## Quotients of del Pezzo surfaces of degree 2.(English)Zbl 1415.14008

Summary: 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.

### MSC:

 14E08 Rationality questions in algebraic geometry 14M20 Rational and unirational varieties 14E07 Birational automorphisms, Cremona group and generalizations
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