Optimal bounds in Bend-and-Break

TL;DR

This work sharpens the Bend-and-Break framework by proving an optimal bound: for a projective variety $X$ with a nef divisor $H$ and a curve $C$ with $K_X\cdot C<0$, every point on $C$ lies on a rational curve $R$ with $H\cdot R \le (\dim X+1)\frac{H\cdot C}{-K_X\cdot C}$, improving the classical $2\dim X$ bound and achieving optimality on $\mathbb{P}^n$ with a hyperplane. It introduces Bend-and-Shatter, a technique showing that a $k$-dimensional family of maps fixing $k$ points yields $k$ non-contracted rational components, which together with MM86/Kollar reductions yields the Bend-and-Break conclusion. The paper then extends these ideas to extremal rays of log canonical pairs, proving that any $(K_X+\Delta)$-negative extremal ray is generated by a rational curve $C$ with $-(K_X+\Delta)\cdot C \le \dim X+1$, with stronger inequalities in birational, klt cases. Collectively, these results provide optimal degree bounds for extremal rays in lc/dlt settings, generalize known toric and LCIQ cases, and deepen the toolkit for the minimal model program via precise control of rational curves and their degenerations.

Abstract

We improve the Bend-and-Break result of Miyaoka and Mori by establishing the optimal degree bound. Our result also yields optimal bounds on lengths of extremal rays of log canonical pairs.

Theorems & Definitions (8)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1: Bend-and-Shatter
• proof
• Proposition 2.2
• proof
• proof : Proof of Theorem \ref{['theo:cleanbandb']}: