Abundance theorem for surfaces over imperfect fields

Abstract

In this paper, we show the abundance theorem for log canonical surfaces over fields of positive characteristic.

Appendix A. Shokurov polytopes in convex geometry

In this section, we summarise some results of Shokurov polytopes in a setting of convex geometry. We fix the following notation.

Notation A.1

Let M and N be torsion free $\mathbb {Z}$-modules and let

$$\begin{aligned} \varphi :M \times N \rightarrow \mathbb {Z}\end{aligned}$$

be a $\mathbb {Z}$-bilinear homomorphism. For $D\in M$ and $C \in N$, we write

$$\begin{aligned} \varphi (D, C)=D\cdot C=C\cdot D \end{aligned}$$

by abuse of notation. Assume that $\dim _{\mathbb {R}}M_{\mathbb {R}}<\infty$, i.e. M is a free $\mathbb {Z}$-module of finite rank. Fix a rational polytope $\mathcal {L} \subset M_{\mathbb {R}}$. Fix an $\mathbb {R}$-linear basis of $M_{\mathbb {R}}$ and we denote the sup norm with respect to this basis by $||\bullet ||$.

Lemma A.2

We use Notation A.1. Fix $K \in M_{\mathbb {Q}}$, $\Delta \in \mathcal {L}$, and $\rho \in \mathbb {R}_{>0}$. Then, there exist positive real numbers $\epsilon , \delta >0$, depending on K, $\Delta$, and $\rho$, which satisfy the following properties.

• (1) For every $C \in N$ such that $-(K+B)\cdot C \le \rho$ for all $B\in \mathcal {L}$, if $(K+\Delta )\cdot C>0$, then $(K+\Delta )\cdot C>\epsilon$.

• (2) If $C\in N$ and $B_0\in \mathcal {L}$ satisfy $||B_0-\Delta ||<\delta$, $(K+B_0)\cdot C\le 0$, and $-(K+B)\cdot C \le \rho$ for all $B\in \mathcal {L}$, then $(K+\Delta )\cdot C\le 0$.

Proof

The assertions follow from the same proof as in [1, Proposition 3.2(1)] or [5, Theorem 4.7.2(1)]. The assertion (2) follows from the same proof as in [5, Theorem 4.7.2(2)] (although the statement of [5, Theorem 4.7.2(2)] is the same as the one of [1, Proposition 3.2(2)], [5, Theorem 4.7.2(2)] fixes minor errors appearing in [1, Proposition 3.2(2)]). $\square$

Proposition A.3

We use Notation A.1. Let $K \in M_{\mathbb {Q}}$ and $\rho \in \mathbb {R}_{>0}$. Fix a subset $\{C_t\}_{t\in T} \subset N$ such that

$$\begin{aligned} -(K+B)\cdot C_t\le \rho \end{aligned}$$

for every $t \in T$ and every $B\in \mathcal {L}$. For any subset $T' \subset T$, we define

$$\begin{aligned} \mathcal {N}_{T'}:=\{B\in \mathcal {L}\,|\,(K+B)\cdot C_t\ge 0 \,\,for\,\, every \,\,t \in T'\}. \end{aligned}$$

Then there exists a finite subset $S \subset T$ such that

$$\begin{aligned} \mathcal {N}_{T}=\mathcal {N}_{S}. \end{aligned}$$

In particular $\mathcal {N}_{T}$ is a rational polytope.

Proof

We can apply the same argument as in [1, Proposition 3.2(3)] by using Lemma A.2 instead of [1, Proposition 3.2(1)(2)]. $\square$