-\epsilon S and restoration divisor \epsilon S [Xu26, Proposition 2.21, Definition 2.23, and Lemma 2.24]. Its steps are (K_W+G-\epsilon S)-negative and (K_W+G)-trivial. The lowered pair is dlt, and its log canonical divisor is pseudoeffective by (11) and (12). The termination required here is therefore supplied by [Xu24, Theorem 1.5]. The threefold MMP package used in this application, including in characteristic five, is recalled in [OAI26, Section 2.3].

The partial-MMP conclusion is either a Mori fiber contraction for the lowered log divisor, trivial for the restored divisor, or nefness of the restored divisor minus \delta times the transform of S for every sufficiently small rational \delta>0. We first track S, the pullbacks of N, and support separation through the steps, and then exclude the fiber-type alternative. Write W_i\dashrightarrow W_{i+1} for a step, with transforms G_i,N_i,S_i. The equality K_{W_i}+G_i\sim_{\mathbb Q}\tau N_i persists by pushforward, so its contracted ray R_i satisfies

In particular S_i cannot be the exceptional divisor of a divisorial step: the negativity lemma would give a contracted curve of negative S_i-degree. Hence S survives throughout the MMP.

Let a:T\to W_i and b:T\to W_{i+1} be a common resolution of a step. The divisor

is b-exceptional. If a curve on T is contracted by b, its image on W_i is a point or a curve contracted to the common contraction target. Thus F has degree zero on every such curve by (13). The negativity lemma applied to F and -F gives

This applies to divisorial contractions and flips. The same argument for the full log divisors gives

Consequently the restored pair remains log canonical.

Equation (14) also preserves nefness and numerical triviality on the support. For the latter assertion, lift a curve in \operatorname{Supp}N_{i+1} to a curve on T dominating it. Its image on W_i lies in \operatorname{Supp}N_i or is a point: the inverse image of the support of an effective \mathbb Q-Cartier divisor is exactly the support of its pullback. The projection formula now gives degree zero. Boundary components contained in \operatorname{Supp}N_i remain contained in the transformed support. If a boundary component is disjoint from that support, its strict transform on T is disjoint from a^*N_i, hence from b^*N_{i+1}. Since that strict transform maps onto its image on W_{i+1}, the transformed boundary component is still disjoint from \operatorname{Supp}N_{i+1}.

These steps also preserve the open complement of the support. Indeed every contracted curve has negative degree against the effective divisor \tau N_i-\epsilon S_i, so is contained in \operatorname{Supp}N_i. A connected positive-dimensional projective fiber contains a curve through each of its closed points. Connectedness of fibers therefore implies that a fiber meeting the complement of \operatorname{Supp}N_i consists of a single point and does not meet the support. Properness and normality show that the contraction is an isomorphism from this complement onto an open subset of its target. For a flip, this open in the target is \mathbb Q-factorial, being isomorphic to the corresponding open of W_i. The new small morphism is therefore also an isomorphism over it.

There is no Mori fiber output. The effective representative \tau N_i-\epsilon S_i persists on each model, as does \tau N_i-tS_i for every rational 0<t\leq\epsilon. A curve in a fiber through a general point outside this effective support has nonnegative degree, whereas the log divisor driving a Mori fiber contraction has negative degree on every contracted curve. This rules out the fiber-type alternative for the lowered class and for every such smaller perturbation of the restored class.

Let Y be the resulting model. The partial-MMP conclusion gives (8) for some rational 0<\delta<\epsilon. Negative MMP steps preserve the \mathbb Q-factorial dlt pair with lowered boundary, so (Y,G_Y-\epsilon S_Y) is dlt. In particular its \mathbb Q-factorial underlying variety Y is klt: every place of zero log discrepancy for a dlt pair has center in its reduced boundary, and removing that effective \mathbb Q-Cartier boundary strictly increases its discrepancy. Here dlt preservation is used for the lowered boundary; for the restored boundary G_Y we use the crepant comparison just proved.

It remains to see why nefness isolates S_Y. If T\ne S_Y is a component of N_Y, numerical triviality of N_Y|_T and (8) imply that -S_Y|_T is nef. On the other hand, S_Y|_T is an effective \mathbb Q-Cartier divisor. If it were nonzero, its intersection with an ample Cartier divisor on the integral projective surface T would be positive, contradicting anti-nefness. Therefore S_Y\cap T=\varnothing. Each other component of G_Y is either such a component of N_Y, or is disjoint from its support. This proves the asserted separation and the local identities (9). The equality N_Y|_{S_Y}\equiv0 gives S_Y|_{S_Y}\equiv0.

