Del Pezzo Surfaces of Degree 5 Over Perfect Fields

Let |$\textbf{k}$| be a perfect field and |$X$| a del Pezzo surface of degree |$5$| over |$\textbf{k}$|⁠. Then one of the following cases holds:

• 1.

  |$\textrm{rk} \, \textrm{NS}(X) = 1$| and |$X$| is obtained by blowing up |$\mathbb{P}^{2}$| in a point |$p=\lbrace p_{1},\dots ,p_{5} \rbrace $| of degree |$5$| with splitting field |$F$| to a del Pezzo surface of degree |$4$|⁠, and then blowing down the strict transform of the conic passing through |$p$|⁠. Moreover, we are in one of the following cases:

  • (a)

    |$\operatorname{Gal}(F/\textbf{k}) \simeq \mathbb{Z}/5\mathbb{Z}$| and |$\operatorname{Aut}_{\textbf{k}}(X)=\langle \widehat{\phi } \rangle \simeq \mathbb{Z}/5\mathbb{Z}$|⁠, where |$\widehat{\phi }$| is the lift of a quadratic birational map of |$\mathbb{P}^{2}_{\overline{\textbf{k}}}$| of order five,

  • (b)

    |$\operatorname{Gal}(F/\textbf{k}) \simeq \operatorname{D}_{5}$| and |$\operatorname{Aut}_{\textbf{k}}(X)=\lbrace \operatorname{id} \rbrace $|⁠,

  • (c)

    |$\operatorname{Gal}(F/\textbf{k}) \simeq \operatorname{GA}(1,5)$| and |$\operatorname{Aut}_{\textbf{k}}(X)=\lbrace \operatorname{id} \rbrace $|⁠,

  • (d)

    |$\operatorname{Gal}(F/\textbf{k}) \simeq \mathcal{A}_{5}$| and |$\operatorname{Aut}_{\textbf{k}}(X)=\lbrace \operatorname{id} \rbrace $|⁠, and

  • (e)

    |$\operatorname{Gal}(F/\textbf{k}) \simeq \operatorname{Sym}_{5}$| and |$\operatorname{Aut}_{\textbf{k}}(X)=\lbrace \operatorname{id} \rbrace $|⁠.

• 2.

  |$\textrm{rk} \, \textrm{NS}(X) = 5$|⁠, |$\textrm{rk} \, \textrm{NS}(X)^{\operatorname{Aut}_{\textbf{k}}(X)} = 1$| and |$X$| is isomorphic to the blow-up of |$\mathbb{P}^{2}$| in the four coordinate points with |$\operatorname{Aut}_{\textbf{k}}(X) \simeq \operatorname{Sym}_{5}$|⁠.

• 3.

  |$\textrm{rk} \, \textrm{NS}(X) \geq 2$|⁠, |$\textrm{rk} \, \textrm{NS}(X)^{\operatorname{Aut}_{\textbf{k}}(X)} = 2$| and |$X$| is one of the following:

  • (a)

    |$X$| is isomorphic to the blow-up of |$\mathcal{Q}^{L}$| in the three |$\textbf{k}$|-rational points |$p_{1}=([1:0],[1:0]), p_{2}=([0:1],[0:1]), p_{3}=([1:1],[1:1]) $|⁠, for some quadratic extension |$L/\textbf{k}$|⁠. Its isomorphism class depends only on the |$\textbf{k}$|-isomorphism class of |$L$|⁠. Moreover, |$\operatorname{Aut}_{\textbf{k}}(X) = \langle \widehat{\alpha },\widehat{\gamma } \rangle \times \langle \widehat{\beta } \rangle \simeq \operatorname{Sym}_{3}\times \mathbb{Z}/2\mathbb{Z}$|⁠, where |$\widehat{\alpha }, \widehat{\beta }, \widehat{\gamma }$| are the lifts of involutions |$\alpha , \beta , \gamma $| of |$\mathcal{Q}^{L}$|⁠.

  • (b)

    |$X$| is isomorphic to the blow-up of |$\mathbb{P}^{2}$| in two points |$r=\lbrace r_{1},r_{2} \rbrace $|⁠, |$s=\lbrace s_{1},s_{2} \rbrace $| of degree |$2$| with splitting field a quadratic extension |$L/\textbf{k}$|⁠. Its isomorphism class depends only on the |$\textbf{k}$|-isomorphism class of |$L$|⁠. Moreover, |$\operatorname{Aut}_{\textbf{k}}(X)=\langle \widehat{\alpha },\widehat{\beta } \rangle \simeq \operatorname{D}_{4} $|⁠, where |$\widehat{\alpha }$| is the lift of an involution |$\alpha $| of |$\mathbb{P}^{2}$| and |$\widehat{\beta }$| the lift of an automorphism |$\beta $| of |$\mathbb{P}^{2}$| of order four.

  • (c)

    |$X$| is isomorphic to the blow-up of |$\mathbb{P}^{2}$| in a point |$q=\lbrace q_{1},q_{2},q_{3} \rbrace $| of degree |$3$| with splitting field |$F$| such that |$\operatorname{Gal}(F/\textbf{k}) \simeq \mathbb{Z}/3\mathbb{Z}$| and in a |$\textbf{k}$|-rational point |$r \in \mathbb{P}^{2}(\textbf{k})$|⁠. Its isomorphism class depends only on the |$\textbf{k}$|-isomorphism class of |$F$|⁠. Moreover, |$\operatorname{Aut}_{\textbf{k}}(X)=\langle \widehat{\alpha } \rangle \times \langle \widehat{\beta } \rangle \simeq \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} $|⁠, where |$\widehat{\alpha }$| is the lift of an automorphism of |$\mathbb{P}^{2}$| of order three and |$\widehat{\beta }$| is the lift of a birational quadratic involution of |$\mathbb{P}^{2}$| with base-point |$q$|⁠.

  • (d)

    |$X$| is isomorphic to the blow-up of |$\mathbb{P}^{2}$| in a point |$q=\lbrace q_{1},q_{2},q_{3} \rbrace $| of degree |$3$| with splitting field |$F$| such that |$\operatorname{Gal}(F/\textbf{k}) \simeq \operatorname{Sym}_{3}$| and in a |$\textbf{k}$|-rational point |$r \in \mathbb{P}^{2}(\textbf{k})$|⁠. Its isomorphism class depends only on the |$\textbf{k}$|-isomorphism class of |$F$|⁠. Moreover, |$\operatorname{Aut}_{\textbf{k}}(X)=\langle \widehat{\alpha } \rangle \simeq \mathbb{Z}/2\mathbb{Z} $|⁠, where |$\widehat{\alpha }$| is the lift of a birational quadratic involution of |$\mathbb{P}^{2}$| with base-point |$q$|⁠.

  • (e)

    |$X$| is isomorphic to the blow-up of |$\mathbb{P}^{2}$| in a point |$q=\lbrace q_{1},q_{2},q_{3},q_{4} \rbrace $| of degree |$4$| with splitting field |$F$| such that |$\operatorname{Gal}(F/\textbf{k}) \simeq \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$|⁠. Its isomorphism class depends only on the |$\textbf{k}$|-isomorphism class of |$F$|⁠. Moreover, |$\operatorname{Aut}_{\textbf{k}}(X)=\langle \widehat{\alpha } \rangle \times \langle \widehat{\beta } \rangle \simeq \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$|⁠, where |$\widehat{\alpha }, \widehat{\beta }$| are the lifts of involutions of |$\mathbb{P}^{2}$|⁠.

  • (f)

    |$X$| is isomorphic to the blow-up of |$\mathbb{P}^{2}$| in a point |$q=\lbrace q_{1},q_{2},q_{3},q_{4} \rbrace $| of degree |$4$| with splitting field |$F$| such that |$\operatorname{Gal}(F/\textbf{k}) \simeq \mathbb{Z}/4\mathbb{Z}$|⁠. Its isomorphism class depends only on the |$\textbf{k}$|-isomorphism class of |$F$|⁠. Moreover, |$\operatorname{Aut}_{\textbf{k}}(X)=\langle \widehat{\alpha } \rangle \simeq \mathbb{Z}/4\mathbb{Z} $|⁠, where |$\widehat{\alpha }$| is the lift of an automorphism of |$\mathbb{P}^{2}$| of order four.

  • (g)

    |$X$| is isomorphic to the blow-up of |$\mathbb{P}^{2}$| in a point |$q=\lbrace q_{1},q_{2},q_{3},q_{4} \rbrace $| of degree |$4$| with splitting field |$F$| such that |$\operatorname{Gal}(F/\textbf{k}) \simeq \operatorname{D}_{4}$|⁠. Its isomorphism class depends only on the |$\textbf{k}$|-isomorphism class of the residue field of |$q$|⁠. Moreover, |$\operatorname{Aut}_{\textbf{k}}(X)=\langle \widehat{\alpha } \rangle \simeq \mathbb{Z}/2\mathbb{Z} $|⁠, where |$\widehat{\alpha }$| is the lift of an involution of |$\mathbb{P}^{2}$|⁠.

  • (h)

    |$X$| is isomorphic to the blow-up of |$\mathbb{P}^{2}$| in a point |$q=\lbrace q_{1},q_{2},q_{3},q_{4} \rbrace $| of degree |$4$| with splitting field |$F$| such that |$\operatorname{Gal}(F/\textbf{k}) \simeq \mathcal{A}_{4}$|⁠. Its isomorphism class depends only on the |$\textbf{k}$|-isomorphism class of |$F$|⁠. Moreover, |$\operatorname{Aut}_{\textbf{k}}(X)=\lbrace \operatorname{id} \rbrace $|⁠.

  • (i)

    |$X$| is isomorphic to the blow-up of |$\mathbb{P}^{2}$| in a point |$q=\lbrace q_{1},q_{2},q_{3},q_{4} \rbrace $| of degree |$4$| with splitting field |$F$| such that |$\operatorname{Gal}(F/\textbf{k}) \simeq \operatorname{Sym}_{4}$|⁠. Its isomorphism class depends only on the |$\textbf{k}$|-isomorphism class of |$F$|⁠. Moreover, |$\operatorname{Aut}_{\textbf{k}}(X)=\lbrace \operatorname{id} \rbrace $|⁠.

  • (j)

    |$X$| is isomorphic to the blow-up of |$\mathcal{Q}^{L}$| in a point |$p=\lbrace p_{1},p_{2},p_{3} \rbrace $| of degree |$3$| with splitting field |$F$| such that |$\operatorname{Gal}(F/\textbf{k}) \simeq \operatorname{Gal}(FL/L) \simeq \mathbb{Z}/3\mathbb{Z}$|⁠, for some quadratic extension |$L/\textbf{k}$|⁠. Its isomorphism class depends on the |$\textbf{k}$|-isomorphism classes of |$L$| and |$F$|⁠. Moreover, |$\operatorname{Aut}_{\textbf{k}}(X)=\langle \widehat{\alpha } \rangle \times \langle \widehat{\Phi _{q}} \rangle \simeq \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$|⁠, where |$\widehat{\alpha }$| is the lift of an automorphism of |$\mathbb{P}^{2}_{L}$| of order three and |$\widehat{\Phi _{q}}$| is the lift of a birational quadratic involution of |$\mathbb{P}^{2}_{L}$| with a base-point |$q$| of degree |$3$|⁠.

