Geodesic
$\mathbb Q$-Complements on Log Surfaces
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebraic geometry: Methods, relations, and applications, Tome 246 (2004), pp. 181-182

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Log surfaces without $\mathbb Q$-complement are classified. 
@article{TM_2004_246_a12,
     author = {S. A. Kudryavtsev and I. Yu. Fedorov},
     title = {$\mathbb Q${-Complements} on {Log} {Surfaces}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {181--182},
     publisher = {mathdoc},
     volume = {246},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2004_246_a12/}
}
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S. A. Kudryavtsev; I. Yu. Fedorov. $\mathbb Q$-Complements on Log Surfaces. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebraic geometry: Methods, relations, and applications, Tome 246 (2004), pp. 181-182. http://geodesic.mathdoc.fr/item/TM_2004_246_a12/