Geodesic
Quotients of Severi--Brauer surfaces
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 501 (2021), pp. 84-88

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We show that a quotient of a non-trivial Severi–Brauer surface $S$ over arbitrary field $\mathbb k$ of characteristic 0 by a finite group $G\subset\operatorname{Aut}(S)$ is $\mathbb k$-rational if and only if $|G|$ is divisible by 3. Otherwise, the quotient is birationally equivalent to $S$.
Keywords: Severi–Brauer surfaces, rationality problems, Brauer group, minimal model program. 
@article{DANMA_2021_501_a15,
     author = {A. S. Trepalin},
     title = {Quotients of {Severi--Brauer} surfaces},
     journal = {Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleni\^a},
     pages = {84--88},
     publisher = {mathdoc},
     volume = {501},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DANMA_2021_501_a15/}
}
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A. S. Trepalin. Quotients of Severi--Brauer surfaces. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 501 (2021), pp. 84-88. http://geodesic.mathdoc.fr/item/DANMA_2021_501_a15/