## Quotients of del Pezzo surfaces of high degree.(English)Zbl 1406.14010

Let $$X$$ be a variety defined over a field $$k$$. $$X$$ is called $$k$$-rational if and only if it is birational to $$\mathbb{P}^n_k$$, for some $$n>0$$. $$X$$ is called geometrically rational if and only if $$\bar{X}=X\otimes_k \bar{k}$$ is $$\bar{k}$$-rational, where $$\bar{k}$$ is the algebraic closure of $$k$$,
This paper studies the problem of when the quotient of a $$k$$-rational surface by a finite group is also $$k$$-rational. In general this is not true. In fact, the set of non $$k$$-rational quotients of rational surfaces is birationally unbounded [the author, Transform. Groups 21, No. 1, 275–295 (2016; Zbl 1375.14123)].
The main result of this paper is the following.
Let $$X$$ be a del Pezzo surface defined over a field $$k$$ of characteristic zero and such that $$X$$ has a $$k$$-point (by a theorem of Manin $$X$$ is $$k$$-rational). Let $$G$$ be a finite group of automorphisms of $$X$$. Suppose that $$K_X^2\geq 5$$. Then the quotient surface $$X/G$$ is also $$k$$-rational. Suppose that $$K_X^2=4$$, the order of $$G$$ is equal to 1,2 or 4 and all non-trivial elements of $$G$$ have only isolated fixed points. Then $$X/G$$ is not $$k$$-rational. In all other cases, $$X/G$$ is $$k$$-rational.
This result combined with previous results of the author implies the following. Let $$X$$ be a smooth rational surface defined over an algebraically closed field $$k$$ such that $$X$$ has a $$k$$-point. Let $$G$$ be a finite group of automorphisms of $$X$$. Then if $$K_X^2 \geq 5$$, the quotient $$X/G$$ is $$k$$-rational.

### MSC:

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

### Keywords:

rational surfaces; Del Pezzo surfaces; finite groups; quotients

Zbl 1375.14123
Full Text:

### References:

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