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Iskovskikh Seminar
March 14, 2013 18:00, Moscow, Steklov Mathematical Institute, room 540

Quotients of del Pezzo surfaces

A. S. Trepalin

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: Any unirational surface is rational over an algebraic closed field. But for nonclosed fields it is not true. For example, any Del Pezzo surface of degre 2, 3 or 4 is unirational but it can be nonrational. One of the important classes of unirational surfaces are quoients of rational surfaces. We will proof the following results about quotients of rational surfaces:
1) If $X$ is a Del Pezzo surface, $X(\Bbbk) \neq \varnothing$, $K_X^2 \geq 5$, $G \subset \mathrm{Aut}(X)$, then $X / G$ — $\Bbbk$-rational.
2) If $X$ is a Del Pezzo surface, $X(\Bbbk) \neq \varnothing$, $K_X^2 = 4$, $G \subset \mathrm{Aut}(X)$, $G \neq \{1\}$, then $X / G$ — $\Bbbk$-rational.
3 If $X$ is a Del Pezzo surface, $X(\Bbbk) \neq \varnothing$, $K_X^2 = 3$, $G \subset \mathrm{Aut}(X)$, $G \neq \{1\}$, $C_3$, then $X / G$ — $\Bbbk$-rational.
4) If $X$ is a Del Pezzo surface, $X(\Bbbk)$ is dense, $K_X^2 =2$, $G \subset \mathrm{Aut}(X)$, $G \neq \{1\}$, $C_2$, $C_2^2$, $C_4$, $D_4$, $Q_8$, then $X / G$ — $\Bbbk$-rational.
5) If $X$ is a Del Pezzo surface, $X(\Bbbk)$ is dense, $K_X^2 =1$, $G \subset \mathrm{Aut}(X)$, $G \neq \{1\}$, $C_2$, $C_3$, $C_6$, $S_3$, then $X / G$ — $\Bbbk$-rational.

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