Finally, all models share the stated open U_X. Resolve the graph of the correspondences among all the models, keeping this smooth open unchanged, to obtain V_0. Composing (14) with N=\mu^*N_0 proves (10). Pullback preserves this equality on every further common resolution.□

Proof. The pair (Y,S_Y) is log canonical since (Y,G_Y) is log canonical and G_Y\geq S_Y. Take a crepant dlt modification extracting only log canonical places. Its boundary \Sigma consists of the strict transform of S_Y and its exceptional divisors, all with coefficient one. Every extracted divisor has center contained in S_Y: outside S_Y the pair is the klt variety Y. Such a divisor has positive order on S_Y, which proves (15). There are no exceptional divisors off S_Y; since Y is \mathbb Q-factorial, the small-morphism argument in the preceding proof shows that f is an isomorphism there.

Crepancy and (9) give (16). Every curve in \Sigma maps to a point or to a curve in S_Y, where S_Y is numerically trivial. The projection formula proves the final numerical assertion. Under the identification in Proposition 4.1, U_X is disjoint from S_Y and hence is unchanged by f.□

Proof. We reproduce the proof because both the finite exceptional set and the nonreduced divisor will occur in our application. Replace Z by Z\cap\operatorname{Supp}E. At z\in Z, write R=\mathcal O_{T,z}, with maximal ideal \mathfrak m, and let t_z be a local equation of E. Normality gives

Here the second assertion uses the dimension-three hypothesis. Indeed, present R=A/I with A regular local of dimension n and essentially of finite type over the field. Each nonmaximal localization of R is normal of dimension at most two, hence Cohen–Macaulay. Catenarity gives \dim A_{\mathfrak q}-\dim R_{\mathfrak p}=n-3 for a prime \mathfrak q above \mathfrak p; the Auslander–Buchsbaum formula therefore gives projective dimension n-3 over A_{\mathfrak q}. Thus \operatorname{Ext}^{n-2}_A(R,A) is supported only at the closed point. It is a finite module, so it has finite length; local duality gives the assertion for H^2_{\mathfrak m}(R).

Choose j\geq1 such that t_z^{j-1} annihilates H^2_{\mathfrak m}(R) at every point of Z. A single j works because Z is finite. For every integer u, the restriction sequences give a commutative diagram

In local trivializations the left vertical map is multiplication by t_z^{j-1}. Local cohomology with support in the finite set Z is the direct sum of the corresponding local groups. 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. Naturality and our choice of j therefore imply

The same j works for every twist: locally the invertible sheaves in the diagram are free of rank one, and the annihilating map is still multiplication by t_z^{j-1}.

Now fix this thickening jE. The ideal of E_{\mathrm{red}}\setminus Z in jE\setminus Z is nilpotent. Lift a chosen generator of \mathcal O_T(E)|_{E_{\mathrm{red}}\setminus Z} to local generating sections on a finite open cover of jE\setminus Z. On overlaps the lifts differ by sections in that nilpotent ideal times the line bundle. For u=p^e sufficiently large, their uth tensor powers agree: in local frames this is the identity (a+b)^{p^e}=a^{p^e}+b^{p^e}, with b^{p^e}=0. They consequently glue to a generator

Let s be its restriction to E\setminus Z. The obstruction to extending s over E is the image of the obstruction to extending s_j over jE. Equation (17), applied to the exact sequences for sections with support in Z, kills that image. Hence s extends to a section of \mathcal O_E(uE).

This extended section has no zeros. On each integral projective surface component F of E_{\mathrm{red}}, it is nonzero because it is a generator off Z. If its restriction to F vanished anywhere, its zero scheme would be a nonempty effective Cartier divisor on F, and would have positive intersection with an ample divisor. This contradicts the numerical triviality of \mathcal O_F(uE). The argument uses only integrality of F, not normality. The section is therefore a generator on the reduction, and then on E, since an element invertible modulo a nilpotent ideal is invertible. Thus \mathcal O_E(uE)\simeq\mathcal O_E.□

Proof. The relations furnished by isolation are

Let \sigma:\Sigma'\to\Sigma be the finite S_2-ification. It is unchanged away from a finite set Z\subset\Sigma [Xu26, Proposition 2.14]. Reduced-boundary adjunction gives an effective rational divisor \Delta' such that (\Sigma',\Delta') is a projective semi-log-canonical surface pair and

We use adjunction in the form of [Xu26, Lemma 2.19]; see also [OAI26, Section 2.5]. The S_2-ification is needed because the reduced boundary itself need not satisfy S_2. Its codimension-one singularities are nodes, and on its normalization the adjunction boundary includes the conductor with coefficient one; the normalized pair is log canonical. These are precisely the semi-log-canonical conditions needed for surface abundance.

