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Elliptic singularities and threefold flops in positive characteristic

Hiromu Tanaka

TL;DR

The paper extends the theory of threefold flops to positive characteristic by proving that any flop of a smooth threefold exists and is smooth over an algebraically closed field of characteristic $p>0$. A central innovation is the development of two-dimensional elliptic singularities over imperfect fields, establishing base-point freeness, very ampleness, and projective normality for associated divisors, and analyzing their canonical models via fundamental cycles and conductor sequences. The approach mirrors characteristic-zero strategies, using generic hyperplane sections and canonical models to reduce to the elliptic-surface analysis, and culminates in vanishing theorems and a precise description of Gorenstein terminal singularities, enabling the flop to be smooth. These results feed into broader goals in Fano classification and birational geometry in positive characteristic, providing essential tools for handling singularities and birational transforms. The work thereby strengthens the toolbox for birational operations in low characteristics and clarifies how elliptic singularities govern the behavior of flips and flops in this setting.

Abstract

Let $X$ be a smooth threefold over an algebraically closed field of positive characteristic. We prove that an arbitrary flop of $X$ is smooth. To this end, we study Gorenstein curves of genus one and two-dimensional elliptic singularities defined over imperfect fields.

Elliptic singularities and threefold flops in positive characteristic

TL;DR

The paper extends the theory of threefold flops to positive characteristic by proving that any flop of a smooth threefold exists and is smooth over an algebraically closed field of characteristic . A central innovation is the development of two-dimensional elliptic singularities over imperfect fields, establishing base-point freeness, very ampleness, and projective normality for associated divisors, and analyzing their canonical models via fundamental cycles and conductor sequences. The approach mirrors characteristic-zero strategies, using generic hyperplane sections and canonical models to reduce to the elliptic-surface analysis, and culminates in vanishing theorems and a precise description of Gorenstein terminal singularities, enabling the flop to be smooth. These results feed into broader goals in Fano classification and birational geometry in positive characteristic, providing essential tools for handling singularities and birational transforms. The work thereby strengthens the toolbox for birational operations in low characteristics and clarifies how elliptic singularities govern the behavior of flips and flops in this setting.

Abstract

Let be a smooth threefold over an algebraically closed field of positive characteristic. We prove that an arbitrary flop of is smooth. To this end, we study Gorenstein curves of genus one and two-dimensional elliptic singularities defined over imperfect fields.
Paper Structure (28 sections, 53 theorems, 216 equations)

This paper contains 28 sections, 53 theorems, 216 equations.

Table of Contents

  1. Introduction
  2. Overviews of proofs
  3. Preliminaries
  4. Notation
  5. Fundamental cycles
  6. Conductor exact sequences
  7. Curves over an imperfect field
  8. Curves of genus zero
  9. Divisors
  10. A characterisation via $\Delta$-genus
  11. Examples
  12. Gorenstein curves of genus one
  13. Base point freeness
  14. Very ampleness and projective normality
  15. Two-dimensional elliptic singularities over imperfect fields
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $k$ be an algebraically closed field of characteristic $p>0$ and let$Y$ be a smooth threefold over $k$. Let $\psi : Y \to Z$ be a flopping contraction, i.e., $\psi$ is a projective birational $k$-morphism to a normal threefold $Z$ over $k$ such that $\dim {\operatorname{Ex}}(\psi)=1$ and $K_Y$

Theorems & Definitions (124)

  • Theorem 1.1: Theorem \ref{['t-flop']}
  • Theorem 1.2: Theorem \ref{['t-terminal3']}
  • Theorem 1.3: Theorem \ref{['t-ell-3']}, Theorem \ref{['t-ell-12']}
  • Theorem 1.4: Theorem \ref{['t-g1-va-prime']}
  • Theorem 1.5: Theorem \ref{['t-g1-va']}
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 114 more