  • (k)

    |$X$| is isomorphic to the blow-up of |$\mathcal{Q}^{L}$| in a point |$p=\lbrace p_{1},p_{2},p_{3} \rbrace $| of degree |$3$| with splitting field |$F$| such that |$\operatorname{Gal}(F/\textbf{k}) \simeq \operatorname{Gal}(FL/L) \simeq \operatorname{Sym}_{3}$|⁠, for some quadratic extension |$L/\textbf{k}$|⁠. Its isomorphism class depends on the |$\textbf{k}$|-isomorphism classes of |$L$| and |$F$|⁠. Moreover, |$\operatorname{Aut}_{\textbf{k}}(X)=\langle \widehat{\Phi _{q}} \rangle \simeq \mathbb{Z}/2\mathbb{Z}$|⁠, where |$\widehat{\Phi _{q}}$| is the lift of a birational quadratic involution of |$\mathbb{P}^{2}_{L}$| with base-point |$q$| of degree |$3$|⁠.

  • (l)

    |$X$| is isomorphic to the blow-up of |$\mathcal{Q}^{L}$| in a point |$p=\lbrace p_{1},p_{2},p_{3} \rbrace $| of degree |$3$| with splitting field |$F$| such that |$\operatorname{Gal}(F/\textbf{k}) \simeq \operatorname{Sym}_{3}$| and |$\operatorname{Gal}(FL/L) \simeq \mathbb{Z}/3\mathbb{Z}$|⁠, for some quadratic extension |$L/\textbf{k}$|⁠. Its isomorphism class depends on the |$\textbf{k}$|-isomorphism classes of |$L$| and |$F$|⁠. Moreover, |$\operatorname{Aut}_{\textbf{k}}(X)=\langle \widehat{\Phi _{q}} \rangle \simeq \mathbb{Z}/2\mathbb{Z}$|⁠, where |$\widehat{\Phi _{q}}$| is the lift of a birational quadratic involution of |$\mathbb{P}^{2}_{L}$| with base-point |$q$| of degree |$3$|⁠.

• 4.

  |$\textrm{rk} \, \textrm{NS}(X) \geq 2$|⁠, |$\textrm{rk} \, \textrm{NS}(X)^{\operatorname{Aut}_{\textbf{k}}(X)} = 3$| and |$X$| is isomorphic to the blow-up of |$\mathbb{P}^{2}$| in two points |$r=\lbrace r_{1},r_{2} \rbrace $|⁠, |$s=\lbrace s_{1},s_{2} \rbrace $| of degree |$2$| whose splitting fields are respectively non |$\textbf{k}$|-isomorphic quadratic extensions |$L/\textbf{k}$|⁠, |$L^{\prime}/\textbf{k}$|⁠. Its isomorphism class depends only on the |$\textbf{k}$|-isomorphism classes of |$L$| and |$L^{\prime}$|⁠. Moreover, |$\operatorname{Aut}_{\textbf{k}}(X)=\langle \widehat{\alpha } \rangle \times \langle \widehat{\beta } \rangle \simeq \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} $|⁠, where |$\widehat{\alpha }, \widehat{\beta }$| are the lifts of two involutions |$\alpha , \beta $| of |$\mathbb{P}^{2}$|⁠.

[16, CH. IV, Corollary 24.5.2] Let |$v: X \rightarrow Z $| be any birational morphism between smooth surfaces. If |$X$| is a del Pezzo surface, then |$Z$| is also a del Pezzo surface.

Let |$X$| be a smooth surface, |$B$| a point or a smooth curve and |$\pi : X \rightarrow B$| a surjective morphism with connected fibres such that |$-K_{X}$| is |$\pi $|-ample. We call |$\pi : X \rightarrow B $| a Mori fibre space if |$\textrm{rk} \, \textrm{NS}(X/B)=1$|⁠, where |$\textrm{NS}(X/B):= \textrm{NS}(X)/\pi ^{\ast }\textrm{NS}(B) $| is the relative Néron–Severi group. A del Pezzo surface |$X$| is said to be an |$\operatorname{Aut}_{\textbf{k}}(X)$|-Mori fibre space if |$\textrm{rk} \, \textrm{NS}(X)^{\operatorname{Aut}_{\textbf{k}}(X)}=1$|⁠.

[[11], [15, Theorem 3.15], [23], [12, §4], see also [24, Theorem 2.1]] Let |$ \textbf{k} $| be a field. Let |$X$| be a smooth del Pezzo surface of degree |$ d \geq 5 $|⁠. If |$ X(\textbf{k}) \neq \varnothing $|⁠, then |$ X $| is rational. This hypothesis is automatically satisfied if |$ d=5 $|⁠.

[20, Lemmas 6.11 and 6.15] Let |$L/\textbf{k}$| be a finite Galois extension. Let |$p_{1},\dots ,p_{4} \in \mathbb{P}^{2}(L)$|⁠, no three collinear, and |$q_{1},\dots ,q_{4} \in \mathbb{P}^{2}(L)$|⁠, no three collinear, be points such that the sets |$\lbrace p_{1},\dots ,p_{4} \rbrace $| and |$\lbrace q_{1},\dots ,q_{4} \rbrace $| are invariant under the Galois action of |$\operatorname{Gal}(L/\textbf{k})$|⁠. Assume that for all |$g \in \operatorname{Gal}(L/\textbf{k})$| there exists |$\sigma \in \operatorname{Sym}_{4}$| such that |$g(p_{i})=p_{\sigma (i)}$| and |$g(q_{i})=q_{\sigma (i)}$| for |$i=1,\dots ,4$|⁠. Then, there exists |$A \in \operatorname{PGL}_{3}(\textbf{k})$| such that |$Ap_{i}=q_{i}$| for |$i=1,\dots ,4$|⁠. Moreover, we have the following specific result for Galois-orbits of size |$4$| and |$2,$| respectively:

• 1.

  If |$\lbrace p_{1},\dots ,p_{4} \rbrace $| is an orbit of size |$4$| of the action of |$\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})$|⁠, then there exists |$ A \in \operatorname{PGL}_{3}(\textbf{k}) $| such that |$Ap_{i}=[1:a_{i}:a_{i}^{2}]$| with |$a_{i} \in \mathbb{A}^{1}(\overline{\textbf{k}})$| forming an orbit |$\lbrace a_{1},\dots ,a_{4} \rbrace \subset \mathbb{A}^{1}(\overline{\textbf{k}}) $| of size |$4$| under the Galois action.

• 2.

  If |$\lbrace p_{1},p_{2} \rbrace $| and |$\lbrace p_{3},p_{4} \rbrace $| form two orbits of size |$2$|⁠, then there exists |$A \in \operatorname{PGL}_{3}(\textbf{k})$| such that |$Ap_{i}=[1:a_{i}:0]$| for |$i=1,2$| and |$Ap_{i}=[1:0:a_{i}]$| for |$i=3,4$| with |$a_{i} \in \mathbb{A}^{1}(\overline{\textbf{k}})$| and |$\lbrace a_{1},a_{2} \rbrace $| as well as |$\lbrace a_{3},a_{4} \rbrace $| form an orbit in |$\mathbb{A}^{1}(\overline{\textbf{k}})$|⁠. In both cases, the field of definition of |$\lbrace p_{1},\dots ,p_{4} \rbrace $| is |$\textbf{k}(a_{1},\dots ,a_{4})$|⁠.

[21, Remark 2.7] Let |$F/\textbf{k}$| be a finite extension. Let |$p_{1},p_{2},p_{3},q_{1},q_{2},q_{3} \in \mathbb{P}^{1}(F)$| such that the sets |$\lbrace p_{1},p_{2},p_{3} \rbrace $| and |$\lbrace q_{1},q_{2},q_{3} \rbrace $| are |$\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})$|-invariant. Suppose that for any |$g \in \operatorname{Gal}(F/\textbf{k})$| there exists |$\sigma \in \operatorname{Sym}_{3}$| such that |$p_{i}^{g}=p_{\sigma (i)}$| and |$q_{i}^{g}=q_{\sigma (i)}$| for |$i=1,2,3$|⁠. Then there exists |$\alpha \in \operatorname{PGL}_{2}(\textbf{k})$| such that |$\alpha (p_{i}) = q_{i}$| for |$i=1,2,3$|⁠.

Let |$L/\textbf{k}$| be a quadratic extension and let |$ p_{1}, p_{2}, p_{3} \in \mathcal{Q}^{L}(\textbf{k}) $| be three |$\textbf{k}$|-rational points contained in pairwise distinct rulings of |$ \mathcal{Q}^{L}_{L} $|⁠. Then there exists |$ \alpha \in \operatorname{Aut}_{\textbf{k}}(\mathcal{Q}^{L}) $| such that |$ \alpha (p_{1}) = \left ([1:0],[1:0]\right ) $|⁠, |$ \alpha (p_{2}) = \left ([0:1],[0:1]\right ) $| and |$ \alpha (p_{3}) = \left ([1:1],[1:1]\right ) $|⁠.

[21, Lemma 3.7] Let |$p=\lbrace p_{1},p_{2},p_{3} \rbrace $| and |$q=\lbrace q_{1},q_{2},q_{3} \rbrace $| be points of degree |$3$| in |$\mathcal{Q}^{L}$|⁠, such that for any |$h \in \operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})$| there exists |$\sigma \in \operatorname{Sym}_{3}$| such that |$p_{i}^{h}=p_{\sigma (i)}$| and |$q_{i}^{h}=q_{\sigma (i)}$| for |$i=1,2,3$|⁠. Suppose that the geometric components of |$p$| (resp. of |$q$|⁠) are in general position on |$\mathcal{Q}^{L}_{\overline{\textbf{k}}}$|⁠. Then there exists |$\alpha \in \operatorname{Aut}_{\textbf{k}}(\mathcal{Q}^{L})$| such that |$\alpha (p_{i})=q_{i}$| for |$i=1,2,3$|⁠.

Let |$X$| be a del Pezzo surface of degree |$5$| and let |$L/\textbf{k}$| be a normal extension of smallest degree such that all the |$(-1)$|-curves of |$X_{\overline{\textbf{k}}}$| are defined over |$L$|⁠. Then the action of |$\operatorname{Aut}_{\textbf{k}}(X)$| on the set of |$(-1)$|-curves of |$X$| induces an isomorphism |$\Psi : \operatorname{Aut}_{\textbf{k}}(X) \overset{\sim }{\longrightarrow } \operatorname{Aut}(\Pi _{L,\rho }) $|⁠, where |$\Pi _{L,\rho }$| denotes the Petersen diagram of |$X_{L}$| equipped with the induced action of |$\operatorname{Gal}(L/\textbf{k})$|⁠, and |$ \operatorname{Aut}(\Pi _{L,\rho }) = \left \lbrace \alpha \in \operatorname{Aut}(\Pi _{L}) \, \vert \, \rho (\operatorname{Gal}(L/\textbf{k})) \circ \alpha = \alpha \circ \rho (\operatorname{Gal}(L/\textbf{k})) \right \rbrace $|⁠.

