The log minimal model program for threefold pairs in characteristics 2 and 3

The log minimal model program (LMMP) for projective pairs of dimension three — existence of flips and minimal models, the cone, contraction and base point free theorems, termination with scaling, and existence of Mori fibre spaces — was established for klt pairs over algebraically closed fields of characteristic $p>5$ by Birkar–Waldron, and extended to characteristic 5 (indeed to every characteristic $p>3$, over perfect fields) by Hacon–Witaszek. The surviving part of the source question asks whether the same program is valid in characteristics 2 and 3. There the methods of all known proofs break down: pl-flipping contractions (flips for pairs with a single coefficient-one boundary component) need not be relatively purely $F$-regular (that is, compatible with iterates of the Frobenius), purely log terminal centres can be non-normal, terminal threefold singularities need not be Cohen–Macaulay, the expected vanishing theorems fail, and Mori fibre spaces exhibit pathologies found only in characteristics 2 and 3; no counterexample to the program itself is known. The problem is to prove, or refute, the three-dimensional LMMP package for pairs over fields of characteristic 2 or 3.

Background

The minimal model program (MMP) is the central organizing scheme of birational geometry. Given a projective variety $X$ over a field together with an effective $\mathbb{Q}$-divisor $\Delta$ (the pair $(X,\Delta)$, with $\Delta$ the boundary), one tries to modify $X$ birationally so that the log canonical class $K_X+\Delta$ becomes nef, where $K_X$ is the canonical divisor. The modification proceeds by contractions of $(K_X+\Delta)$-negative extremal rays — the one-dimensional faces of the closed cone of curves $\overline{NE}(X)$, the closure of the cone spanned by classes of complete curves (its dual is the cone of nef divisors, those intersecting every curve non-negatively). A contraction may be divisorial (an exceptional divisor is contracted) or small (only a locus of codimension at least two is contracted); a small contraction must be corrected by a flip, a birational surgery replacing the contracted locus by a new one on which $-(K_X+\Delta)$ is relatively ample. If $K_X+\Delta$ is pseudo-effective (a limit of effective divisors), the expected output is a minimal model on which $K_X+\Delta$ is nef; otherwise the expected output is a Mori fibre space, a contraction $X\to S$ with $\dim S<\dim X$, relative Picard number $\rho(X/S)=1$ (the rank of the group of line bundles on $X$ modulo those pulled back from $S$) and $-(K_X+\Delta)$ ample over $S$, which reduces classification questions to lower-dimensional bases and fibres.

The singularities of a pair are measured by discrepancies: writing $K_Y=f^*(K_X+\Delta)+\sum_E a_E E$ on a log resolution $f\colon Y\to X$, the pair is klt (Kawamata log terminal) when all $a_E>-1$, and dlt (divisorial log terminal) under the mild relaxation that allows coefficients equal to $1$ and normal-crossing behaviour on a dense open subset; klt and dlt pairs are the standard classes in which the MMP is run. A normal variety is $\mathbb{Q}$-factorial when every Weil divisor is $\mathbb{Q}$-Cartier. The analytic backbone of the program in characteristic zero consists of the cone theorem (generation of $\overline{NE}(X)$ by the $(K_X+\Delta)$-non-negative part plus countably many rational curves $\Gamma_i$ with $-6\le(K_X+\Delta)\cdot\Gamma_i<0$ in dimension three, without accumulation in the negative part), the contraction theorem (each such ray is contractible), the base point free theorem (a nef divisor $D$ with $D-(K_X+\Delta)$ nef and big is semi-ample, i.e. some multiple defines a morphism), and termination with scaling (running the program on $K_X+\Delta+\lambda A$, with the scaling parameter $\lambda$ decreased at each step against a fixed ample divisor $A$, stops in finitely many steps). In characteristic zero this package is known for projective pairs in all dimensions.

Over fields of positive characteristic $p$, the classical proofs collapse: the Kodaira and Kawamata–Viehweg vanishing theorems (cohomology vanishing for ample, respectively nef and big, line bundles) fail, and the cone and base point free theorems must instead be built out of the existence of flips. For threefolds over algebraically closed fields, Hacon and Xu established the existence of flips for threefold pairs in characteristic $p>5$, Cascini–Tanaka–Xu proved base point freeness in the same range for pairs that are strongly $F$-regular (a Frobenius condition recalled below), and Birkar and Waldron completed the package: for projective klt pairs of dimension three over an algebraically closed field of characteristic $p>5$ they proved the cone theorem, the base point free theorem, the contraction theorem, finiteness of minimal models, termination with scaling, and the existence of Mori fibre spaces, by methods that deliberately avoid canonical bundle formulas.

