Log-canonical threefold abundance in numerical dimension one

The explanation

Take a line times a plane, the product of a projective line and a projective plane. A divisor on it is a surface inside it, counted with a multiplicity. Let L be the surface sitting above one point of the line. Let H be the sum of L and the surface above a line in the plane. Then L·H·H = 1 but L·L·H = 0, since two different copies of L never meet. A nef divisor with these two kinds of numbers has numerical dimension one. Here L comes from a map to the line, and its copies are the fibers.

Abundance asks for such maps in general. On a shape X there is a special divisor, the canonical divisor K_X. Nef means it meets every curve with nonnegative degree. Abundance predicts that a nef canonical divisor is semiampleSource 1. That means some multiple of it is cut out by enough functions with prescribed poles to define a map with no base points. Semiampleness turns positive numbers into an actual mapSource 2. When the numerical dimension is one, the map goes to a curve. The log version adds a boundary B, a weighted sum of surfaces, and asks the same of K_X + B. Log canonical is the mildest class of singular pairs where this makes sense.

Over the complex numbers, threefolds have been settled for decades. Miyaoka proved this case for minimal threefolds in 1988Source 3. Kawamata gave another proof in 1992. Keel, Matsuki and McKernan proved log abundance for log canonical threefold pairs in 1994Source 4. This family works in positive characteristic. That means arithmetic where some prime p counts as zero. With p = 5, adding 1 five times gives 0. Many tools from complex geometry fail in that world. Zheng Xu proved key cases for characteristic above 3. These include nonvanishing and the case of nef dimension at most twoSource 5. Nef dimension counts how many directions the divisor can see. Xu then proved the case of numerical dimension twoSource 6. One case stayed open: numerical dimension one with nef dimension threeSource 7.

The companion paper treats terminal threefolds, which have the mildest singularities. It claims that if K_X is nef of numerical dimension one, then K_X is semiample. So it gives a map to a curveSource 8. The danger it rules out is a divisor with the right numbers but only one section, which could never give a mapSource 9.

The main paper claims the log case. For a projective log canonical threefold pair in characteristic above 3, a nef K_X + B of numerical dimension one is semiampleSource 10. The threefold X need not be terminal or Q-factorial, a technical tidiness conditionSource 11. The boundary coefficients can be any rationals that log canonicity allows.

The obstacle is the boundary. The companion's tool, a ban on a certain kind of curve-like divisor, speaks about the canonical bundle aloneSource 12,Source 13. Near the log divisor, the boundary and the resolution's extra pieces get in the way. So the proof needs a relation for a power of the canonical bundle thereSource 14. It first shows each boundary piece either lies inside the divisor's support or misses it entirely. A bend-and-break argument with rational curves supplies that split. The forbidden divisor then appears, giving a contradiction.

How sure is this? No Lean formalization is listed for either paper. Both rest on the positive-characteristic minimal model program and many earlier results. No specialist review has been published.

What it doesn't show. This does not cover characteristics 2 and 3, or numerical dimension zero.