The group |$\operatorname{Aut}_{\textbf{k}}(X)$| acts on the set of |$(-1)$|-curves of |$X$| preserving the intersection form, so we get a homomorphism |$\Psi : \operatorname{Aut}_{\textbf{k}}(X) \rightarrow \operatorname{Aut}(\Pi _{L,\rho })$|⁠. Let |$\pi : X_{L} \rightarrow \mathbb{P}^{2}_{L}$| be the contraction of |$E_{1},\dots ,E_{4}$|⁠. Then an element of the kernel of |$\Psi $| descends via |$\pi $| to an automorphism of |$\mathbb{P}^{2}_{L}$| that fixes four points in general position, so is trivial, and so |$\Psi $| is injective. Now let |$\phi \in \operatorname{Aut}(\Pi _{L,\rho })$|⁠. Then, since |$\Theta _{L}$| defines an isomorphism, there exists a unique |$f \in \operatorname{Aut}(X_{L})$| such that |$\Theta _{L}(f)=\phi $|⁠; and because |$\Theta _{L}$| is |$\operatorname{Gal}(L/\textbf{k})$|-equivariant, we get: |$\Theta _{L}(g \circ f \circ g^{-1})=\rho (g) \circ \Theta _{L}(f) \circ \rho (g)^{-1}=\rho (g) \circ \phi \circ \rho (g)^{-1}=\phi =\Theta _{L}(f)$| for all |$g \in \operatorname{Gal}(L/\textbf{k})$|⁠. Because |$\Theta _{L}$| is injective, this implies |$g \circ f \circ g^{-1}=f$| for all |$g \in \operatorname{Gal}(L/\textbf{k})$|⁠, and so |$f \in \operatorname{Aut}_{\textbf{k}}(X)$| is defined over |$\textbf{k}$|⁠.

As an abstract group, |$\operatorname{Aut}(\Pi _{\overline{\textbf{k}},\rho }) $| is isomorphic to the centralizer of the subgroup |$\rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k}))$| in |$\operatorname{Sym}_{5}$|⁠, denoted by |$C_{\operatorname{Sym}_{5}}(\rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})))$|⁠. We will describe the geometric actions of |$\operatorname{Aut}(\Pi _{\overline{\textbf{k}},\rho })$| in each case in Sections 3.2, 3.3, 3.4, and 3.5, and will realize geometrically the generators of the group |$\operatorname{Aut}_{\textbf{k}}(X)$|⁠.

Let |$X$| be a del Pezzo surface of degree |$5$| such that |$ \rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})) = \langle (34)\rangle \simeq \mathbb{Z}/2\mathbb{Z} $| in |$\textrm{Sym}_{5}$| as indicated in Figure 2b. Then the following holds.

• 1.

  There exist a quadratic extension |$ L/\textbf{k} $| and a birational morphism |$ \eta : X \longrightarrow \mathcal{Q}^{L} $| contracting three disjoint |$(-1)$|-curves defined over |$\textbf{k}$| onto |$ p_{1} = \left ([1:0],[1:0]\right ), p_{2} = \left ([0:1],[0:1]\right ) $| and |$ p_{3} = \left ([1:1],[1:1]\right ) $|⁠.

• 2.

  Any two such surfaces are isomorphic if and only if the respective quadratic extensions are |$\textbf{k}$|-isomorphic.

• 3.

  |$ \operatorname{Aut}_{\textbf{k}}(X) = \langle \widehat{\alpha },\widehat{\gamma } \rangle \times \langle \widehat{\beta } \rangle \simeq \operatorname{Sym}_{3} \times \mathbb{Z}/2\mathbb{Z} $|⁠, where |$\eta \widehat{\alpha }\eta ^{-1}, \eta \widehat{\beta }\eta ^{-1}, \eta \widehat{\gamma }\eta ^{-1}$| are involutions of |$\mathcal{Q}^{L}$|⁠.

• 4.

  |$ \textrm{rk} \, \textrm{NS}(X)^{\operatorname{Aut}_{\textbf{k}}(X)} = 2 $|⁠, so in particular |$ X \longrightarrow \ast $| is not an |$ \operatorname{Aut}_{\textbf{k}}(X) $|-Mori fibre space, and the |$\operatorname{Aut}_{\textbf{k}}(X)$|-minimal models of |$X$| are the quadric |$\mathcal{Q}^{L}$| and the rational del Pezzo surface |$Y$| of degree |$6$| obtained by blowing up |$\mathbb{F}_{0} \simeq \mathbb{P}^{1} \times \mathbb{P}^{1}$| in a point of degree |$2$| whose splitting field is |$L$|⁠.

(1) The incidence diagram |$ \Pi _{\overline{\textbf{k}},\rho } $| presents exactly four |$(-1)$|-curves defined over |$\textbf{k}$| of |$X$|⁠, three of which are disjoint namely |$ E_{1}, E_{2} $| and |$ D_{34} $|⁠. Their contraction yields a birational morphism |$ \eta : X \rightarrow Z $| onto a rational del Pezzo surface of degree |$8$| with three |$\textbf{k}$|-rational points, |$ p_{1},p_{2},p_{3} $|⁠, and the map |$ \eta $| conjugates the |$ \operatorname{Gal}(\overline{\textbf{k}}/\textbf{k}) $|-action on |$ X_{\overline{\textbf{k}}} $| to an action that exchanges the fibrations of |$Z_{\overline{\textbf{k}}}$|⁠. Thus, |$ Z \simeq \mathcal{Q}^{L} $| for some quadratic extension |$ L/\textbf{k} $| by [21, Lemma 3.2(1)], and by Lemma 2.2.3 we can assume that |$ p_{1} = \left ([1:0],[1:0]\right ), p_{2} = \left ([0:1],[0:1]\right ) $| and |$ p_{3} = \left ([1:1],[1:1]\right ) $|⁠. For any of the |$\textbf{k}$|-points |$ p_{i} \in \mathcal{Q}^{L}(\textbf{k}) $|⁠, |$ i=1,2,3 $|⁠, there is a birational map |$ \psi : \mathcal{Q}^{L} \dashrightarrow \mathbb{P}^{2} $| that is the composition of the blow-up of |$ p_{i} $| with the contraction of a curve onto a point of degree |$2$| in |$ \mathbb{P}^{2} $| whose splitting field is |$L$|⁠. The surface |$X$| is therefore isomorphic to the blow-up of |$ \mathbb{P}^{2} $| in two |$\textbf{k}$|-rational points |$ r_{1}, r_{2} $| and one point of degree two |$ r=\lbrace r_{3},r_{4} \rbrace $| whose splitting field is |$L$| (see Figure 3).

(2) Any two triplets of |$\textbf{k}$|-rational points on |$ \mathcal{Q}^{L} $| whose components are contained in pairwise distinct rulings of |$ \mathcal{Q}^{L}_{L} $| can be sent onto each other by an element of |$ \operatorname{Aut}_{\textbf{k}}(\mathcal{Q}^{L}) $| by Lemma 2.2.3. It follows that any two del Pezzo surfaces satisfying our hypothesis are isomorphic if and only if there exist birational contractions |$\eta , \eta ^{\prime}$| onto isomorphic del Pezzo surfaces |$ \mathcal{Q}^{L} $| and |$ \mathcal{Q}^{L^{\prime}} $| of degree |$ 8 $|⁠. This is the case if and only if |$L$| and |$L^{\prime}$| are |$\textbf{k}$|-isomorphic by [21, Lemma 3.2(3)].

(4) From 1 we have a contraction |$\eta : X \rightarrow \mathcal{Q}^{L}$| onto three rational points, so thanks to Lemma 2.2.3 we get |$\textrm{rk} \, \textrm{NS}(X)^{\operatorname{Aut}_{\textbf{k}}(X)}=2$|⁠. Now, note that the curve |$D_{12}$| is stabilized by |$\operatorname{Aut}_{\textbf{k}}(X)$| (see point 3) and can then be contracted equivariantly. The contraction of |$D_{12}$| gives a birational morphism to the rational del Pezzo surface |$Y$| of degree |$6$| in [21, Lemma 4.10], which is obtained by blowing up |$\mathbb{F}_{0} \simeq \mathbb{P}^{1} \times \mathbb{P}^{1}$| in a point of degree |$2$| with splitting field |$L$| and whose |$\textbf{k}$|-automorphism group acts transitively on the set of |$(-1)$|-curves, implying that |$Y \rightarrow \ast $| is an |$\operatorname{Aut}_{\textbf{k}}(X)$|-Mori fibre space.

We note that via the birational map |$\psi : \mathcal{Q}^{L} \dashrightarrow \mathbb{P}^{2}$| constructed in the proof of Proposition 3.2.1, (1) we could also have lifted |$\alpha ,\beta ,\gamma $| to |$\textbf{k}$|-birational involutions of |$\mathbb{P}^{2}$|⁠.

Let |$X$| be a del Pezzo surface of degree |$5$| such that |$ \rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})) = \langle (12)(34) \rangle \simeq \mathbb{Z}/2\mathbb{Z} $| in |$\operatorname{Sym}_{5}$| as indicated in Figure 2c. Then the following holds:

• 1.

  There exist a quadratic extension |$L/\textbf{k}$| and two points |$ r=\lbrace r_{1},r_{2} \rbrace , s=\lbrace s_{1},s_{2} \rbrace $| in |$\mathbb{P}^{2}$| of degree |$2$| with splitting field |$L$| such that |$X$| is isomorphic to the blow-up of |$ \mathbb{P}^{2} $| in |$r$| and |$s$|⁠.

• 2.

  Any two such surfaces are isomorphic if and only if the respective quadratic field extensions are |$\textbf{k}$|-isomorphic.

• 3.

  |$ \operatorname{Aut}_{\textbf{k}}(X)=\langle \widehat{\alpha },\widehat{\beta } \rangle \simeq \operatorname{D}_{4} $|⁠, where |$\widehat{\alpha }$| and |$\widehat{\beta }$| are respectively the lifts of an involution |$\alpha $| and an automorphism |$\beta $| of order four of |$\mathbb{P}^{2}$|⁠.

• 4.

  |$ \textrm{rk} \, \textrm{NS}(X)^{\operatorname{Aut}_{\textbf{k}}(X)} = 2 $|⁠, so in particular |$ X \rightarrow \ast $| is not an |$ \operatorname{Aut}_{\textbf{k}}(X) $|-Mori fibre space, but |$X$| carries an |$\operatorname{Aut}_{\textbf{k}}(X)$|-Mori fibre space structure |$X \rightarrow \mathbb{P}^{1}$| given by the linear system of conics passing through the points |$r$| and |$s$| on |$\mathbb{P}^{2}$|⁠.

(1) The Petersen graph |$ \Pi _{\overline{\textbf{k}},\rho } $| of |$X$| presents three disjoint |$(-1)$|-curves: the |$\textbf{k}$|-curve |$D_{34}$| and the irreducible curve |$E$| whose geometric components are |$E_{1},E_{2}$|⁠. The contraction of |$D_{34}$| and |$E$| yields a birational morphism |$ \eta : X \rightarrow Z $| to a rational del Pezzo surface |$Z$| of degree |$8$| such that |$\textrm{rk} \, \textrm{NS}(Z) = 1$| and hence |$ Z \simeq \mathcal{Q}^{L} $| for some quadratic extension |$ L=\mathbf{k}(a_{1}) $| by [21, Lemma 3.2(1)]. Let |$a_{2}=a_{1}^{g}$| be the conjugate of |$a_{1}$| under the action of |$\operatorname{Gal}(L/\textbf{k})=:\langle g \rangle $|⁠. The morphism |$\eta $| contracts the curve |$D_{34}$| onto a |$\textbf{k}$|-rational point |$q \in \mathcal{Q}^{L}(\textbf{k})$| and the curve |$E$| onto a point |$ p=\lbrace p_{1},p_{2} \rbrace $| of degree |$2$| whose geometric components are exchanged by the |$\operatorname{Gal}(L/\textbf{k})$|-action. Moreover, there is a birational map |$ \psi : \mathcal{Q}^{L} \dashrightarrow \mathbb{P}^{2} $| that is the composition of the blow-up of |$q$| with the contraction of a curve onto a point of degree |$2$| in |$ \mathbb{P}^{2} $| whose splitting field is |$L$|⁠. The surface |$X$| is therefore isomorphic to the blow-up of |$\mathbb{P}^{2}$| in two points |$ r,s $| of degree |$2$| whose splitting field is |$L$|⁠, and we can assume that |$ r=\lbrace [a_{1}:0:1],[a_{2}:0:1] \rbrace $|⁠, |$ s=\lbrace [a_{1}:1:0],[a_{2}:1:0] \rbrace $| by Lemma 2.2.1, (2).

