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 Mosc. Math. J., 2018, Volume 18, Number 3, Pages 557–597 (Mi mmj686)

Quotients of del Pezzo surfaces of degree $2$

Andrey Trepalinab

a Institute for Information Transmission Problems, 19 Bolshoy Karetnyi side-str., Moscow 127994, Russia
b Laboratory of Algebraic Geometry, National Research University Higher School of Economics, 6 Usacheva str., Moscow 119048, Russia

Abstract: Let $\Bbbk$ be any field of characteristic zero, $X$ be a del Pezzo surface of degree $2$ and $G$ be a group acting on $X$. In this paper we study $\Bbbk$-rationality questions for the quotient surface $X / G$. If there are no smooth $\Bbbk$-points on $X / G$ then $X / G$ is obviously non-$\Bbbk$-rational. Assume that the set of smooth $\Bbbk$-points on the quotient is not empty. We find a list of groups such that the quotient surface can be non-$\Bbbk$-rational. For these groups we construct examples of both $\Bbbk$-rational and non-$\Bbbk$-rational quotients of both $\Bbbk$-rational and non-$\Bbbk$-rational del Pezzo surfaces of degree $2$ such that the $G$-invariant Picard number of $X$ is $1$. For all other groups we show that the quotient $X / G$ is always $\Bbbk$-rational.

Key words and phrases: Rationality problems, del Pezzo surfaces, Minimal model program, Cremona group.

DOI: https://doi.org/10.17323/1609-4514-2018-18-3-557-597

Full text: http://www.mathjournals.org/.../2018-018-003-007.html
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MSC: 14E08, 14M20, 14E07
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Citation: Andrey Trepalin, “Quotients of del Pezzo surfaces of degree $2$”, Mosc. Math. J., 18:3 (2018), 557–597

Citation in format AMSBIB
\Bibitem{Tre18} \by Andrey~Trepalin \paper Quotients of del Pezzo surfaces of degree~$2$ \jour Mosc. Math.~J. \yr 2018 \vol 18 \issue 3 \pages 557--597 \mathnet{http://mi.mathnet.ru/mmj686} \crossref{https://doi.org/10.17323/1609-4514-2018-18-3-557-597} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000456105800007} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85053834923} 

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• http://mi.mathnet.ru/eng/mmj/v18/i3/p557

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This publication is cited in the following articles:
1. A. Trepalin, “Quotients of del Pezzo surfaces”, Int. J. Math., 30:12 (2019), 1950068