The surface \Sigma' is projective because it is finite over \Sigma. The last class in (19) is numerically trivial, by (18) and the projection formula for curves. Posva’s abundance theorem for projective semi-log-canonical surfaces in positive characteristic therefore makes K_{\Sigma'}+\Delta' semiample [Pos23, Theorem 3.1]. A semiample numerically trivial rational line bundle is \mathbb Q-linearly trivial: a globally generated Cartier multiple defines a morphism contracting every curve, hence constant on each connected component. Its line bundle is trivial there.

Since \tau s is positive and rational, clearing denominators in (19) yields a positive Cartier multiple M of f^*S_Y with \sigma^*\mathcal O_\Sigma(M) trivial. In particular,

The divisor M is effective, its reduction is \Sigma, and it is numerically trivial on every component of its support. Applying Lemma 5.1 on the normal projective threefold Y_d proves torsion on the full Cartier scheme M.□

Proof. Equation (3) and Proposition 3.1 give the hypotheses of Proposition 4.1. Choose its isolated model (Y,G_Y,N_Y,S_Y) and the reduced dlt modification f:(Y_d,\Sigma)\to(Y,S_Y) of Corollary 4.2. Let M be the positive Cartier multiple of f^*S_Y supplied by Proposition 5.2.

The canonical section and its support. Let V_0 be a smooth projective common resolution of X and Y_d, chosen to dominate the intervening models. Write

These models share the open set

where P_{\mathrm{out}} is the union of the boundary components disjoint from \operatorname{Supp}N_0. We choose the common resolution to be an isomorphism over this open. Projective resolution and resolution of divisor supports in dimension three permit this choice [CP09, CJS13]. The pullback equalities through the partial MMP give

The integer m was chosen so that N_0\sim m(K_X+B) and both mK_X and mB are Cartier. It gives a rational section \theta of \omega_{V_0}^{\otimes m} with

Indeed, the right side is an integral divisor linearly equivalent to mK_{V_0}. Put E_0=\operatorname{Supp}q^*N_0. Every prime component in (21) is either contained in E_0 or disjoint from it. For q^*B this follows from boundary separation on X. For a q-exceptional divisor, its irreducible center lies in

A terminal threefold is regular in codimension two, so \operatorname{Sing}X is finite [OAI26, Lemma 2.3]. The center is therefore contained in \operatorname{Supp}N_0, or is disjoint from it. Taking inverse images gives the assertion for the exceptional divisor.

Let F_0 be the union of the components of \operatorname{div}(\theta) outside E_0. Thus

The disjointness is the point of this description. Components of q^*N_0 can cancel in (21); we retain all of E_0 when choosing the open set preserved by the subsequent resolution.

Removing the prime-to-p exponent. Write m=p^b m' with b\geq0 and (m',p)=1. Choose a nonzero rational generator \eta of \omega_{V_0}^{\otimes p^b} and write \theta=u\eta^{m'}, where u\in k(V_0)^\times. Adjoin one root v of v^{m'}=u and normalize V_0 in the resulting field. This is a finite separable cover, possibly of degree smaller than m'. Over V_0\setminus\operatorname{Supp}\operatorname{div}(\theta), it is an integral component of the torsor of m'th roots of the invertible section \theta. This description is independent of the rational generator \eta. Since m' is invertible in k, that torsor is finite étale.

Resolve the normalization and the total inverse image of Z_0, preserving the regular étale open over V_0\setminus Z_0. We obtain a smooth projective threefold and a generically finite separable morphism \pi:V\to V_0 for which the total pullback support of N_0 is simple normal crossings. The morphisms used to compare divisors are

Here q,h_0,f are birational; \pi is generically finite and separable. Set \rho=q\circ\pi and h=h_0\circ\pi. By (20), \rho^*N_0=(f\circ h)^*N_Y, while the Cartier divisor supplied by Proposition 5.2 pulls back as h^*M.

