---
abstract: 'In this paper, by using the trace map of Frobenius, we consider problems on extending sections for positive characteristic threefolds.'
address: 'Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502 Japan'
author:
- Hiromu Tanaka
title: The trace map of Frobenius and extending sections for threefolds
---

Introduction
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In characteristic zero, by the Kodaira vanishing theorem and its generalizations, we can establish some results on adjoint divisors, such as the Kawamata–Shokurov basepoint free theorem (see, for example, [@KM Theorem 3.3]) and the Hacon–M^c^Kernan extension theorem ([@HM Theorem 5.4.21]). These theorems claim, under suitable conditions, that an adjoint divisor $m(K_X+\Delta+A)$ has good properties where $m\in\mathbb Z_{>0}$, $(X, \Delta)$ is a pair and $A$ is an ample divisor. In this paper, we only consider the following very simple situation: $X$ is a smooth projective variety, $\Delta=S$ is a smooth prime divisor and $A$ is an ample Cartier divisor. The following fact immediately follows from the Kodaira vanishing theorem.

\[0char-zero\] Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be a smooth projective variety over $k$. Let $S$ be a smooth prime divisor on $X$ and let $A$ be an ample Cartier divisor on $X$ such 