Let $k$ be a field. Recall that a del Pezzo surface $S$ over $k$ is a smooth projective surface with ample anticanonical divisor $-K_S$. We define its degree to be $d = K_S^2$.

We say that $S$ is minimal if every birational morphism $S \to S'$ is an isomorphism. We say that $S$ is rational if it is birational to $\mathbb{P}^2_k$ over $k$.

Manin proved in his book that every minimal del Pezzo surface of degree $3$ (i.e. every cubic surface) is non-rational. My question is whether the same is true for del Pezzo surfaces of degree $d=2$ or $1$.

Is every minimal del Pezzo surface of degree $2$ or $1$ non-rational?

Certainly the answer should be yes, but I can't find a proof in the literature. Note that by a result of Iskovskih, minimal surfaces either have Picard number $2$ and a conic bundle, or Picard number $1$. In the case that the Picard number is $2$, non-rationality is proven in Corollary 1.7 of

Iskovskih - Rational surfaces with a sheaf of rational curves and with a positive square of canonical class.

Thus one may assume that $\mathrm{Pic}(S) \cong \mathbb{Z}$, generated by $-K_S$. Manin's proof does not naively generalise to the case of $d=2,1$, as he uses many facts about cubic surfaces (e.g. he uses the rational map given by sending a pair of points $(P,Q)$ to the third intersection point of $S$ with the line though $P,Q$).

I was thinking that perhaps one could reduce the case of $d=2,1$ to cubic surfaces. For example, if $d=2$ and the Galois action on the lines is maximal (i.e. given by the Weyl group $W(\mathbf{E}_7)$), then there exists a finite field extension $K$ of $k$ such that $S_K$ contains a line $L$, and the blow-down of $L$ is a minimal cubic surface. Thus $S_K$ is non-rational by Manin's result, hence $S$ is non-rational. But it is not clear to me that such an extension $K/k$ always exists in general.

  • $\begingroup$ Yes, minimal del Pezzo surfaces of degree less then 5 are non-rational. Regrettably, I can't find any reference. I remember that del Pezzo surfaces of degree 1 are superrigit. $\endgroup$ – Edward Teach Sep 12 '16 at 15:43
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    $\begingroup$ This follows from Thm 3.3.1 in "Rational varieties: algebra, geometry and arithmetic" by Manin and Tsfasman. Russian version is freely abaliable online mathnet.ru/php/… $\endgroup$ – SashaP Sep 12 '16 at 19:55
  • $\begingroup$ Thanks for the useful reference! The original reference seems to be Manin, "Rational surfaces over perfect fields II". If you include this as an answer, I will accept it. $\endgroup$ – Daniel Loughran Sep 13 '16 at 8:14
  • $\begingroup$ My Russian not being what it used to be, I would also very much appreciate a short sketch of Manin's proof! $\endgroup$ – RP_ Sep 13 '16 at 16:59

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