Geodesic
Conic bundles with big discriminant loci
Izvestiya. Mathematics , Tome 68 (2004) no. 2, pp. 429-434

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We prove that conic bundles with sufficiently big discriminant locus cannot be birationally transformed into fibrations whose generic fibre has numerically trivial canonical divisor. 
@article{IM2_2004_68_2_a9,
     author = {I. A. Cheltsov},
     title = {Conic bundles with big discriminant loci},
     journal = {Izvestiya. Mathematics },
     pages = {429--434},
     publisher = {mathdoc},
     volume = {68},
     number = {2},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2004_68_2_a9/}
}
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I. A. Cheltsov. Conic bundles with big discriminant loci. Izvestiya. Mathematics , Tome 68 (2004) no. 2, pp. 429-434. http://geodesic.mathdoc.fr/item/IM2_2004_68_2_a9/