# Moscow Mathematical Journal

Volume 18, Issue 3, July–September 2018 pp. 557–597.

Quotients of del Pezzo Surfaces of Degree 2

**Authors**:
Andrey Trepalin (1)

**Author institution:**(1) Institute for Information Transmission Problems, 19 Bolshoy Karetnyi side-str., Moscow 127994, Russia

Laboratory of Algebraic Geometry, National Research University Higher School of Economics, 6 Usacheva str., Moscow 119048, Russia

**Summary: **

Let 𝕜 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 𝕜-rationality questions for the quotient surface *X*/*G*. If there are
no smooth 𝕜-points on *X*/*G* then *X*/*G* is obviously non-𝕜-rational.
Assume that the set of smooth 𝕜-points on the quotient is not empty.
We find a list of groups such that the quotient surface can be non-𝕜-rational. For these groups we construct examples of both 𝕜-rational and
non-𝕜-rational quotients of both 𝕜-rational and non-𝕜-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 𝕜-rational.

2010 Math. Subj. Class. 14E08, 14M20, 14E07.

**Keywords:**Rationality problems, del Pezzo surfaces, Minimal model program, Cremona group.

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