The positive-characteristic proofs use the Frobenius morphism $F\colon X\to X$, which raises functions to their $p$-th powers. A pair is strongly $F$-regular when every perturbation by a small effective divisor splits off some iterate of $F$; this is the Frobenius-theoretic counterpart of the klt condition, and the two notions agree for surfaces in the large-characteristic regime on which these proofs rest, but not in small characteristic. The Hacon–Xu flip construction reduces to pl-flips — flips for plt (purely log terminal) pairs, the class between klt and dlt in which exactly one boundary component $S$ has coefficient $1$, taken with $-S$ relatively ample over the contraction — and requires pl-flipping contractions to be relatively purely $F$-regular; this is the main input that holds for $p>5$ and, as Hacon–Witaszek note, is not true in general when $p\le 5$. The singularity theory of small characteristics degenerates in parallel: in characteristic 2 there exist purely log terminal threefold pairs whose codimension-one log canonical centre (the image of the boundary component of coefficient $1$) is not normal, klt threefold singularities that are neither rational (a resolution carries no higher direct images of the structure sheaf) nor Cohen–Macaulay (the local rings have depth equal to their dimension), and klt Fano threefolds with non-vanishing intermediate cohomology (Cascini–Tanaka); non-normal purely log terminal centres exist in every characteristic $p\ge 3$ in dimension $2p+2$ (Bernasconi); terminal threefold singularities that are not Cohen–Macaulay exist precisely in (mixed) characteristics $2,3,5$ (Totaro); and in characteristics 2 and 3 there are Mori fibre spaces with klt total space but non-klt base, and Mori fibre spaces carrying relatively non-trivial torsion line bundles (Tanaka).

Subsequent to the source paper, Hacon and Witaszek settled the characteristic-5 case: flips exist for $\mathbb{Q}$-factorial dlt threefold pairs over perfect fields of characteristic 5, and consequently, for threefold pairs over perfect fields of every characteristic $p>3$, the MMP with scaling can be run and terminates with a log minimal model or a Mori fibre space, the base point free theorem holds for klt pairs, and the cone theorem holds for projective $\mathbb{Q}$-factorial dlt pairs. Their argument replaces the missing $F$-regularity by $m$-complements (effective boundaries $B^{+}\ge\Delta$ for which $m(K_X+B^{+})$ is linearly trivial over the base, with mild singularities) with $m\in\{1,2,3,4,6\}$, together with qdlt modifications (birational replacements whose singularities are log canonical and, at each log canonical centre, quotient singularities); they note that the key statement — that $6$-complements force relative $F$-regularity for surface pairs — is the step they are unable to generalise to characteristic 3, and express the hope that their methods will be useful for characteristics 2 and 3. In the same period they proved that the relative dlt MMP over $\mathbb{Q}$-factorial bases is valid in all characteristics $p>0$.

The original question — whether the Birkar–Waldron package extends from $p>5$ to $p\le 5$ — thus survives precisely in characteristics 2 and 3, for which neither a proof of any component of the package nor a counterexample to it is known.

Problem Statement

Let $k$ be an algebraically closed field of characteristic $2$ or $3$. Determine whether the following package of statements — the log minimal model program in dimension three — holds over $k$; each item is stated with the hypotheses under which the characteristic-$\ge 5$ source theorems (Birkar–Waldron, Hacon–Witaszek) establish it, and $\overline{NE}(X)_{K_X+\Delta\ge 0}$ denotes the subcone of the closed cone of curves on which $K_X+\Delta$ is non-negative:

1. Flips. If $(X,\Delta)$ is a $\mathbb{Q}$-factorial dlt pair of dimension $3$ over $k$ and $f\colon X\to Z$ is a $(K_X+\Delta)$-flipping contraction (a small projective contraction such that $-(K_X+\Delta)$ is relatively ample over $Z$), then the flip $f^{+}\colon X^{+}\to Z$ exists.

2. Cone theorem. If $(X,\Delta)$ is a projective $\mathbb{Q}$-factorial dlt pair of dimension $3$ over $k$, then there exist countably many rational curves $\Gamma_i$ on $X$ such that

$$\overline{NE}(X)=\overline{NE}(X)_{K_X+\Delta\ge 0}+\sum_i\mathbb{R}[\Gamma_i],\qquad -6\le (K_X+\Delta)\cdot\Gamma_i<0,$$

and the rays $\mathbb{R}[\Gamma_i]$ do not accumulate inside $\overline{NE}(X)_{K_X+\Delta<0}$.

3. Contraction theorem. If $(X,\Delta)$ is a projective $\mathbb{Q}$-factorial dlt pair of dimension $3$ over $k$, then every $(K_X+\Delta)$-negative extremal ray $R$ of $\overline{NE}(X)$ is contractible: there exists a projective contraction $X\to T$ whose contracted curves are exactly those with class in $R$.

4. Base point free theorem. If $(X,\Delta)$ is a projective klt pair of dimension $3$ over $k$ and $D$ is a nef $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor on $X$ such that $D-(K_X+\Delta)$ is nef and big, then $D$ is semi-ample.

5. Termination with scaling and Mori fibre spaces. If $(X,\Delta)$ is a projective klt pair of dimension $3$ over $k$ and $A$ is a fixed ample divisor, then the $(K_X+\Delta)$-MMP with scaling of $A$ terminates after finitely many steps; if $K_X+\Delta$ is pseudo-effective it terminates with a log minimal model, and otherwise with a Mori fibre space.

References

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