This paper analyzes the class of Mori dream spaces-varieties \(X\) whose Cox ring \(\mathcal{R}(X) = \bigoplus_{\mathcal{L} \in \operatorname{Pic}(X)} H^0(X,\mathcal{L})\) is a finitely generated algebra. The authors particularly study the case of projective toric bundles, which have the advantage of displaying more nuanced behavior than toric varieties, but nevertheless having a nice combinatorial interpretation. For example, in [Algebra Number Theory 6, No. 5, 995–1017 (2012; Zbl 1261.14002)], J. González et al. show there is a projective toric bundle which is not a Mori dream space (see Theorem 1.4, 1.5 and Example 1.7), whereas all toric varieties are automatically Mori dream spaces. George and Manon use the combinatorial tools developed in [K. Kaveh and C. Manon, Math. Z. 302, No. 3, 1367–1392 (2022; Zbl 1510.14036)] to answer the following question:
Question 1.1. Given a toric vector bundle \(\mathcal{E}\) such that \(\mathbb{P}\mathcal{E}\) is a Mori dream space, when is \(\mathbb{P}(\mathcal{E} \oplus \mathcal{E})\), or more generally \(\mathbb{P}(\mathcal{E} \otimes V)\), a Mori dream space?
The authors link this to the question of when a toric flag bundle is a Mori dream space.
Theorem 1.3. Let \(\mathcal{E}\) be a toric vector bundle, then the projectivized toric vector bundle \(\mathbb{P}(\mathcal{E} \otimes V)\) is a Mori dream space for all \(\dim(V) \leq \ell\) if and only if the toric flag bundle \(\mathcal{F}\mathcal{L}_I(\mathcal{E})\) is a Mori dream space for all \(I\) with \(\max(I) \leq \ell\). Moreover, the full flag bundle \(\mathcal{F}\mathcal{L}(E)\) is a Mori dream space if and only if \(\mathbb{P}(\mathcal{E} \otimes V)\) is a Mori dream space for all finite dimensional vector spaces V. In particular, the projectivization \(\mathbb{P}(\mathcal{E} \oplus \cdots \oplus \mathcal{E})\) of the sum \(\mathcal{E} \oplus \cdots \oplus \mathcal{E}\) with \(\ell\) summands is a Mori dream space if and only if \(\mathcal{F}\mathcal{L}_I(\mathcal{E})\) is a Mori dream space for all \(I\) with \(\max(I) \leq \ell\).
The authors are able to use this theorem to produce explicit examples of flag bundles which are Mori dream spaces, as well as explicit non-examples of Mori dream spaces. A nice general consequence they show is that the tangent bundle is a Mori dream space whenever the base toric variety is a product of projective spaces. Using a result of T. Kaneyama [Nagoya Math. J. 111, 25–40 (1988; Zbl 0820.14010)], they are also able to show a pleasant result about vector bundles on projective space.
Corollary 1.9. If \(\mathcal{F}\) be an irreducible toric vector bundle of rank \(n\) on \(\mathbb{P}^n\), then \(\mathbb{P}(\mathcal{F})\) is a Mori dream space.
The authors also study the algebraic properties of the Cox rings of projective bundles/toric flag bundles which satisfy Theorem 1.3. In particular, if \(\mathcal{E}\) satisfies Theorem 1.3, then there is a family of finitely generated Cox rings \(\mathcal{R}(\mathbb{P}(\mathcal{E} \otimes V))\) as \(V\) varies, and we would be interested in what properties are preserved as \(V\) varies.
Theorem 1.8. Let \(\mathcal{E}\) be a toric vector bundle over \(X\) with general fiber \(E\). If the Cox ring of \(\mathbb{P}(\mathcal{E} \otimes E)\) is generated in degree \(d\), then the Cox ring of \(\mathbb{P}(\mathcal{E} \otimes V)\) is generated in degree \(d\) for all finite dimensional vector spaces \(V\).
This establishes an analogue to a classical representation theoretic result of H. Weyl [The classical groups, their invariants and representations. Reprint of the second edition (1946) of the 1939 original. Princeton, NJ: Princeton University Press (1997; Zbl 1024.20501)] to the context of toric vector bundles.