(2) Applying Lemma 2.2.1, (2) to |$ \lbrace [a_{1}:0:1],[a_{2}:0:1],[a_{1}:1:0],[a_{2}:1:0] \rbrace $| and |$ \lbrace q_{1},q_{2},q_{3},q_{4} \rbrace $| for some |$ q_{1},q_{2},q_{3},q_{4} \in \mathbb{P}^{2}(L) $|⁠, no three collinear, we see that we can send any two points |$ t=\lbrace q_{1},q_{2}\rbrace $|⁠, |$u=\lbrace q_{3},q_{4}\rbrace $| in |$\mathbb{P}^{2} $| of degree |$2$| whose splitting field is |$L$| onto |$ r=\lbrace [a_{1}:0:1],[a_{2}:0:1] \rbrace $|⁠, |$ s=\lbrace [a_{1}:1:0],[a_{2}:1:0] \rbrace $| by an element from |$ \textrm{PGL}_{3}(\textbf{k}) $|⁠. It follows that any such two surfaces satisfying our hypothesis are isomorphic if and only if the respective quadratic extensions are |$\textbf{k}$|-isomorphic.

(4) The set |$\{E_{1},E_{2},E_{3},E_{4}\}$| forms one |$\operatorname{Aut}_{\textbf{k}}(X)$|-orbit and hence can be equivariantly contracted onto |$\mathbb{P}^{2}$|⁠, giving |$\textrm{rk} \, \textrm{NS}(X)^{\operatorname{Aut}_{\textbf{k}}(X)}=2$|⁠. Moreover, the conic bundle induced by |$2H-E_{1}-E_{2}-E_{3}-E_{4}$| on |$X_{\overline{\textbf{k}}}$| (see Figure 1d), where |$H$| denotes the pullback of a general line of |$\mathbb{P}^{2}_{\overline{\textbf{k}}}$|⁠, is defined over |$\textbf{k}$| and preserved by the action of |$\operatorname{Aut}_{\textbf{k}}(X)$|⁠. Since |$\textrm{rk} \, \textrm{NS}(X)^{\operatorname{Aut}_{\textbf{k}}(X)}=2$|⁠, the corresponding conic bundle structure |$X \rightarrow \mathbb{P}^{1}$| given by the linear system of conics through |$r$| and |$s$| on |$\mathbb{P}^{2}$| is an |$\operatorname{Aut}_{\textbf{k}}(X)$|-Mori fibre space.

We note that via the birational map |$\psi ^{-1}: \mathbb{P}^{2} \dashrightarrow \mathcal{Q}^{L}$| given in the proof of Proposition 3.2.3, (1) we could also have lifted |$\alpha , \beta $| to birational transformations of |$\mathcal{Q}^{L}$|⁠.

Let |$X$| be a del Pezzo surface of degree |$5$| such that |$\rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})) = \langle (12)\rangle \times \langle (34)\rangle \simeq \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} $| in |$\operatorname{Sym}_{5}$| as indicated in Figure 2d. Then there exist quadratic extensions |$L=\textbf{k}(a_{1})$| and |$L^{\prime}=\textbf{k}(b_{1})$| that are not |$\textbf{k}$|-isomorphic, with |$ t^{2}+at+\tilde{a}=(t-a_{1})(t-a_{2}), \, t^{2}+bt+\tilde{b}=(t-b_{1})(t-b_{2}) \in \textbf{k}[t] $| the minimal polynomials of |$a_{1},b_{1}$|⁠, and such that the following holds:

• 1.

  there is a birational morphism |$\varepsilon : X \longrightarrow \mathbb{P}^{2} $| contracting two irreducible curves |$E, F$| onto two points |$ \varepsilon (E)=\lbrace [b_{1}:0:1],[b_{2}:0:1] \rbrace , \, \varepsilon (F)=\lbrace [a_{1}:1:0],[a_{2}:1:0] \rbrace $| of degree |$2$| whose splitting fields are respectively |$ L^{\prime}=\textbf{k}(b_{1}), L=\textbf{k}(a_{1}) $|⁠.

• 2.

  Any two such surfaces |$X$| and |$\tilde{X}$| are isomorphic if and only if |$\tilde{L}, \tilde{L^{\prime}}$| are respectively |$\textbf{k}$|-isomorphic to |$L,L^{\prime}$|⁠.

• 3.

  |$\operatorname{Aut}_{\textbf{k}}(X)=\langle \widehat{\alpha } \rangle \times \langle \widehat{\beta } \rangle \simeq \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} $|⁠, where |$\widehat{\alpha }, \widehat{\beta }$| are the lifts of two involutions |$\alpha , \beta $| of |$\mathbb{P}^{2}$|⁠.

• 4.

  |$ \textrm{rk} \, \textrm{NS}(X)^{\operatorname{Aut}_{\textbf{k}}(X)}=3 $| and |$ \varepsilon \operatorname{Aut}_{\textbf{k}}(X) \varepsilon ^{-1} = \operatorname{Aut}_{\textbf{k}}(\mathbb{P}^{2},\lbrace [b_{1}:0:1],[b_{2}:0:1] \rbrace ,\lbrace [a_{1}:1:0],[a_{2}:1:0] \rbrace ) $|⁠. In particular, |$ X \longrightarrow \ast $| is not an |$\operatorname{Aut}_{\textbf{k}}(X)$|-Mori fibre space and the |$\operatorname{Aut}_{\textbf{k}}(X)$|-minimal models of |$X$| are |$\mathbb{P}^{2}$| and the quadric |$\mathcal{Q}^{L}$|⁠.

(1) The Petersen graph |$ \Pi _{\overline{\textbf{k}},\rho } $| of |$X$| presents three disjoint |$(-1)$|-curves: the |$\textbf{k}$|-curve |$D_{34}$| and the curve denoted |$E$| whose geometric components are |$E_{1}, E_{2}$|⁠. Their contraction yields a birational morphism |$ \eta : X \rightarrow Z $| onto a rational del Pezzo surface |$Z$| of degree |$8$| such that |$\textrm{rk} \, \textrm{NS}(Z) = 1$| and hence |$ Z \simeq \mathcal{Q}^{L} $| for some quadratic extension |$ L/\textbf{k} $| by [21, Lemma 3.2(1)].

Fig. 4

Blow-up model for a del Pezzo surface as in Proposition 3.2.5.

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The Figure 4 shows the induced |$\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})$|-action on the image by |$\eta $| of |$\Pi _{\overline{\textbf{k}}}$|⁠. Then the image of the generator |$g$| of |$\operatorname{Gal}(L/\textbf{k})$| in |$\operatorname{Sym}_{5}$| is either |$(34)$| or |$(12)(34)$|⁠. It follows that the splitting field of |$ \eta (E)=\lbrace q_{1},q_{2} \rbrace :=q $| in |$\mathcal{Q}^{L}$| is a quadratic extension |$L^{\prime}$| not |$\textbf{k}$|-isomorphic to |$L$|⁠, such that the generator |$g^{\prime}$| of |$\operatorname{Gal}(L^{\prime}/\textbf{k})$| induces the action of |$(12)$| on |$\Pi _{\overline{\textbf{k}}}$|⁠. Let |$a_{1} \in L, b_{1} \in L^{\prime}$| such that |$L=\textbf{k}(a_{1})$| and |$L^{\prime}=\textbf{k}(b_{1})$|⁠. As in Propositions 3.2.1 and 3.2.3, there is a birational map |$ \psi : \mathcal{Q}^{L} \dashrightarrow \mathbb{P}^{2} $| that is the composition of the blow-up of |$p$| with the contraction of the curve |$F$| whose geometric components are |$E_{3},E_{4}$| onto a point |$r=\lbrace r_{1},r_{2} \rbrace $| of degree |$2$| in |$\mathbb{P}^{2}$| whose splitting field is |$L$| (see Figure 4). The surface |$X$| is therefore isomorphic to the blow-up |$\varepsilon : X \rightarrow \mathbb{P}^{2}$| of |$\mathbb{P}^{2}$| in the two points |$s:=\psi (q)$| and |$r$| of degree |$2$| whose splitting fields are respectively |$L^{\prime}$| and |$L$|⁠, and we can assume that |$ s=\lbrace [b_{1}:0:1],[b_{2}:0:1] \rbrace $| and |$ r=\lbrace [a_{1}:1:0],[a_{2}:1:0] \rbrace $| by Lemma 2.2.1.

(2) Applying Lemma 2.2.1, (2) to the sets |$ \lbrace [a_{1}:1:0],[a_{2}:1:0],[b_{1}:0:1],[b_{2}:0:1] \rbrace $| and |$ \lbrace w_{1},w_{2},w_{3},w_{4} \rbrace $| for some |$ w_{1},w_{2},w_{3},w_{4} \in \mathbb{P}^{2}(LL^{\prime}) $|⁠, no three collinear, we see that we can send any two points |$ t=\lbrace w_{1},w_{2}\rbrace $|⁠, |$u=\lbrace w_{3},w_{4}\rbrace $| of degree |$2$| in |$\mathbb{P}^{2} $| whose splitting fields are |$L, L^{\prime}$| onto |$ r=\lbrace [a_{1}:1:0],[a_{2}:1:0] \rbrace $|⁠, |$ s=\lbrace [b_{1}:0:1],[b_{2}:0:1] \rbrace $| by an element from |$ \textrm{PGL}_{3}(\textbf{k}) $|⁠. This gives the assertion.

(4) The sets |$\{E_{1},E_{2}\}$| and |$\{E_{3},E_{4}\}$| form two |$\operatorname{Aut}_{\textbf{k}}(X)$|-orbits and hence can be equivariantly contracted onto |$\mathbb{P}^{2}$|⁠, giving |$\textrm{rk} \, \textrm{NS}(X)^{\operatorname{Aut}_{\textbf{k}}(X)}=3$|⁠. Note that on the other hand we can also equivariantly contract the fixed curve |$D_{12}$| (resp. |$D_{34}$|⁠) and the orbit |$\lbrace E_{3},E_{4} \rbrace $| (resp. |$\lbrace E_{1},E_{2} \rbrace $|⁠) onto |$\mathcal{Q}^{L}$|⁠.

Let |$X$| be a del Pezzo surface of degree |$5$| such that |$\rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})) = \lbrace \operatorname{id} \rbrace $| in |$ \operatorname{Sym}_{5} $| as indicated in Figure 2a. Then:

• 1.

  there is one isomorphism class of such del Pezzo surfaces, obtained by blowing up |$\mathbb{P}^{2}$| in four |$\textbf{k}$|-rational points |$p_{1}, p_{2}, p_{3}, p_{4}$| in general position.

• 2.

  |$\operatorname{Aut}_{\textbf{k}}(X) = \langle G, \widehat{\sigma } \rangle \simeq \operatorname{Sym}_{5} $|⁠, where |$G$| is the lift of the subgroup |$\operatorname{Aut}_{\textbf{k}}(\mathbb{P}^{2},\lbrace p_{1},p_{2},p_{3},p_{4} \rbrace ) $| and |$\widehat{\sigma }$| is the lift of the standard quadratic involution |$\sigma : [x:y:z] {\scriptsize{\vdash}}\kern-5pt{\dashrightarrow} [yz:xz:xy]$|⁠.

