# Effective Adjunction Theory

## TL;DR

This paper explores the effectivity of adjoint divisors on algebraic varieties, establishing criteria for uniruledness and non-vanishing of sections in polarized manifolds, advancing understanding of their geometric properties.

## Contribution

It proves new characterizations of uniruled varieties and non-vanishing results for adjoint line bundles on polarized manifolds.

## Key findings

- Uniruledness characterized by vanishing of certain sections.
- Non-vanishing of sections for $K_X+tL$ when pseudo-effective.
- Provides criteria linking effectivity and geometric properties.

## Abstract

Here we investigate the property of effectivity for adjoint divisors. Among others, we prove the following results: (i) A normal projective variety $X$ with at most canonical singularities is uniruled if and only if for each very ample Cartier divisor $H$ on $X$ we have $H^0(X, m_0K_X+H)=0$ for some $m_0=m_0(H)>0$. (ii) Let $(X,L)$ be a polarized manifold of dimension $4$ and let $t$ be an integer with $t \ge 3$. If $K_X+tL$ is pseudo-effective, then $H^0(X, K_X+tL) \ne 0$.