By Fulton W.

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16 below). 15. Any Mori fibration W X ! 12. 13 and where ˛ and ˇ are isomorphisms of fibrations. Proof. 13. This is an easy caseby-case study; here are the steps. Starting from a Mori fibration W X ! W where W is a point, the only links we can perform are links of type I or II centered at a real point or two conjugate non-real points. X /, so the choice of the point is not relevant. Blowing-up a real point in P2 or two non-real points in Q3;1 gives rise to a link of type I to F1 or D6 . The remaining cases correspond to the stereographic projection Q3;1 Ü P2 and its converse.

We denote by N the maximum of the integers ni . If N Ä 1, we are done because all links of type II between Fj and Fj 0 with j; j 0 Ä 1 are standard. We can thus assume N 2, which implies that there exists i such that ni D N , ni 1 < N; ni C1 Ä N . qi /. For j 2 fi 1; i; i C 1g, we denote by j W Fnj Ü Fn0j a Sarkisov link centered at qj ; qj . We obtain then the following commutative diagram Fn i ✤ ✤ ✤ Fn0i 'i 1 i 1 1 'i ❴ ❴ ❴G Fn ❴ ❴ ❴G Fn i i C1 ✤ ✤ ✤ i ✤ i C1 1 ✤ ✤ 'i0 1 'i0 ❴ ❴ ❴ G Fn 0 ❴ ❴ ❴ G Fn 0 ; i i C1 where 'i0 1 ; 'i0 are Sarkisov links.

R// of degree 5. R// is indeed generated by projectivities and standard quintic transformations. 1. Let 'W Q3;1 Ü Q3;1 be a birational map that decomposes as ' D '3 '2 '1 , where 'i W Xi 1 Ü Xi is a Sarkisov link for each i , where X0 D Q3;1 D X2 , X1 D D6 . If '2 has a real base-point, then ' can be written as ' D 2 1 , where 1 ; . 2 / 1 are links of type II from Q3;1 to P2 . Proof. '1 / 1 ; '3 is the blow-up of two conjugate non-real points. '1 / 1 ' '3 Q3;1 ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴G Q3;1 : The map ' has thus exactly three base-points, two of them being non-real and one being real; we denote them by p1 ; pN1 , q.