The rational section v\eta is a section of \pi^*\omega_{V_0}^{\otimes p^b}. The determinant of the differential is a nonzero section of the rational line \omega_V\otimes\pi^*\omega_{V_0}^{-1}; denote it by J_\pi. Its nonvanishing at the generic point follows from separability. Consequently

Both factors are invertible on \pi^{-1}(V_0\setminus Z_0): the first is a root of an invertible section, and the second is the Jacobian of an étale morphism. Thus the divisor of \psi is supported on \pi^{-1}(Z_0). By (22), each of its prime components lies in \operatorname{Supp}\rho^*N_0 or is disjoint from that support.

A connected primitive Cartier divisor. Put M_V=h^*M and choose a connected component C of \operatorname{Supp}M_V. Since M is a positive multiple of f^*S_Y, and S_Y is disjoint from the other components of N_Y, the set C is also a connected component of \operatorname{Supp}\rho^*N_0. Write the part of M_V supported on C as

The divisor D is Cartier because V is smooth, and its multiplicities are positive with gcd one. Its support is connected and simple normal crossings.

Every curve in \operatorname{Supp}M_V maps under h to a point or to a curve in \Sigma. The projection formula and Proposition 5.2 therefore show that M_V is numerically trivial on every component of its support. Near C we have M_V=aD, so D|_{D_\alpha}\equiv0 for every \alpha. For a curve not contained in \operatorname{Supp}D, effectivity gives nonnegative intersection with D. For a curve in its support, the intersection is zero. Thus D is nef, and every very ample divisor H_V satisfies

Torsion also survives on the entire Cartier scheme. Choose t>0 with \mathcal O_M(tM)\simeq\mathcal O_M. The pullback divisor M_V is scheme-theoretically V\times_{Y_d}M: its local equation is the pullback of a local equation of M, which is a nonzero divisor since V is integral and h is dominant. Pulling back the trivialization gives \mathcal O_{M_V}(tM_V)\simeq\mathcal O_{M_V}; no flatness assumption on h is needed. Restricting along D\subset M_V, and using M_V=aD near C, gives

Hence \mathcal O_D(D) is torsion.

The section \psi of (23) has, on a neighborhood W of C, divisor supported on D. Indeed, its other components are disjoint from C and can be removed from that neighborhood. Thus there are integers b_\alpha such that

Numerical proportionality and very general points. It remains to show that no D-trivial curve passes through a very general point of V. Put N_V=\rho^*N_0. It is a nonzero nef divisor, numerically trivial on every component of its support by the projection formula. Since \operatorname{Supp}D is contained in that support, we have

Together with (25), the Hodge index theorem gives

We verify that this is numerical equivalence on every curve, rather than only on a fixed hyperplane section, following [OAI26, Section 2.8].

Let \Gamma\subset V be any integral curve. For l\gg0, there is an integral surface T\in|lH_V| containing \Gamma. To see this, twist the ideal of \Gamma sufficiently far and then multiply its generators by very ample sections. The resulting linear system has no fixed divisorial component and gives a birational map onto its image off \Gamma. Bertini irreducibility gives an irreducible general member, which is generically reduced on that birational open. As a Cartier divisor on smooth V, it has no embedded components, and is therefore integral. On a smooth projective resolution \widetilde T\to T, let d,n,e be the pullbacks of D|_T,N_V|_T,H_V|_T, respectively. Their intersections are

Hence d-cn has square zero and is orthogonal to e. The Hodge index form is negative definite on the orthogonal complement of the positive-square class e, so d-cn is numerically zero. The projective surjection \widetilde T\to T supplies an integral curve dominating \Gamma: take the closure of a closed point of the fiber over its generic point. The projection formula then gives (D-cN_V)\cdot\Gamma=0. This proves (28).

Finally let Z_i\subset X be the countably many proper closed subsets excluded by the nef reduction of N_0. Remove from V the non-quasi-finite locus of \rho and all the sets \rho^{-1}(Z_i). These are proper closed subsets, since \rho is dominant and generically finite. Their complement contains very general points because k is uncountable. Every curve through any point of this complement maps to a curve on X. If it were D-trivial, then (28) and the projection formula would make its image N_0-trivial, contradicting maximal nef dimension. All properties in Theorem 2.1 have now been established.□

Proof of Theorem 1.1. Theorem 2.1 contradicts Proposition 6.1. The maximal-nef-dimension case is therefore impossible. By Proposition 2.3, it remains to apply abundance in nef dimension at most two [Xu24, Theorem 1.8]. This gives semiampleness, and faithful descent along the algebraically closed field extension gives it over the original field.□