Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On minimal 3-folds with $K^3\geq 86$

Meng Chen, Sicheng Ding

TL;DR

The article resolves a long-standing question about when the 5-canonical map of a minimal projective 3-fold X becomes birational by proving that $\mathrm{Vol}(X)=K_X^3\ge 86$ suffices for $|5K_X|$ to be birational. It replaces earlier volume bounds with a moving-divisor strategy: construct a divisor $L$ with $h^0(X,L)\ge 5$ and ensure $D=2K_X-L$ is pseudo-effective, enabling a Tankeev-based induction and a case split by $d_L$. The work meticulously treats the cases $d_L=1,2,3$, proving non-vanishing of adjoint linear systems, and, when necessary, exploiting a $(1,2)$-surface fibration to conclude birationality. Overall, the results sharpen the canonical-volume threshold and introduce robust methods to handle low-codimension non-klt centers in higher-dimensional birational geometry, with potential further refinements for specific $p_g$ and χ configurations.

Abstract

We prove that the $5$-canonical map of every minimal projective $3$-fold $X$ with $K_X^3\geq 86$ is stably birational onto its image, which loosens previous requirements $K_X^3>4355^3$ and $K_X^3>12^3$ respectively given by Todorov and Chen. The essential technical ingredient of this paper is an efficient utilization of a moving divisor which grows from global $2$-forms by virtue of Chen-Jiang.

On minimal 3-folds with $K^3\geq 86$

TL;DR

The article resolves a long-standing question about when the 5-canonical map of a minimal projective 3-fold X becomes birational by proving that suffices for to be birational. It replaces earlier volume bounds with a moving-divisor strategy: construct a divisor with and ensure is pseudo-effective, enabling a Tankeev-based induction and a case split by . The work meticulously treats the cases , proving non-vanishing of adjoint linear systems, and, when necessary, exploiting a -surface fibration to conclude birationality. Overall, the results sharpen the canonical-volume threshold and introduce robust methods to handle low-codimension non-klt centers in higher-dimensional birational geometry, with potential further refinements for specific and χ configurations.

Abstract

We prove that the -canonical map of every minimal projective -fold with is stably birational onto its image, which loosens previous requirements and respectively given by Todorov and Chen. The essential technical ingredient of this paper is an efficient utilization of a moving divisor which grows from global -forms by virtue of Chen-Jiang.

Paper Structure

This paper contains 13 sections, 13 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and conventions
  4. Technical preparation
  5. The birationality of $\varphi_{5,X}$ (Part I: $d_L=1$ and $3$)
  6. Set-up for $\varphi_{L,X}$
  7. The case with $d_L=1$
  8. The non-vanishing of $H^0(X',5K_{X'}-M')$ when $d_L\geq 2$
  9. The case with $d_L=3$
  10. The birationality of $\varphi_{5,X}$ (Part II: $d_L=2$)
  11. The case with large canonical degree
  12. The case with small canonical degree
  13. Proof of Theorem \ref{['mainthm']}

Key Result

Theorem 1.2

Let $X$ be a minimal projective 3-fold with the canonical volume $\text{\rm Vol}(X)\geq 86$. Then $|mK_X|$ gives a birational map for all $m\geq 5$.

Theorems & Definitions (28)

  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 18 more