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On the Mori theory and Newton–Okounkov bodies of Bott–Samelson varieties

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Abstract

We prove that on a Bott–Samelson variety X every movable divisor is nef. This enables us to consider Zariski decompositions of effective divisors, which in turn yields a description of the Mori chamber decomposition of the effective cone. This amounts to information on all possible birational morphisms from X. Applying this result, we prove the rational polyhedrality of the global Newton–Okounkov cone of a Bott–Samelson variety with respect to the so called ‘horizontal’ flag. In fact, we prove the stronger property of the finite generation of the corresponding global value semigroup.

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Acknowledgements

We thank Marcel Maslovarić for valuable discussions. We also wish to thank O. Debarre for suggesting the current, slicker, proof of Lemma 2.8. Finally, we are grateful to the anonymous referee for helpful remarks and suggestions for improvements.

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Authors and Affiliations

  1. Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstraße 3-5, 37073, Göttingen, Germany

    Georg Merz & Henrik Seppänen

  2. Mathematisches Institut, Universität Bayreuth, 95447, Bayreuth, Germany

    David Schmitz

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  1. Georg Merz
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  2. David Schmitz
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  3. Henrik Seppänen
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Correspondence to Henrik Seppänen.

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The first author was supported by DFG Research Training Group 1493 “Mathematical Structures in Modern Quantum Physics”.

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Merz, G., Schmitz, D. & Seppänen, H. On the Mori theory and Newton–Okounkov bodies of Bott–Samelson varieties. Math. Z. 300, 1203–1240 (2022). https://doi.org/10.1007/s00209-021-02812-9

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  • DOI: https://doi.org/10.1007/s00209-021-02812-9

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