Geodesic
Mori Structures on a~Fano Threefold of Index~2 and Degree~1
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebraic geometry: Methods, relations, and applications, Tome 246 (2004), pp. 116-141

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It is proved that the Mori structures on a nonsingular Fano threefold of index 2 and degree 1 are represented precisely by this Fano variety itself and by fibrations into del Pezzo surfaces of degree 1 that emerge from the blowups of curves of arithmetic genus 1 and degree 1. In particular, such a Fano variety is nonrational and all its birational automorphisms are regular. 
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     author = {M. M. Grinenko},
     title = {Mori {Structures} on {a~Fano} {Threefold} of {Index~2} and {Degree~1}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
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M. M. Grinenko. Mori Structures on a~Fano Threefold of Index~2 and Degree~1. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebraic geometry: Methods, relations, and applications, Tome 246 (2004), pp. 116-141. http://geodesic.mathdoc.fr/item/TM_2004_246_a7/