Abundance mod p?

0:00Characteristic p

Our last episode was about abundance: a nef canonical class should be semiample, so that its sections define a map. This episode moves to characteristic p, where varieties are cut out by equations over a field in which a prime p equals zero. There, familiar tools break. The derivative of x^p is p·x^(p−1), which is zero. And Kodaira’s vanishing theorem can fail. Still, the minimal model program for threefolds now works when p is greater than five, by Hacon and Xu, Birkar, and Birkar and Waldron, and for p equal to five, by Hacon and Witaszek. Abundance for threefolds in characteristic p has been assembled piece by piece, by Chenyang Xu and Lei Zhang, Witaszek, and others. Most recently, Zheng Xu proved nonvanishing when p is greater than three, and then the case of numerical dimension two.

0:55Numerical dimension one

The numerical dimension of a nef class L on a threefold measures how many copies of L can still meet. Numerical dimension one means that L·H² > 0, but L²·H = 0, for an ample class H. Here’s the typical example. On the product of a line and a plane, pull back a point of the line. Its intersection numbers are one and zero. It’s semiample: it comes from the map to the line, and every fiber is full of curves on which L has degree zero. The danger is a class with the same numbers, but only one section up to scale, and no degree-zero curve through a very general point. Its nef dimension would be three. Zheng Xu’s results, covering positive Kodaira dimension and nef dimension at most two, left exactly this case open, in numerical dimension one.

1:48The claim

Result 035 has two OpenAI preprints. A 46-page paper, dated September 24, treats projective, ℚ-factorial terminal threefolds over an algebraically closed field of characteristic greater than three. If K is nef of numerical dimension one, it’s semiample, with Kodaira dimension one. A 16-page paper, dated October 5, extends this to log canonical pairs with rational boundary, without assuming that the variety is terminal or ℚ-factorial. That would settle the numerical-dimension-one case of threefold log abundance when p is greater than three.

2:28How the proof goes

Suppose a counterexample exists. The terminal paper first puts it in a precise shape: a smooth threefold V with a connected nef cycle D. The cycle is numerically trivial on each of its components, its normal bundle is torsion, and near D, a p-power of the canonical bundle is a divisor supported on D. Then it studies forms on thicker and thicker neighborhoods of D. The tangent bundle is strongly semistable, since otherwise a foliation would give curves of degree zero through very general points. A careful count of jets pins down the canonical bundle near D, and how deep powers of D’s normal bundle stay trivial: a limiting ratio of p/(p + 1). The finish is pure characteristic p. Cartier descent, built on the Frobenius, shows that every line bundle on the punctured formal neighborhood of D is a p-th power. But the numerical conditions give a degree map from those line bundles to the integers that is not zero. A p-th power has degree divisible by p, so every degree is divisible by every power of p, and must be zero. That’s the contradiction. The log canonical paper builds the same forbidden shape from a pair with boundary. Bend-and-break makes each boundary component lie inside the cycle or away from it, a partial minimal model program isolates one component, surface abundance gives torsion there, and a finite cover removes the part of the exponent prime to p.

3:59Where it stands

These are preprints, not yet peer reviewed, with no Lean proof, and neither has been revised. They build on Zheng Xu’s published work, and the log canonical paper takes the terminal paper’s obstruction as an input. We found no public assessment. If they hold, this case of abundance is settled for threefolds in every characteristic greater than three, matching what Miyaoka, Kawamata, and Keel, Matsuki and McKernan proved over the complex numbers more than thirty years ago.