Geodesic
The Castelnuovo criterion of rationality
Matematičeskie zametki, Tome 11 (1972) no. 1, pp. 27-32

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A new proof is provided for the rationality criterion for algebraic surfaces over an arbitrary base field, using $l$-adic cohomologies. 
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     author = {G. Kurke},
     title = {The {Castelnuovo} criterion of rationality},
     journal = {Matemati\v{c}eskie zametki},
     pages = {27--32},
     publisher = {mathdoc},
     volume = {11},
     number = {1},
     year = {1972},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1972_11_1_a3/}
}
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G. Kurke. The Castelnuovo criterion of rationality. Matematičeskie zametki, Tome 11 (1972) no. 1, pp. 27-32. http://geodesic.mathdoc.fr/item/MZM_1972_11_1_a3/