• 3.

  |$ \textrm{rk} \, \textrm{NS}(X)^{\operatorname{Aut}_{\textbf{k}}(X)} = 1 $| and |$ X \longrightarrow \ast $| is an |$ \operatorname{Aut}_{\textbf{k}}(X) $|-Mori fibre space.

Since |$\rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k}))=\lbrace \operatorname{id}\rbrace $|⁠, all the |$(-1)$|-curves on the Petersen diagram of |$X_{\overline{\textbf{k}}}$| are defined over |$\textbf{k}$|⁠. Then the statement is proven analogously to the classical statement over algebraically closed fields and can be found in [3, Proposition 6.3.7] or in [10, §8.5.4] for instance.

Let |$X$| be a del Pezzo surface of degree |$5$| such that |$\rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})) = \langle (123) \rangle \simeq \mathbb{Z}/3\mathbb{Z} $| in |$ \operatorname{Sym}_{5} $| as indicated in Figure 2f. Then the following holds:

• 1.

  there exist a point |$q=\lbrace q_{1},q_{2},q_{3} \rbrace $| in |$\mathbb{P}^{2}$| of degree |$3$| with splitting field |$L$| such that |$ \operatorname{Gal}(L/\textbf{k}) \simeq \mathbb{Z}/3\mathbb{Z} $| and a |$\textbf{k}$|-rational point |$r \in \mathbb{P}^{2}(\textbf{k}) $|⁠, such that |$X$| is isomorphic to the blow-up of |$\mathbb{P}^{2}$| in |$r$| and |$q$|⁠.

• 2.

  Any two such surfaces are isomorphic if and only if the corresponding field extensions are |$\textbf{k}$|-isomorphic.

• 3.

  |$\operatorname{Aut}_{\textbf{k}}(X)=\langle \widehat{\alpha } \rangle \times \langle \widehat{\beta } \rangle \simeq \mathbb{Z}/6\mathbb{Z} $|⁠, where |$\widehat{\alpha }$| is the lift of an automorphism of |$\mathbb{P}^{2}$| of order three and |$\widehat{\beta }$| is the lift of a birational quadratic involution of |$\mathbb{P}^{2}$| with base-point |$q$|⁠.

• 4.

  |$ \textrm{rk} \, \textrm{NS}(X)^{\operatorname{Aut}_{\textbf{k}}(X)} = 2 $|⁠, so in particular |$ X \longrightarrow \ast $| is not an |$ \operatorname{Aut}_{\textbf{k}}(X) $|-Mori fibre space, and the |$\operatorname{Aut}_{\textbf{k}}(X)$|-minimal models of |$X$| are the del Pezzo surface |$Y$| of degree |$6$| obtained by blowing up |$\mathbb{P}^{2}$| in |$q$| and |$\mathbb{F}_{0} \simeq \mathbb{P}^{1} \times \mathbb{P}^{1}$|⁠.

(1) The Petersen diagram |$\Pi _{\overline{\textbf{k}},\rho }$| of |$X$| contains only one |$(-1)$|-curve defined over |$\textbf{k}$| namely |$E_{4}$|⁠. Its contraction yields a birational morphism |$\eta : X \rightarrow Y $| to a rational del Pezzo surface |$Y$| of degree |$6$| as in [21, Lemma 4.2], where the set of |$(-1)$|-curves of |$Y$| is the union of two curves |$C_{1}$| and |$C_{2}$| whose geometric components are respectively |$\lbrace D_{13},D_{12},D_{23} \rbrace $| and |$\lbrace E_{1},E_{2},E_{3} \rbrace $|⁠, on which |$\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})$| acts cyclically (see also Figure 5). Then there exists a point |$q=\lbrace q_{1},q_{2},q_{3} \rbrace $| in |$\mathbb{P}^{2}$| of degree |$3$| with splitting field |$L$| such that |$\operatorname{Gal}(L/\textbf{k})\simeq \mathbb{Z}/3\mathbb{Z}$| and such that |$Y$| is isomorphic to the blow-up of |$\mathbb{P}^{2}$| in |$q$|⁠. Therefore, |$X$| is isomorphic to the blow-up of |$\mathbb{P}^{2}$| in a point |$q$| of degree |$3$| and a |$\textbf{k}$|-rational point |$r\in \mathbb{P}^{2}(\textbf{k})$| (see Figure 5).

(2) The fact that any two del Pezzo surfaces satisfying our hypothesis are isomorphic if and only if the respective splitting fields are |$\textbf{k}$|-isomorphic follows from Lemma 2.2.1.

(3) The action of |$\operatorname{Aut}_{\textbf{k}}(X)$| on the set of |$(-1)$|-curves of |$X$| gives an isomorphism |$\Psi : \operatorname{Aut}_{\textbf{k}}(X) \overset{\sim }{\longrightarrow } \operatorname{Aut}(\Pi _{L,\rho }) $| by Lemma 3.1.1. In this case, since |$\operatorname{Aut}(\Pi _{L,\rho }) \simeq \lbrace \sigma \in \operatorname{Sym}_{5} \, \vert \, (123) \circ \sigma = \sigma \circ (123) \rbrace = C_{\operatorname{Sym}_{5}}(\langle (123) \rangle )$|⁠, we directly get |$\operatorname{Aut}(\Pi _{L,\rho }) \simeq \langle (123),(45) \rangle \simeq \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$|⁠. We now construct the corresponding geometric actions on |$X$|⁠. Let us define the map |$\phi _{q}:=\pi _{2}\circ \pi _{1}^{-1} \in \operatorname{Bir}_{\textbf{k}}(\mathbb{P}^{2})$|⁠, where |$\pi _{1},\pi _{2}: Y \rightarrow \mathbb{P}^{2} $| are respectively the contractions of the curves |$C_{1}, C_{2}$| (see Figure 5). This is a birational map of degree |$2$|⁠. By Lemma 2.2.1 we can assume that |$\phi _{q}$| fixes the rational point |$r$| and that it contracts the line through |$q_{i},q_{j}$| onto |$q_{k}$|⁠, where |$\lbrace i,j,k \rbrace = \lbrace 1,2,3 \rbrace $|⁠. These conditions imply that |$\phi _{q}$| is an involution and that it lifts to an automorphism |$\widehat{\beta }$| of X defined over |$\textbf{k}$| inducing a rotation of order two on the hexagon of |$Y$| and acting as |$(45)$| on the |$(-1)$|-curves of |$X$|⁠. Since |$\langle \sigma =(123) \rangle = \mathbb{Z}/3\mathbb{Z} \simeq \operatorname{Gal}(L/\textbf{k}) $|⁠, Lemma 2.2.1 guarantees that there exists |$\alpha ^{\prime} \in \operatorname{Aut}_{\textbf{k}}(\mathbb{P}^{2})$| such that |$\alpha ^{\prime}(q_{i})=q_{\sigma (i)}$|⁠, |$i=1,2,3$|⁠, and |$\alpha ^{\prime}(r)=r$|⁠. Then |$\alpha ^{\prime 3}$| and |$(\alpha ^{\prime}\circ \phi _{q}\circ \alpha ^{\prime -1})\circ \phi _{q}$| are linear and fix |$ r,q_{1},q_{2},q_{3} $|⁠, and hence |$\alpha ^{\prime}$| is of order |$3$| and |$\alpha ^{\prime}$| and |$\phi _{q}$| commute. The lift |$\widehat{\alpha }$| of |$\alpha ^{\prime}$| is an automorphism of |$X$| commuting with |$\widehat{\beta }$|⁠; it induces a rotation of order |$3$| on the hexagon of |$Y$| and it thus acts as |$(123)$| on |$\Pi _{L,\rho }$|⁠. In conclusion, |$\operatorname{Aut}_{\textbf{k}}(X)=\langle \widehat{\alpha }\rangle \times \langle \widehat{\beta }\rangle \simeq \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} $|⁠.

(4) The curve |$E_{4}$| is stabilized by |$\operatorname{Aut}_{\textbf{k}}(X)$| (see point 3) and can then be contracted equivariantly. The contraction of |$E_{4}$| gives a birational morphism to the rational del Pezzo surface |$Y$| of degree |$6$| whose |$\textbf{k}$|-automorphism group acts transitively on the hexagon, implying that |$Y \rightarrow \ast $| is an |$\operatorname{Aut}_{\textbf{k}}(X)$|-Mori fibre space. It follows that |$\textrm{rk} \, \textrm{NS}(X)^{\operatorname{Aut}_{\textbf{k}}(X)}=2$|⁠. Note that we can also equivariantly contract the |$\operatorname{Aut}_{\textbf{k}}(X)$|-orbit |$\lbrace D_{24},D_{34},D_{14} \rbrace $| onto the surface |$\mathbb{F}_{0} \simeq \mathbb{P}^{1} \times \mathbb{P}^{1}$|⁠, which finally yields the claim.

We construct a del Pezzo surface as in Proposition 3.3.2. From [21, Example 4.4], let |$\textbf{k} = \mathbb{F}_{2}$| and |$L/\textbf{k}$| be the splitting field of |$P(X)=X^{3}+X+1$|⁠, that is, |$\vert L \vert = 8$|⁠. Then |$ 3=\vert L:\textbf{k} \vert =\vert \operatorname{Gal}(L/\textbf{k}) \vert $| and |$ \sigma : a \mapsto a^{2} $| generates |$\operatorname{Gal}(L/\textbf{k})$| (see, for instance, [17, Theorem 6.5]). If |$\zeta $| is a root of |$P(X)$|⁠, then |$\sigma (\zeta ^{4})=\zeta $| and hence the point |$\lbrace [1:\zeta :\zeta ^{4}],[1:\zeta ^{2}:\zeta ],[1:\zeta ^{4}:\zeta ^{2}] \rbrace $| is of degree |$3$|⁠, its geometric components are not collinear and they are cyclically permuted by |$\sigma $|⁠. We take |$ r=[1:1:1] \in \mathbb{P}^{2}(\textbf{k}) $| as a |$\textbf{k}$|-rational point.

Let |$X$| be a del Pezzo surface of degree |$5$| such that |$ \rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})) = \langle (123),(12) \rangle \simeq \operatorname{Sym}_{3} $| in |$ \operatorname{Sym}_{5} $| as indicated in Figure 2k. Then the following holds:

• 1.

  There exist a point |$ q = \lbrace q_{1},q_{2},q_{3} \rbrace $| in |$ \mathbb{P}^{2} $| of degree |$3$| with splitting field |$L$| such that |$ \operatorname{Gal}(L/\textbf{k}) \simeq \operatorname{Sym}_{3} $| and a rational point |$ r \in \mathbb{P}^{2}(\textbf{k}) $| such that |$X$| is isomorphic to the blow-up of |$ \mathbb{P}^{2} $| in |$q$| and |$r$|⁠.

• 2.

  Any two such surfaces are isomorphic if and only if the corresponding field extensions are |$\textbf{k}$|-isomorphic.

• 3.

  |$ \operatorname{Aut}_{\textbf{k}}(X)=\langle \widehat{\alpha } \rangle \simeq \mathbb{Z}/2\mathbb{Z} $|⁠, where |$\widehat{\alpha }$| is the lift of a birational quadratic involution |$\phi _{q} $| with base-point |$q$|⁠.

