Log-canonical threefold abundance in numerical dimension one

1 · The essentials

These manuscripts study three-dimensional geometric spaces and a question about turning numerical information into geometric structure. The main manuscript considers a space together with an additional weighted boundary. Under specific assumptions, it shows that a distinguished expression attached to the space and boundary has enough globally defined sections, after taking a suitable multiple, to have no common zero. This property is called semiampleness. The result concerns a narrowly specified case: the expression has numerical dimension one, and the underlying field has characteristic greater than three. The main manuscript does not require two additional restrictions on the space that appear in the companion manuscript. The result is shown in the manuscript, but it is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine lamps whose beams collectively illuminate every point of a room. Semiampleness resembles being able to choose, after a suitable scaling, a collection of lamps that leaves no point unlit; the numerical assumptions are not themselves that collection.

Why it matters

The main result establishes the stated numerical-dimension-one case of log abundance for projective log canonical threefold pairs in characteristic greater than three. Its scope includes an effective rational boundary and does not require the original space to be terminal or ℚ-factorial. The companion manuscript treats the related boundary-free setting under those additional restrictions.

2 · Going further

The precise problem concerns a projective log canonical threefold pair $(X, B)$ over an algebraically closed field of characteristic $p > 3$ . The boundary $B$ is effective with rational coefficients, and the adjoint divisor $D = K_{X} + B$ is assumed to be ℚ-Cartier and nef, with numerical dimension $ν (D) = 1$ . The main manuscript shows that $D$ is semiample: a suitable positive integer multiple is Cartier and has a base-point-free complete linear system.

The companion manuscript considers a projective ℚ-factorial terminal threefold $X$ over an algebraically closed field of characteristic $p > 3$ . Under the assumptions that $K_{X}$ is nef and $ν (K_{X}) = 1$ , it shows that $K_{X}$ is semiample and that $κ (X, K_{X}) = 1$ . The main manuscript instead allows a log canonical pair with an effective rational boundary and explicitly dispenses with terminality and ℚ-factoriality of the original $X$ .

The supplied abstracts state the conclusions but give no description of the argument. Consequently, they do not support an account of the main technique, intermediate steps, or whether one manuscript's argument is used in the other. Their stated relationship is the common numerical-dimension-one abundance question, treated under different hypotheses.

Useful notions

Projective varieties and dimension
Divisors and rational boundaries
Singularities of pairs
Nefness and numerical dimension
Linear systems and semiampleness

What was known before / what is new

Before

The supplied material gives no historical account or chronology of earlier results. It includes a companion manuscript showing abundance in numerical dimension one for projective ℚ-factorial terminal threefolds in characteristic greater than three.

New

The main manuscript states the numerical-dimension-one log abundance result for projective log canonical threefold pairs with effective rational boundary. Unlike the companion statement, it does not require the original variety to be terminal or ℚ-factorial. Its conclusion is semiampleness of the nef, ℚ-Cartier adjoint $K_{X} + B$ .

Limits and assumptions

The main statement requires projectivity, dimension three, log canonical singularities of the pair, an effective rational boundary, and an algebraically closed field of characteristic $p > 3$ . It also requires $K_{X} + B$ to be ℚ-Cartier, nef, and of numerical dimension exactly one. The supplied abstracts do not address other numerical dimensions, other characteristics, or the removal of the remaining hypotheses, and they identify no further open problems. No argument details are supplied. The main result is not formalized in Lean and has not been formally verified.

Mini glossary

Threefold: An algebraic variety of dimension three.
Effective rational boundary: A boundary divisor whose coefficients are nonnegative rational numbers.
ℚ-Cartier divisor: A divisor for which some positive integer multiple is Cartier.
Nef: A divisor is nef if its intersection with every curve is nonnegative.
Semiample: A divisor is semiample if some positive integer multiple is Cartier and its global sections have no common zero.

Keywords : Log abundance · Threefolds · Numerical dimension one

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Proves log abundance for projective log canonical threefold pairs over algebraically closed fields of characteristic p > 3 when the effective boundary is rational and the ℚ-Cartier adjoint is nef of numerical dimension one. The adjoint is semiample, without requiring the original variety to be terminal or ℚ-factorial.

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

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Log abundance in numerical dimension one for threefolds in positive characteristic

05/10/2026

To be verified

We prove the numerical-dimension-one case of log abundance for threefolds over algebraically closed fields of characteristic p > 3. If $(X, B)$ is a projective log canonical threefold pair with effective rational boundary, and $K_{X} + B$ is ℚ-Cartier, nef, and of numerical dimension one, then $K_{X} + B$ is semiample. Neither terminality nor ℚ-factoriality of X is required.

PDF GitHub folder

Abundance in numerical dimension one for terminal threefolds in positive characteristic

24/09/2026

To be verified

Let X be a projective ℚ-factorial terminal threefold over an algebraically closed field of characteristic p > 3. We prove the numerical-dimension-one case of abundance: if K_X is nef with $ν (K_{X}) = 1$ , then K_X is semiample and $κ (X, K_{X}) = 1$ .

PDF GitHub folder