# Newton-Okounkov bodies on projective bundles over curves

## TL;DR

This paper computes Newton-Okounkov bodies for projective bundles over curves, using Wolfe's estimates, and characterizes semi-stable vector bundles through these geometric objects.

## Contribution

It provides explicit computations of Newton-Okounkov bodies on these varieties and links them to the stability of vector bundles, offering new geometric insights.

## Key findings

- Explicit Newton-Okounkov bodies for projective bundles over curves
- Characterization of semi-stability via Newton-Okounkov bodies
- Application of Wolfe's estimates to volume computations

## Abstract

In this article, we study Newton-Okounkov bodies on projective vector bundles over curves. Inspired by Wolfe's estimates used to compute the volume function on these varieties, we compute all Newton-Okounkov bodies with respect to linear flags. Moreover, we characterize semi-stable vector bundles over curves via Newton-Okounkov bodies.