Although toric varieties are Mori dream spaces and their Cox rings are as simple as possible (i.e. they are polynomial rings), the work of González-Hering-Payne-Süß shows that in general we cannot expect that a projectivized vector bundle over a toric variety retains the Mori dream property. On the other hand, earlier work of Hausen-Süß gives an explicit presentation of the Cox ring for the projectivized tangent bundle over a smooth toric variety, which is finitely generated (so in this case we retain the Mori dream property). This dichotomy brings natural questions: when does the projectivized tangent bundle over a smooth Mori dream space remains a Mori dream space? What is its Cox ring? I will report on a work in progress (joint with Jeong-Seop Kim, Sunchon National University) where we provide an answer for del Pezzo surfaces along with some connections to well-known problems in geometry of Fano varieties.