Let $S$ be a del Pezzo surface of degree six over $\mathbb{C}$. Then $S$ is the blow-up of $\mathbb{P}^2$ in three general points $p\_1,p\_2,p\_3$.


Is it true that its automorphism group is $((\mathbb{C}^{\*})^{2}\rtimes S\_2)\times S\_3$?


Here $(\mathbb{C}^{\*})^{2}$ are the automorphisms of $\mathbb{P}^2$ fixing $p\_1,p\_2,p\_3$, $S\_2$ is the group generated by the standard Cremona centered at $p\_1,p\_2,p\_3$, and $S\_3$ are the permutations of $p\_1,p\_2,p\_3$.



Automorphisms of del Pezzo surfaces
Not exactly. The quadratic transformation commutes with the action of $S\_3$, and they both act on $(\mathbb{C}^\*)^2$; so the automorphism group is $(\mathbb{C}^{\*})^{2}\rtimes (S\_3\times S\_2)$. You'll find a detailed study of the automorphisms of del Pezzo surfaces in Chapter 8 of Dolgachev's book "Classical Algebraic Geometry: a modern view".