• 4.

  |$ \textrm{rk} \, \textrm{NS}(X)^{\operatorname{Aut}_{\textbf{k}}(X)} = 2 $|⁠, so in particular |$ X \longrightarrow \ast $| is not an |$ \operatorname{Aut}_{\textbf{k}}(X) $|-Mori fibre space, and the |$\operatorname{Aut}_{\textbf{k}}(X)$|-minimal models of |$X$| are the rational del Pezzo surface |$Y$| of degree |$6$| obtained by blowing up |$\mathbb{P}^{2}$| in |$q$| and |$\mathbb{F}_{0} \simeq \mathbb{P}^{1} \times \mathbb{P}^{1}$|⁠.

(1) The incidence diagram |$ \Pi _{\overline{\textbf{k}},\rho } $| of |$X$| contains only one |$(-1)$|-curve that is defined over |$ \textbf{k} $| namely |$E_{4}$|⁠. Its contraction yields a birational morphism |$ \eta : X \rightarrow Y $| onto a rational del Pezzo surface |$Y$| of degree |$6$| as in [21, Lemma 4.3]. Then there exists a point |$ q=\lbrace q_{1},q_{2},q_{3} \rbrace $| in |$ \mathbb{P}^{2} $| of degree |$3$| with splitting field |$L$| such that |$ \operatorname{Gal}(L/\textbf{k}) \simeq \operatorname{Sym}_{3} $| and whose geometric components are in general position in |$\mathbb{P}^{2}_{L}$|⁠, such that |$Y$| is isomorphic to the blow-up of |$ \mathbb{P}^{2} $| in |$q$|⁠. Finally, |$X$| is isomorphic to the blow-up of |$\mathbb{P}^{2}$| in a point |$q$| of degree |$3$| and in a |$\textbf{k}$|-rational point |$r$|⁠.

(2) The fact that any such two del Pezzo surfaces are isomorphic if and only if the respective splitting fields are |$\textbf{k}$|-isomorphic follows from Lemma 2.2.1.

(3) By Lemma 3.1.1, the action of |$ \operatorname{Aut}_{\textbf{k}}(X) $| on the set of |$(-1)$|-curves of |$X$| yields an isomorphism |$ \Psi : \operatorname{Aut}_{\textbf{k}}(X) \overset{\sim }{\longrightarrow } \operatorname{Aut}(\Pi _{L,\rho }) $|⁠. Since |$\operatorname{Aut}(\Pi _{L,\rho }) \simeq \lbrace \sigma \in \operatorname{Sym}_{5} \, \vert \, (123) \circ \sigma = \sigma \circ (123) \, \textrm{and} \, (12) \circ \sigma = \sigma \circ (12) \rbrace = C_{\operatorname{Sym}_{5}}(\langle (123),(12) \rangle )$|⁠, we get |$\operatorname{Aut}(\Pi _{L,\rho }) \simeq \langle (45) \rangle \simeq \mathbb{Z}/2\mathbb{Z}$|⁠. To finish, we construct the corresponding geometric action. From Proposition 3.3.2(3) there exists a birational quadratic involution |$\phi _{q} \in \operatorname{Bir}_{\textbf{k}}(\mathbb{P}^{2})$| that lifts to an automorphism |$\widehat{\alpha }$| of |$X$| inducing the same action as |$(45)$| on the |$(-1)$|-curves of |$X$|⁠. In conclusion, |$ \operatorname{Aut}_{\textbf{k}}(X) = \langle \widehat{\alpha } \rangle \simeq \mathbb{Z}/2\mathbb{Z} $|⁠.

(4) The proof is analogous to the one of Proposition 3.3.2(4).

We construct a del Pezzo surface as in Proposition 3.3.4. Indeed, from [21, Example 4.5], let |$ \textbf{k}:= \mathbb{Q} $|⁠, let |$ \zeta := \sqrt [3]{2} $| and |$ \omega := e^{\frac{2 \pi i}{3}} $|⁠. Then |$ L:= \textbf{k}(\zeta ,\omega ) $| is a Galois extension of |$\textbf{k}$| of degree |$6$| and |$ \operatorname{Gal}(L/\textbf{k}) \simeq \operatorname{Sym}_{3} $| is the group of |$ \textbf{k} $|-isomorphisms |$ [ (\zeta ,\omega ) \overset{\sigma _{1}}{\longmapsto } (\zeta ,\omega ), (\zeta ,\omega ) \overset{\sigma _{2}}{\longmapsto } (\omega \zeta ,\omega ), (\zeta ,\omega ) \overset{\sigma _{3}}{\longmapsto } (\zeta ,\omega ^{2}), (\zeta ,\omega ) \overset{\sigma _{4}}{\longmapsto } (\omega \zeta ,\omega ^{2}), (\zeta ,\omega ) \overset{\sigma _{5}}{\longmapsto } (\omega ^{2} \zeta ,\omega ), (\zeta ,\omega ) \overset{\sigma _{6}}{\longmapsto } (\omega ^{2} \zeta ,\omega ^{2}) ] $| (see [17, Example 2.21]). The point |$ q = \lbrace [\zeta :\zeta ^{2}:1], [\omega \zeta :\omega ^{2}\zeta ^{2}:1], [\omega ^{2}\zeta :\omega \zeta ^{2}:1] \rbrace $| is of degree |$3$|⁠, its geometric components are not collinear and any non-trivial element of |$ \operatorname{Gal}(L/\textbf{k}) $| permutes them non-trivially. We take |$ r=[1:1:1] \in \mathbb{P}^{2}(\textbf{k}) $| as a |$\textbf{k}$|-rational point.

We note that in Proposition 3.3.2 (1) (resp. 3.3.4 (1)), we could also have contracted the curve of |$X$| whose geometric components are |$\lbrace D_{14},D_{24},D_{34} \rbrace $| onto a rational del Pezzo surface |$Z$| of degree |$8$| with |$\textrm{rk} \, \textrm{NS}(Z) = 2$|⁠, namely |$Z \simeq \mathbb{F}_{0} \simeq \mathbb{P}^{1} \times \mathbb{P}^{1}$| by [21, Lemma 3.2]; that is |$X$| is also obtained by blowing up |$\mathbb{F}_{0}$| in a point of degree |$3$| whose splitting field |$L$| is such that |$\operatorname{Gal}(L/\textbf{k}) \simeq \mathbb{Z}/3\mathbb{Z}$| (resp. |$\operatorname{Gal}(L/\textbf{k}) \simeq \operatorname{Sym}_{3}$|⁠).

Let |$X$| be a del Pezzo surface of degree |$5$| such that |$\rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k}))$| is one of the subgroups of |$\operatorname{Sym}_{5}$| occuring in Table 1, and corresponding to Figures 2e, 2g, 2i, 2h, and 2j, respectively.

Then the following holds:

• 1.

  There exists a point |$p = \lbrace p_{1},p_{2},p_{3},p_{4} \rbrace $| in |$\mathbb{P}^{2}$| of degree |$4$| with splitting field |$L$| such that |$\operatorname{Gal}(L/\textbf{k}) \simeq \Gamma $|⁠, where |$\Gamma $| is given in Table 1, and such that |$X$| is isomorphic to the blow-up of |$\mathbb{P}^{2}$| in |$p$|⁠.

• 2.

  Any two such surfaces are isomorphic if and only if the corresponding residue fields are |$\textbf{k}$|-isomorphic, which is equivalent to having |$\textbf{k}$|-isomorphic splitting fields except when |$\operatorname{Gal}(L/\textbf{k}) \simeq \operatorname{D}_{4}$|⁠.

• 3.

  |$\operatorname{Aut}_{\textbf{k}}(X) \simeq G$|⁠, where |$G$| is described in Table 1, and it is generated by the lifts of some automorphisms of |$\mathbb{P}^{2}$|⁠.

• 4.

  |$\textrm{rk} \, \textrm{NS}(X)^{\operatorname{Aut}_{\textbf{k}}(X)} = 2$|⁠, so in particular |$X \rightarrow \ast $| is not an |$\operatorname{Aut}_{\textbf{k}}(X)$|-Mori fibre space, but |$X$| carries an |$\operatorname{Aut}_{\textbf{k}}(X)$|-Mori fibre space structure |$X \rightarrow \mathbb{P}^{1}$| given by the linear system of conics passing through |$p$|⁠.

Let us give the proof for a del Pezzo surface |$X$| as in Figure 2g. (1) The union |$E=E_{1} \cup E_{2} \cup E_{3} \cup E_{4}$| makes up a |$\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})$|-orbit and hence defines an irreducible |$\textbf{k}$|-curve. Its contraction yields a birational morphism |$\eta : X \longrightarrow \mathbb{P}^{2}$| onto a point |$p=\lbrace p_{1},p_{2},p_{3},p_{4} \rbrace $| of degree |$4$| whose geometric components are in general position in |$\mathbb{P}^{2}_{\overline{\textbf{k}}}$| (see Figure 6).

Fig. 6

Blow-up model for a del Pezzo surface as in Proposition 3.3.7, (2g).

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Let |$L$| be any splitting field of |$p$|⁠. Since |$\rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k}))=\langle (1234)\rangle =\mathbb{Z}/4\mathbb{Z} $|⁠, the action of |$\operatorname{Gal}(L/\textbf{k})$| on |$\lbrace p_{1},p_{2},p_{3},p_{4} \rbrace $| induces an exact sequence |$ 1 \longrightarrow H \longrightarrow \operatorname{Gal}(L/\textbf{k}) \longrightarrow \mathbb{Z}/4\mathbb{Z} \longrightarrow 1 $|⁠. The field |$ L^{\prime}:=\lbrace a \in L \mid h(a)=a \,\, \forall \, h \in H\rbrace $| is an intermediate field between |$L$| and |$\textbf{k}$|⁠, over which |$p_{1},p_{2},p_{3},p_{4} $| are rational. The minimality of |$L$| implies that |$L^{\prime}=L$| and hence |$H=\operatorname{Gal}(L/L^{\prime})=\lbrace \textrm{id} \rbrace $| [17, Corollary 2.10], and so |$\operatorname{Gal}(L/\textbf{k}) \simeq \mathbb{Z}/4\mathbb{Z} $|⁠.

(2) The claim follows from Lemma 2.2.1, (1) and we can even assume that the blown-up point |$p$| is of the form |$\lbrace [1:a_{1}:a_{1}^{2}],[1:a_{2}:a_{2}^{2}],[1:a_{3}:a_{3}^{2}],[1:a_{4}:a_{4}^{2}] \rbrace $|⁠, with |$a_{i} \in L$| forming an orbit |$\lbrace a_{1},a_{2},a_{3},a_{4} \rbrace \subset L $| of size four under the Galois action.

