P1 bundle over MDS

Theorem 3.2. Let X be a Mori Dream Space. Choose D1, … , Dr Weil divisors generating the torsion free part of Cl(X), and let L be a nontrivial Cartier divisor which is in the subgroup generated by the Di . Let Y = PX(OX(L) ⊕ OX), which is a P1 bundle over X with projection π : Y → X. Then 1. The divisors π ∗Di, E∞, where E∞ is the section of X at infinity form a Z-basis for the torsion free part of Cl(Y ). 2. Cox(Y ; π ∗D1, … , π ∗Dr , E∞) ∼= Cox(X; D1, … , Dr )[s, t]. 3. Y is also a Mori Dream Space. (page) (Brown, 2013, p. 658)