Let $\pi\colon X\to\mathbb{P}^2$ be the blow-up of the projective plane at nine points in very general position. Is it true that ${\rm Aut}(X)$ is the trivial group? If so, is there an easy argument to prove it?
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4$\begingroup$ Yes, it's true. I have to admit I don't know the details of the proof, but here is a reference: MR0563788 (81d:14020) Gizatullin, M. H. Rational G-surfaces. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980), no. 1, 110–144, 239. $\endgroup$– user5117Commented Oct 21, 2013 at 14:55
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1$\begingroup$ I should add that I don't have access to that paper, and the review doesn't make it clear that the paper contains the theorem the OP asked about: I'm going purely on the reference on page 3 of the paper MR2682184 (2011i:14034) Totaro, Burt. The cone conjecture for Calabi-Yau pairs in dimension 2. Duke Math. J. 154 (2010), no. 2, 241–263. $\endgroup$– user5117Commented Oct 21, 2013 at 15:14
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Yes to your first question. See the paper
Koitabashi - Automorphism groups of generic rational surfaces.
As for your second question, whilst Kotabashi uses some standard techniques on the geometry of surfaces and abelian varieties, I would not personally say that his proof is "easy". I will let you form your own opinion on the matter.
answered Oct 21, 2013 at 16:59
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$\begingroup$ Many thanks, Koitabashi paper is very clear. $\endgroup$– user41695Commented Oct 22, 2013 at 16:22