(3) By Lemma 3.1.1 the action of |$\operatorname{Aut}_{\textbf{k}}(X)$| on the set of |$(-1)$|-curves of |$X$| yields an isomorphism |$\Psi : \operatorname{Aut}_{\textbf{k}}(X) \overset{\sim }{\longrightarrow } \operatorname{Aut}(\Pi _{L,\rho })$|⁠. Since |$\operatorname{Aut}(\Pi _{L,\rho }) \simeq \lbrace \sigma \in \operatorname{Sym}_{5} \, \vert \, (1234) \circ \sigma = \sigma \circ (1234) \rbrace = C_{\operatorname{Sym}_{5}}(\langle (1234) \rangle )$|⁠, we then get |$\operatorname{Aut}(\Pi _{L,\rho }) \simeq \langle (1234) \rangle \simeq \mathbb{Z}/4\mathbb{Z}$|⁠. To finish we construct explicitly the corresponding geometric action: if |$\langle \sigma =(1234)\rangle =\mathbb{Z}/4\mathbb{Z} \simeq \operatorname{Gal}(L/\textbf{k})$|⁠, Lemma 2.2.1 guarantees that there exists |$\alpha \in \operatorname{Aut}_{\textbf{k}}(\mathbb{P}^{2})$| such that |$\alpha (p_{i})=p_{\sigma (i)}$|⁠, for |$i=1,2,3,4$|⁠. Then |$\alpha ^{4}$| is linear and fixes |$p_{1},p_{2},p_{3},p_{4}$|⁠, and hence |$\alpha $| is an automorphism of order four. The lift |$\widehat{\alpha }$| of |$\alpha $| is an automorphism of |$X$| defined over |$\textbf{k}$| that acts as |$(1234)$| on |$\Pi _{L}$|⁠. In conclusion, |$\operatorname{Aut}_{\textbf{k}}(X) = \langle \widehat{\alpha }\rangle \simeq \mathbb{Z}/4\mathbb{Z} $|⁠.

(4) The set |$\{ E_{1},E_{2},E_{3},E_{4} \}$| forms one |$\operatorname{Aut}_{\textbf{k}}(X)$|-orbit and hence can be equivariantly contracted onto |$\mathbb{P}^{2}$|⁠, giving |$\textrm{rk} \, \textrm{NS}(X)^{\operatorname{Aut}_{\textbf{k}}(X)} = 2$|⁠. Moreover, the conic bundle induced by |$2H-E_{1}-E_{2}-E_{3}-E_{4}$| on |$X$| (see Figure 1d), where |$H$| denotes the pullback of a general line of |$\mathbb{P}^{2}$|⁠, is preserved by the action of |$\operatorname{Aut}_{\textbf{k}}(X)$|⁠. Since |$\textrm{rk} \textrm{NS}(X)^{\operatorname{Aut}_{\textbf{k}}(X)} = 2$|⁠, the corresponding conic bundle structure |$X \rightarrow \mathbb{P}^{1}$| given by the linear system of conics through |$p$| on |$\mathbb{P}^{2}$| is an |$\operatorname{Aut}_{\textbf{k}}(X)$|-Mori fibre space.

• 1.
  We construct a del Pezzo surface as in Proposition 3.3.7, (2e): let |$P(X)=X^{4}+aX^{2}+b$| be an irreducible polynomial over |$\mathbb{Q}$|⁠. We denote |$\pm \alpha $| and |$\pm \beta $| its roots in a splitting field |$L$|⁠. Then |$\operatorname{Gal}(L/\mathbb{Q})$| is isomorphic to |$\mathbb{Z}/4\mathbb{Z}$| or to |$\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$| or to |$\operatorname{D}_{4}$|⁠. Note that |$ \alpha ^{2}-\beta ^{2} \notin \mathbb{Q}$| otherwise |$P$| would be reducible, and we have the following characterization: |$\operatorname{Gal}(L/\mathbb{Q})=\langle \tau \circ \sigma \rangle \times \langle \sigma ^{2} \rangle \simeq \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} \Longleftrightarrow \alpha \beta \in \mathbb{Q} $|⁠, where |$ \tau : \alpha \mapsto -\alpha , \beta \mapsto \beta $| and |$ \sigma : \alpha \mapsto \beta , \beta \mapsto -\alpha $|⁠. In the latter case,

  $$ \begin{equation*} p=\lbrace [1:\alpha:\alpha^{2}],[1:-\alpha:\alpha^{2}],[1:\beta:\beta^{2}],[1:-\beta:\beta^{2}] \rbrace \end{equation*} $$

  is a point of degree |$4$| in |$\mathbb{P}^{2}$| whose geometric components are defined over |$L$| and form an orbit of size four under the |$\operatorname{Gal}(L/\mathbb{Q})$|-action. One can consider |$P(X)=X^{4}-6X^{2}+4$| over |$\mathbb{Q}$| with roots |$ \pm \alpha = \pm \sqrt{3+\sqrt{5}} $| and |$ \pm \beta = \pm \sqrt{3-\sqrt{5}} $| for a specific example.
• 2.

  We construct a del Pezzo surface as in Proposition 3.3.7, (2g). Indeed, let |$\textbf{k} = \mathbb{F}_{2}$| and |$L/\textbf{k}$| be the splitting field of |$P(X)=X^{4}+X+1$|⁠, that is, |$\vert L \vert = 16$|⁠. Then |$4=\vert L:\textbf{k} \vert =\vert \operatorname{Gal}(L/\textbf{k}) \vert $| and |$\sigma : s \mapsto s^{2} $| generates |$\operatorname{Gal}(L/\textbf{k})$|⁠. If |$\zeta $| is a root of |$P(X)$|⁠, then |$\sigma (\zeta ^{8})=\zeta $| and hence the point |$\lbrace [1:\zeta :\zeta ^{2}],[1:\zeta ^{2}:\zeta ^{4}],[1:\zeta ^{4}:\zeta ^{8}],[1:\zeta ^{8}:\zeta ^{16}] \rbrace $| is of degree |$4$|⁠, its geometric components are not collinear and they are cyclically permuted by |$\sigma $|⁠.

• 3.

  We construct a del Pezzo surface as in Proposition 3.3.7, (2i): let |$\textbf{k}=\mathbb{Q}$| and let |$L$| be a splitting field of the polynomial |$X^{4}-6 \in \textbf{k}[X]$|⁠. Then |$L=\textbf{k}(\sqrt [4]{6},i)$| and |$ \vert L:\textbf{k} \vert =8$|⁠. The Galois group |$\Gamma :=\operatorname{Gal}(L/\textbf{k})$| contains the |$\mathbb{Q}$|-isomorphisms |$r$| and |$s$| being such that |$r(\sqrt [4]{6})=i\sqrt [4]{6}$|⁠, |$r(i)=i$|⁠, |$s(\sqrt [4]{6})= \sqrt [4]{6}$| and |$s(i)=-i$|⁠, with |$\vert \operatorname{Gal}(L/\textbf{k}) \vert = \vert L:\textbf{k} \vert = 8$| because |$L/\textbf{k}$| is Galois. Thus, |$\operatorname{Gal}(L/\textbf{k}) = \langle r,s \, \vert \, r^{4}=s^{2}=\operatorname{id} \,, \, s \circ r \circ s=r^{-1} \rangle \simeq \operatorname{D}_{4} $| and |$ p=\lbrace [1:\sqrt [4]{6}:\sqrt{6}],[1:-\sqrt [4]{6}:\sqrt{6}],[1:i\sqrt [4]{6}:-\sqrt{6}],[1:-i\sqrt [4]{6}:-\sqrt{6}] \rbrace $| is a point of degree |$4$| in |$\mathbb{P}^{2}$| whose geometric components are non-trivially permuted by any non-trivial element in |$\Gamma $|⁠.

• 4.

  We construct a del Pezzo surface as in Proposition 3.3.7, (2h). Indeed, let |$\textbf{k}=\mathbb{Q}$|⁠, let |$P(X)=X^{4}+8X+12 \in \textbf{k}[X]$| and |$L$| be a splitting field of |$P$| over |$\textbf{k}$|⁠. Then |$P$| is separable and |$\textrm{disc}(P)=576^{2} $| so |$\Gamma :=\operatorname{Gal}(L/\textbf{k})$| is a subgroup of |$\mathcal{A}_{4}$|⁠, and |$P$| is irreducible so |$\Gamma $| acts transitively on the set of roots of |$P$|⁠. We have |$P(X) \equiv X^{4}+3X+7=(X+1)(X^{3}+4X^{2}+X+2) \operatorname{mod} \, 5$|⁠, where |$X^{3}+4X^{2}+X+2$| is irreducible over |$\mathbb{F}_{5}$| with Galois group |$\operatorname{Gal}(\mathbb{F}_{5^{3}}/\mathbb{F}_{5}) \simeq \mathbb{Z}/3\mathbb{Z} $| generated by |$x \mapsto x^{5}$|⁠. It follows that |$\Gamma $| contains |$\mathbb{Z}/3\mathbb{Z}$| as a subgroup. Moreover, |$P(X) \equiv (X^{2}+4X+7)(X^{2}+13X+9) \operatorname{mod} \, 17$|⁠, where |$X^{2}+4X+7$| is irreducible over |$\mathbb{F}_{17}$| and its Galois group is |$\operatorname{Gal}(\mathbb{F}_{17^{2}}/\mathbb{F}_{17}) \simeq \mathbb{Z}/2\mathbb{Z} $| generated by |$x \mapsto x^{17} $|⁠. Then |$\Gamma $| contains |$\mathbb{Z}/2\mathbb{Z}$| as a subgroup as well. In conclusion, |$\Gamma $| is a subgroup of |$\mathcal{A}_{4}$| acting transitively on the roots of |$P$|⁠, and both |$2$| and |$3$| divide |$\vert \Gamma \vert $|⁠. This implies |$\Gamma \simeq \mathcal{A}_{4}$|⁠. Let us denote by |$a_{1},a_{2},a_{3},a_{4}$| the roots of |$P$| in |$L$|⁠. Then |$p=\lbrace [1:a_{1}:a_{1}^{2}],[1:a_{2}:a_{2}^{2}],[1:a_{3}:a_{3}^{2}],[1:a_{4}:a_{4}^{2}] \rbrace $| defines a point of degree |$4$| whose geometric components are non-trivially permuted by any non-trivial element of |$\Gamma $|⁠.

Note that if |$p, p^{\prime}$| are two points of degree |$4$| in general position on |$\mathbb{P}^{2}$|⁠, then |$p$| and |$p^{\prime}$| are projectively equivalent, that is, there exists |$\alpha \in \operatorname{PGL}_{3}(\textbf{k})$| with |$\alpha (p) = p^{\prime}$|⁠, exactly if |$p$| and |$p^{\prime}$| have |$\textbf{k}$|-isomorphic residue fields thanks to Lemma 2.2.1. This is equivalent to having |$\textbf{k}$|-isomorphic splitting fields, say |$L/\textbf{k}$|⁠, |$L^{\prime}/\textbf{k}$|⁠, except when |$\operatorname{Gal}(L/\textbf{k}) \simeq \textrm{D}_{4}$| (see Proposition 3.3.7, 2i,(2)).

Let |$X$| be a del Pezzo surface of degree |$5$| such that |$ \rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})) = \langle (123)(45) \rangle \simeq \mathbb{Z}/6\mathbb{Z} $| in |$ \textrm{Sym}_{5} $| as indicated in Figure 2l. Then the following holds.

• 1.

  There exist a quadratic extension |$ L/\textbf{k} $| and a birational morphism |$ \eta : X \longrightarrow \mathcal{Q}^{L} $| contracting an irreducible curve onto a point |$ p=\lbrace p_{1},p_{2},p_{3} \rbrace $| of degree |$3$| whose geometric components are contained in pairwise distinct rulings of |$ \mathcal{Q}^{L}_{L} $|⁠, and with splitting field |$ F $| such that |$ \operatorname{Gal}(F/\textbf{k}) \simeq \operatorname{Gal}(FL/L) \simeq \mathbb{Z}/3\mathbb{Z} $|⁠, where |$FL$| is a splitting field of |$p$| over |$L$|⁠.

• 2.

