# A Reider-type Result for Smooth Projective Toric Surfaces

## TL;DR

This paper establishes numerical criteria for when the adjoint line bundle on a smooth projective toric surface is nef or ample, based on the associated lattice polytope.

## Contribution

It provides new necessary and sufficient conditions for the positivity of the adjoint series on toric surfaces using polytope analysis.

## Key findings

- Criteria for nefness of |K_X+L|
- Criteria for ampleness of |K_X+L|
- Polytope-based numerical conditions

## Abstract

Let $L$ be an ample line bundle over a smooth projective toric surface $X$. Then $L$ corresponds to a very ample lattice polytope $P$ that encodes many geometric properties of $L$. In this article, by studying $P$, we will give some necessary and sufficient numerical criteria for the adjoint series $|K_X+L|$ to be either nef or (very) ample.