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Varieties with pseudoeffective canonical divisor and their Kodaira dimension

Gilberto Bini

TL;DR

This paper studies Mori's conjecture that a smooth complex projective variety $X$ is uniruled precisely when its Kodaira dimension $kod(X)$ is negative. It develops a framework connecting $kod(X)$ to Castelnuovo-Mumford regularity of multiples of $K_X$ via Koszul cohomology, using a generic complete intersection curve $C\subset X$ and a spectral sequence argument to translate cohomological data into regularity bounds. Under the assumption that $K_X$ is pseudoeffective, the authors prove that $kod(X)\ge0$ by establishing CM-regularity bounds $reg_A((1-s)K_X)\ge d+1$ (for $s\ge s_A$) and extracting a nonvanishing Koszul obstruction, thereby ruling out negative Kodaira dimension in dimension $d\ge3$. Together with the BDPP nonvanishing result, this yields that $kod(X)=-\infty$ iff $X$ is uniruled in dimension $d\ge3$, with reductions to curves and surfaces for smaller dimensions. The work clarifies the nonvanishing direction of Mori's conjecture under a precise KO-spectral framework and highlights the role of Green-Lazarsfeld-type Koszul cohomology in birational geometry.

Abstract

Let X be a smooth, projective variety over the field of complex numbers. Here we focus on a conjecture attributed to Shigefumi Mori, which claims that X is uniruled if and only if the Kodaira dimension of X is negative.

Varieties with pseudoeffective canonical divisor and their Kodaira dimension

TL;DR

This paper studies Mori's conjecture that a smooth complex projective variety is uniruled precisely when its Kodaira dimension is negative. It develops a framework connecting to Castelnuovo-Mumford regularity of multiples of via Koszul cohomology, using a generic complete intersection curve and a spectral sequence argument to translate cohomological data into regularity bounds. Under the assumption that is pseudoeffective, the authors prove that by establishing CM-regularity bounds (for ) and extracting a nonvanishing Koszul obstruction, thereby ruling out negative Kodaira dimension in dimension . Together with the BDPP nonvanishing result, this yields that iff is uniruled in dimension , with reductions to curves and surfaces for smaller dimensions. The work clarifies the nonvanishing direction of Mori's conjecture under a precise KO-spectral framework and highlights the role of Green-Lazarsfeld-type Koszul cohomology in birational geometry.

Abstract

Let X be a smooth, projective variety over the field of complex numbers. Here we focus on a conjecture attributed to Shigefumi Mori, which claims that X is uniruled if and only if the Kodaira dimension of X is negative.
Paper Structure (5 sections, 4 theorems, 10 equations)

This paper contains 5 sections, 4 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Ample Curves and Castelnuovo-Mumford regularity
  4. Nonnegative Kodaira Dimension
  5. Uniruled Varieties

Key Result

Lemma 2.1

Following the notation adopted above, if the numerical dimension $\nu(X)$ of $X$ is different from $0$, the divisor ${K_X}_{|_C}$ is ample on $C$.

Theorems & Definitions (8)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof