Extremal rays of non-integral $L$-length

TL;DR

This paper investigates the structure of smooth complex projective varieties with line bundles when a specific invariant related to extremal rays takes non-integer values, revealing new geometric insights.

Contribution

It characterizes the structure of pre-polarized manifolds for non-integral values of the invariant u_L(R), extending understanding of extremal rays in algebraic geometry.

Findings

Describes the structure of u_L(R) for non-integer values.

Provides classification results for extremal rays with non-integral u_L(R).

Enhances understanding of the geometry of line bundles on smooth projective varieties.

Abstract

Let $X$ be a smooth complex projective variety and let $L$ be a line bundle on it. We describe the structure of the pre-polarized manifold $(X,L)$ for non integral values of the invariant $\tau_L(R):=-K_X\cdot\Gamma/(L \cdot \Gamma)$, where $\Gamma$ is a minimal curve of an extremal ray $R:=\mathbb R_+[\Gamma]$ on $X$ such that $L \cdot R>0$.