When one numerical direction becomes a fibration in positive characteristic

← All digests

Family 035 · Algebraic and complex geometry
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-12

1. The question

Imagine that the canonical geometry of a three-dimensional algebraic variety has only one direction of numerical growth. Should that direction come from a genuine map to a curve?

The abundance conjecture predicts that a nef canonical divisor, including a specified boundary when appropriate, is generated by sections after taking a positive multiple. Family 035 addresses the numerical-dimension-one case over fields of positive characteristic. Its first manuscript, M1, claims the result for log canonical threefold pairs. The second, M2, treats terminal threefolds without boundary and develops the structural obstruction used by M1.

The distinction between “one numerical direction” and “a map to a curve” is the problem itself. A small intersection dimension does not automatically produce a family of curves on which the divisor has degree zero, let alone global sections defining a morphism.

2. Background and essential definitions

The ambient geometry

A threefold is an algebraic variety of dimension three. Projective means that it embeds in projective space; normality rules out certain defects in its local rings while allowing controlled singularities. The field $k$ is algebraically closed and has characteristic $p>3$, so $p$ is a prime, beginning with five, and adding the number one to itself $p$ times gives zero.

The canonical divisor $K_X$ represents top-degree differential forms on the smooth part of $X$. A boundary $B$ is an effective rational combination of hypersurfaces, and $L=K_X+B$ is the log canonical divisor. A divisor is $\mathbb Q$-Cartier when some positive integer multiple defines a line bundle. All section statements use such multiples.

The pair $(X,B)$ is log canonical if, on resolutions, the coefficients of the boundary obtained from

$$K_Y+B_Y=\pi^*(K_X+B)$$

are at most one. This condition measures the allowed singularities of both the space and its boundary. Terminal singularities are a more restrictive condition on a boundary-free variety. A variety is $\mathbb Q$-factorial if every Weil divisor has a Cartier positive multiple. M1 does not require either terminality or $\mathbb Q$-factoriality of the original threefold.

Three dimensions associated with a divisor

A divisor $L$ is nef when $L\cdot C\geq0$ for every curve $C$. For a nef divisor on a threefold and an ample divisor $H$, numerical dimension one means

$$L\cdot H^2>0,\qquad L^2\cdot H=0.$$

The first intersection detects a nontrivial positive direction; the second says that two copies of $L$ do not produce positive growth. This condition is independent of the ample divisor used to test it. Write it as $\nu(L)=1$.

The Iitaka dimension $\kappa(L)$ instead measures the largest dimension of a rational image obtained from ratios of global sections of $mL$. It is $-\infty$ if all positive section spaces vanish. A single section only gives nonvanishing; obtaining two sections with a nonconstant ratio can produce a positive-dimensional image.

The nef dimension $n(L)$ comes from the nef reduction, which groups together directions traced by curves of $L$-degree zero through very general points. “Very general” means outside a countable union of proper closed subsets, after working over an uncountable algebraically closed field when needed. On a threefold, $n(L)=3$ means there are no such zero-degree curves through a very general point.

These invariants ask different questions. The difficult configuration in these papers is

$$\nu(L)=1,\qquad\kappa(L)=0,\qquad n(L)=3.$$

Numerical growth is small, there are too few section ratios to define a curve map, but zero-degree curves do not sweep out the space.

Semiampleness and its geometric meaning

A line bundle is semiample if a positive power is generated by global sections at every point. For nef $L$ with $\nu(L)=1$, this yields a morphism with connected fibers

$$f:X\longrightarrow C,\qquad mL\sim f^*A,$$

where $C$ is a normal projective curve and $A$ is ample. Here $\sim$ denotes linear equivalence, hence an identity of the associated line bundles. The fibers are surfaces, and $L$ has degree zero on their curves. This is the actual geometric structure sought in the opening question.

3. Development history and previous partial results

Over the complex numbers, Miyaoka’s numerical-dimension-one theorem studied effective pluricanonical divisors and their neighborhoods. Kawamata’s abundance theorem completed the minimal-threefold picture and gave a related approach using finite covers and infinitesimal deformations. Keel–Matsuki–McKernan established log abundance for threefolds, with the correction cited in M1.

Positive characteristic requires different tools. Frobenius can help with line bundles and differential forms, but characteristic-zero vanishing arguments cannot simply be imported. Keel’s base-point-freeness criterion and later work of Cascini–Tanaka–Xu established important foundations. Hacon–Xu, Birkar, and subsequent authors developed the threefold minimal model program for characteristic greater than five; Hacon–Witaszek treated characteristic five.

Nonvanishing also has a substantial prior history. Xu–Zhang proved it for terminal minimal threefolds in characteristic greater than five. Witaszek’s canonical-bundle-formula work proved further nonvanishing and low-nef-dimension abundance results. These are advances toward obtaining and using sections, rather than consequences of the new manuscripts.

Zheng Xu’s characteristic-greater-than-three theorem supplies nonvanishing for pseudoeffective lc adjoints and abundance in the low-nef-dimension cases used here. His subsequent numerical-dimension-two result treats a different slice of the problem. M2 explicitly identifies the remaining task in its setting: exclude numerical dimension one with Kodaira dimension zero and maximal nef dimension.

Thus the new step is not obtaining the first section from scratch. It is forcing additional section growth, then applying the established passage from positive Kodaira dimension to semiampleness. M1 extends the companion’s obstruction to arbitrary rational lc boundaries.

4. What the new manuscripts claim

M1: the log canonical statement

Let $k$ be algebraically closed of characteristic $p>3$, and let $(X,B)$ be a normal projective log canonical threefold pair. Assume that $B\geq0$ is rational and that $L=K_X+B$ is $\mathbb Q$-Cartier, nef, and of numerical dimension one. M1’s Theorem 1.1 states that $L$ is semiample.

There is no terminality or $\mathbb Q$-factoriality assumption on $X$, and boundary coefficients are arbitrary rational coefficients compatible with log canonicity. Auxiliary birational models may satisfy stronger conditions, but these should not be added to the original theorem’s hypotheses.

The conclusion gives $\kappa(L)=1$ and the curve fibration described above. It does not assert that a particular specified low multiple already generates the line.

M2: the boundary-free case and its obstruction

M2 assumes that $X$ is normal, projective, $\mathbb Q$-factorial, and terminal, with $K_X$ nef and $\nu(K_X)=1$. Its Theorem 1.1 claims a positive Cartier multiple with at least two linearly independent sections, and hence $\kappa(K_X)=1$ and semiampleness.

No restriction is imposed on irregularity, the Albanese map, or the first Betti number. The manuscript works directly in characteristic $p$; it does not assume that $X$ can be lifted to characteristic zero.

For M1, the crucial companion output is a nonexistence theorem for a specially prepared effective divisor on a smooth threefold. It combines numerical dimension one, maximal nef dimension, a torsion normal line on the whole possibly nonreduced divisor, and a local canonical relation. M1 constructs precisely this configuration from a hypothetical counterexample with boundary.

5. Why the advance matters

Importance: 4/5, provisional. The result would close a significant numerical-dimension case of threefold log abundance in positive characteristic, including characteristic five and the broader lc singularities. It connects intersection theory with an actual fibration in a setting where familiar characteristic-zero methods are unavailable.

The scope is substantial but specific: dimension three, $p>3$, and numerical dimension one. The score reflects that central contribution within birational geometry without treating it as an all-dimensional or all-characteristic abundance theorem.

The companion’s mechanism also has independent explanatory interest. It turns a geometric failure to find enough sections into a conflict between divisibility of line bundles and integer-valued intersection degree.

6. The main idea

Isolate the problematic divisor

Prior nonvanishing gives an effective Cartier divisor $N\sim m(K_X+B)$. Assume that the desired abundance fails. The earlier low-nef-dimension results reduce attention to $n(N)=3$.

M1 first separates the boundary from $N$: each boundary component is either part of its support or disjoint from it. The argument uses bend-and-break, a technique that extracts rational curves from families of curves with negative canonical degree. In this situation a failure of separation would produce zero-$N$-degree curves through very general points, contradicting maximal nef dimension.

A sequence of controlled birational operations then isolates a component, and surface abundance gives torsion along it. “Torsion” means that some positive tensor power of the restricted line is trivial. The papers retain this property on the full Cartier thickening, including multiplicities, rather than only on the reduced support.

Study all finite neighborhoods

On a smooth model the construction yields a connected divisor $D$, with primitive multiplicities, which is nef and numerically trivial on each of its components. Its normal line $\mathcal O_D(D)$ is torsion, and a power of the canonical bundle has a local divisor supported on $D$.

M2 studies successively thicker infinitesimal neighborhoods of $D$. These retain derivatives transverse to the divisor even though their underlying point set stays the same. Estimates for differential forms and finite jets control cohomology when poles of finite order along $D$ are allowed.

The resulting object is a punctured formal neighborhood: it remembers all finite thickenings while allowing the local equation of the divisor to be inverted. It is an algebraic formal construction, not simply an ordinary open subset of the original threefold.

Divisibility conflicts with integer degree

Cartier descent is a characteristic-$p$ method relating suitable differential data to Frobenius pullback. M2 uses it to claim that every line bundle on this formal neighborhood has a $p$-th root. Its Picard group, the group of line bundles under tensor product, would therefore be $p$-divisible.

At the same time, intersections with an ample divisor give a nonzero homomorphism from this group to $\mathbb Z$. The incompatibility is elementary once these constructions are available. If every group element can be divided by $p$ repeatedly, every integer in the homomorphism’s image must be divisible by $p^e$ for every $e\geq1$. Only zero has that property.

The geometric constructions instead give a positive degree, proportional to $D\cdot H^2>0$. The contradiction rules out the prepared configuration, forcing positive Iitaka dimension and then semiampleness through the cited earlier theorem.

7. Scope, limitations, and open questions

The characteristic restriction is $p>3$; neither characteristic two nor three is covered. The ground field is algebraically closed. Extensions to more general fields require separate descent and singularity arguments.

Numerical dimension one must not be replaced by nef dimension one. The hard part explicitly allows nef dimension three before excluding it. Nor does the result assert abundance for every nef divisor: the canonical-plus-boundary form is essential.

The proof’s stronger singularity conditions occur on intermediate models. M1’s final conclusion concerns the original lc pair, while M2 retains its terminal and $\mathbb Q$-factorial hypotheses. The argument uses established positive-characteristic minimal-model, nonvanishing, and surface-abundance inputs in addition to the companion obstruction.

The statements give no uniform bound on the generating multiple or a classification of the resulting surface fibers. Such effective bounds, smaller characteristics, and higher-dimensional analogues are further questions. The claims here are reported from the manuscripts without proof verification.

8. Sources and further reading

M1 §§1–2 state the log canonical theorem and the exact companion obstruction; §§3–6 explain boundary separation and the construction of the prohibited divisor. M2 §§1–2 describe the terminal case and prepared cycle; §§3–6 develop the formal-neighborhood and divisibility mechanism.

The linked papers of Miyaoka, Kawamata, Witaszek, and Zheng Xu locate the new claims among prior partial solutions. For optional terminology, see nef line bundles and Frobenius endomorphisms.

Source manuscripts

• M1. OpenAI, Log abundance in numerical dimension one for threefolds in positive characteristic (October 5, 2026).
• M2. OpenAI, Abundance in numerical dimension one for terminal threefolds in positive characteristic (September 24, 2026).

Sources consulted on 12 October 2026 at repository revision fd4aeeb. Download the manuscript citations.