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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