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Liftings of ideals in positive characteristic to those in characteristic zero: up to dimension three

Shihoko Ishii

Abstract

We study a pair consisting of a smooth variety of arbitrary dimension over a field of positive characteristic and a multi-ideal with a real exponent. We prove that the set of log discrepancies for a fixed exponent is discrete. Additionally, we show that the set of log canonical thresholds (lcts) of multi-ideals on a smooth variety in positive characteristic is contained within the set of lcts of multi-ideals on a smooth variety over the complex number field. As a result, we find that the accumulation points of log canonical thresholds are rational if all the exponents are rational. We also obtain ACC for the set of lcts of multi-ideals on a smooth varieties in positive characteristic. This version is rewritten for up to dimension three.

Liftings of ideals in positive characteristic to those in characteristic zero: up to dimension three

Abstract

We study a pair consisting of a smooth variety of arbitrary dimension over a field of positive characteristic and a multi-ideal with a real exponent. We prove that the set of log discrepancies for a fixed exponent is discrete. Additionally, we show that the set of log canonical thresholds (lcts) of multi-ideals on a smooth variety in positive characteristic is contained within the set of lcts of multi-ideals on a smooth variety over the complex number field. As a result, we find that the accumulation points of log canonical thresholds are rational if all the exponents are rational. We also obtain ACC for the set of lcts of multi-ideals on a smooth varieties in positive characteristic. This version is rewritten for up to dimension three.

Paper Structure

This paper contains 5 sections, 23 theorems, 307 equations.

Table of Contents

  1. Introduction
  2. Basics on liftings
  3. log discrepancies for multi-ideals and log canonical thresholds of ideals
  4. Construction of a weak lifting of the prime divisor
  5. Proof of the Main Theorem and applications

Key Result

Theorem 1.1

Let $N$ be an integer $\geq 2$, and $k$ an algebraically closed field of characteristic $p>0$. Let $A={\mathbb{A}}_k^N$ be the affine space over $k$ of dimension $N$, and $0$ the origin of $A$. Let $E$ be a prime divisor over $A$ with the center at $0$, and ${\mathfrak{a}}, {\mathfrak{a}}_1,\ldots, Then, there exists a prime divisor $F_{\mathbb{C}}$ over the affine space $A_{\mathbb{C}}={\mathbb{

Theorems & Definitions (72)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • proof
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • ...and 62 more