# Newton–Okounkov bodies on projective bundles over curves

## 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.

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## Notes

1. 1.

In other words, $$\mu _{\max }(E)=\sigma _1 \ge \cdots \ge \sigma _r=\mu _{\min }(E)$$ can be viewed as the components of the vector $$\varvec{\sigma }=(\sigma _1,\ldots ,\sigma _r)=(\underbrace{\mu _\ell ,\ldots ,\mu _\ell }_{r_\ell \text { times}},\underbrace{\mu _{\ell -1},\ldots ,\mu _{\ell -1}}_{r_{\ell -1} \text { times}},\ldots ,\underbrace{\mu _1,\ldots ,\mu _1}_{r_1 \text { times}})\in \mathbb {Q}^r$$.

2. 2.

In positive characteristic one should take into account iterates of the absolute Frobenius as in [2, 4]. It is worth mentioning that in positive characteristic the semistability of a vector bundle over a curve does not imply the semi-stability of its symmetric powers.

3. 3.

In fact, this statement remains true even if we do not assume that $$E\not \subseteq {\mathbf {B}}_+(\eta )$$. See [15, Prop. 1.6].

4. 4.

Unlike Chen, we do not normalize the measure in order to have $$\lambda (\Delta _{r-1})=1$$.

5. 5.

We note that if $$C\cong \mathbb {P}^1$$ then all the slopes $$\mu _i=\mu (Q_i)$$ are integer numbers and hence the inequality “$$>2g-1$$” becomes “$$\ge 0$$”.

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## Acknowledgements

I would like to express my gratitude to my thesis supervisors, Stéphane Druel and Catriona Maclean, for their advice, helpful discussions and encouragement throughout the preparation of this article. I also thank Bruno Laurent, Laurent Manivel and Bonala Narasimha Chary for fruitful discussions. Finally, I would like to thank the anonymous referee for a very helpful and detailed report.

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Correspondence to Pedro Montero.

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Montero, P. Newton–Okounkov bodies on projective bundles over curves. Math. Z. 291, 1357–1379 (2019). https://doi.org/10.1007/s00209-018-2173-3