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

Trepalin, Andrey

Mosc. Math. J. 18, No. 3, 557-597 (2018).

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

Keywords:

rationality problems; del Pezzo surfaces; minimal model program; Cremona group