  Any two such del Pezzo surfaces are isomorphic if and only if the corresponding field extensions of degree two and three are |$\textbf{k}$|-isomorphic.

• 3.

  |$ \operatorname{Aut}_{\textbf{k}}(X)=\langle \widehat{\alpha } \rangle \times \langle \widehat{\Phi _{q}} \rangle \simeq \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} $|⁠, where |$\widehat{\alpha }$| is the lift of an automorphism of |$\mathbb{P}^{2}_{L}$| of order three and |$\widehat{\Phi _{q}}$| is the lift of a quadratic birational involution of |$\mathbb{P}^{2}_{L}$| with base-point |$q$| of degree |$3$|⁠.

• 4.

  |$ \textrm{rk} \, \textrm{NS}(X)^{\operatorname{Aut}_{\textbf{k}}(X)} = 2 $|⁠, so in particular |$ X \rightarrow \ast $| is not an |$ \operatorname{Aut}_{\textbf{k}}(X) $|-Mori fibre space, and the |$\operatorname{Aut}_{\textbf{k}}(X)$|-minimal models of |$X$| are the quadric |$\mathcal{Q}^{L}$| and the rational del Pezzo surface |$Y$| of degree |$6$| such that |$Y_{L}$| is isomorphic to the blow-up of |$\mathbb{P}^{2}_{L}$| in a point |$q$| of degree |$3$| with splitting field |$E$| such that |$\operatorname{Gal}(E/L) \simeq \mathbb{Z}/3\mathbb{Z}$|⁠.

(1) There is only one |$(-1)$|-curve defined over |$ \textbf{k} $| on |$ X_{\overline{\textbf{k}}} $| namely |$ E_{4} $|⁠. Its contraction yields a birational morphism |$ \varepsilon : X \rightarrow Y $| onto a rational del Pezzo surface |$Y$| of degree |$6$| as in [21, Lemma 4.6 and Remark B.3], where the |$\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})$|-action on the hexagon of |$Y_{\overline{\textbf{k}}}$| is indicated in Figure 7. On the other hand, |$ \Pi _{\overline{\textbf{k}}} $| contains an orbit of three disjoint |$(-1)$|-curves |$ D_{34},D_{24},D_{14} $| (see Figure 2l). The contraction of the irreducible curve |$D$| with geometric components |$D_{34},D_{24},D_{14}$| yields a birational morphism |$ \eta : X \rightarrow Z $| onto a rational del Pezzo surface of degree |$8$|⁠. It contracts |$D$| onto a point |$p=\lbrace p_{1},p_{2},p_{3} \rbrace $| of degree |$3$| and it conjugates the |$ \operatorname{Gal}(\overline{\textbf{k}}/\textbf{k}) $|-action on |$X_{\overline{\textbf{k}}}$| to an action that exchanges the fibrations of |$Z_{\overline{\textbf{k}}}$|⁠, so |$ \textrm{rk} \, \textrm{NS}(Z) = 1 $|⁠. Hence, |$ Z \simeq \mathcal{Q}^{L} $| for some quadratic extension |$ L/\textbf{k} $| [21, Lemma 3.2(1)]. Figure 8 shows the action of |$\rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k}))=\langle (123)(45) \rangle $| on the image by |$ \eta $| of the incidence diagram |$ \Pi _{\overline{\textbf{k}}} $|⁠. Then |$ \eta $| conjugates the |$ \operatorname{Gal}(\overline{\textbf{k}}/L) $|-action on |$ \mathcal{Q}^{L}_{\overline{\textbf{k}}} $| to an action on |$ \Pi _{\overline{\textbf{k}}} $| with |$ \rho (\operatorname{Gal}(\overline{\textbf{k}}/L)) = \mathbb{Z}/3\mathbb{Z} $|⁠. Let |$ F/\textbf{k} $| be any splitting field of |$ p \in \mathcal{Q}^{L} $| and let us show that |$\operatorname{Gal}(F/\textbf{k})\simeq \mathbb{Z}/3\mathbb{Z}$|⁠. Let |$ M/\textbf{k} $| be a minimal normal extension containing |$F$| such that |$ \operatorname{Gal}(\overline{\textbf{k}}/M) $| acts trivially on |$ \Pi _{\overline{\textbf{k}}} $|⁠. The |$ \operatorname{Gal}(\overline{\textbf{k}}/\textbf{k}) $|-action on |$ \Pi _{\overline{\textbf{k}}} $| induces a short exact sequence |$ 1 \rightarrow \operatorname{Gal}(\overline{\textbf{k}}/M) \rightarrow \operatorname{Gal}(\overline{\textbf{k}}/\textbf{k}) \rightarrow \mathbb{Z}/6\mathbb{Z} \rightarrow 1 $|⁠, which implies |$ \operatorname{Gal}(M/\textbf{k}) \simeq \operatorname{Gal}(\overline{\textbf{k}}/\textbf{k}) / \operatorname{Gal}(\overline{\textbf{k}}/M) \simeq \mathbb{Z}/6\mathbb{Z} $|⁠. The extension |$F/\textbf{k}$| is normal, so |$\operatorname{Gal}(F/\textbf{k}) \subseteq \operatorname{Gal}(M/\textbf{k})$|⁠. Then |$ \mathbb{Z}/3\mathbb{Z} \subseteq \operatorname{Gal}(F/\textbf{k}) $| implies |$ \operatorname{Gal}(F/\textbf{k}) \simeq \mathbb{Z}/3\mathbb{Z} $| since |$ F/\textbf{k} $| is a splitting field of |$ p=\lbrace p_{1},p_{2},p_{3} \rbrace $|⁠. Then |$ FL/L $| is a splitting field of |$p$| over |$L$|⁠, |$ FL/L $| is Galois and |$ \operatorname{Gal}(FL/L) \simeq \operatorname{Gal}(F/F \cap L) = \operatorname{Gal}(F/\textbf{k}) \simeq \mathbb{Z}/3\mathbb{Z} $| (see [17, Theorem 5.5]).

Fig. 7

The |$ \operatorname{Gal}(\overline{\textbf{k}}/\textbf{k}) $|-action on |$ Y_{\overline{\textbf{k}}} $|⁠.

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Fig. 8

Blow-up model for a del Pezzo surface as in Proposition 3.4.1.

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(2) Two del Pezzo surfaces |$ \mathcal{Q}^{L} $| and |$ \mathcal{Q}^{L^{\prime}} $| are isomorphic if and only if the corresponding quadratic extensions |$L$| and |$L^{\prime}$| are |$\textbf{k}$|-isomorphic [21, Lemma 3.2(3)]. By Lemma 2.2.4, |$ \operatorname{Aut}_{\textbf{k}}(\mathcal{Q}^{L}) $| acts transitively on the set of points of degree |$3$| in |$ \mathcal{Q}^{L} $| with |$\textbf{k}$|-isomorphic splitting fields and whose geometric components are in general position. This yields the claim.

(3&4) The action of |$ \operatorname{Aut}_{\textbf{k}}(X) $| on the set of |$ (-1) $|-curves of |$X$| yields an isomorphism |$ \Psi : \operatorname{Aut}_{\textbf{k}}(X) \overset{\sim }{\longrightarrow } \operatorname{Aut}(\Pi _{\overline{\textbf{k}},\rho }) $| by Lemma 3.1.1. Since |$\operatorname{Aut}(\Pi _{\overline{\textbf{k}},\rho }) \simeq \lbrace \sigma \in \operatorname{Sym}_{5} \, \vert \, (123)(45) \circ \sigma = \sigma \circ (123)(45) \rbrace = C_{\operatorname{Sym}_{5}}(\langle (123)(45) \rangle )$|⁠, we get |$\operatorname{Aut}(\Pi _{\overline{\textbf{k}},\rho }) \simeq \langle (123) \rangle \times \langle (45) \rangle \simeq \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$|⁠. We now construct the corresponding geometric actions. From part (1) we know that |$Y_{L}$| is isomorphic to the del Pezzo surface of degree |$6$| from [21, Lemma 4.2], which is the blow-up |$ \pi : Y_{L} \rightarrow \mathbb{P}^{2}_{L} $| of a point |$q=\lbrace q_{1},q_{2},q_{3} \rbrace $| of degree |$3$|⁠, and so that |$X_{L}$| is isomorphic to the del Pezzo surface from Proposition 3.3.2. Since |$ Y $| is rational, |$ \operatorname{Gal}(L/\textbf{k}) = \langle g \rangle $| has a fixed point |$ r \in Y(\textbf{k}) $|⁠. Let |$ \phi _{q} \in \operatorname{Bir}_{L}(\mathbb{P}^{2}) $| be the quadratic involution from Proposition 3.3.2(3), such that |$ \phi _{q} $| fixes |$ \pi (r) \in \mathbb{P}^{2}_{L}(L) \simeq \mathbb{P}^{2}(L) $| and |$ \Phi _{q}:= \pi ^{-1} \phi _{q} \pi \in \operatorname{Aut}_{L}(Y) $| induces a rotation of order two on the hexagon of |$ Y_{L} $|⁠. Then |$ \Phi _{q}(g\Phi _{q}g) \in \operatorname{Aut}_{L}(Y) $| preserves the edges of the hexagon and fixes |$r$|⁠. It therefore descends to an element of |$ \operatorname{Aut}_{L}(\mathbb{P}^{2},q_{1},q_{2},q_{3}) $| fixing |$ \pi (r) $| and is hence equal to the identity. It follows that |$ \Phi _{q} \in \operatorname{Aut}_{\textbf{k}}(Y) $| and it induces a rotation of order two on the hexagon of |$ Y $|⁠. Thus, it lifts by |$ \varepsilon $| to an automorphism |$ \widehat{\Phi _{q}} $| of |$X$| defined over |$ \textbf{k} $| that acts as |$(45)$| on |$ \Pi _{\overline{\textbf{k}}} $|⁠. Now let |$ \tilde{\alpha } \in \operatorname{Aut}_{L}(\mathbb{P}^{2},\lbrace q_{1},q_{2},q_{3} \rbrace ) $| be the automorphism of order three from Proposition 3.3.2(3) such that |$ \tilde{\alpha } $| fixes |$ \pi (r) \in \mathbb{P}^{2}(L) $|⁠, |$\tilde{\alpha }: q_{1} \mapsto q_{2} \mapsto q_{3}$| and |$ \alpha := \pi ^{-1} \tilde{\alpha } \pi \in \textrm{Aut}_{L}(Y) $| induces a rotation of order three on the hexagon of |$ Y_{L} $|⁠. We can argue as above that |$ \alpha \in \operatorname{Aut}_{\textbf{k}}(Y) $| and thus it lifts to an automorphism |$ \widehat{\alpha } $| of |$X$| defined over |$\textbf{k}$| that acts as |$(123)$| on |$ \Pi _{\overline{\textbf{k}}} $|⁠. In conclusion, |$ \operatorname{Aut}_{\textbf{k}}(X)=\langle \widehat{\alpha } \rangle \times \langle \widehat{\Phi _{q}} \rangle \simeq \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} $|⁠. For the last assertion, we have a contraction |$\eta : X \rightarrow \mathcal{Q}^{L}$| onto a point of degree |$3$| from (1), so thanks to Lemma 2.2.4, we get |$ \textrm{rk} \, \textrm{NS}(X)^{\operatorname{Aut}_{\textbf{k}}(X)} = 2 $|⁠. Now, note that the curve |$E_{4}$| is stabilized by |$\operatorname{Aut}_{\textbf{k}}(X)$| and can then be contracted equivariantly. It gives an equivariant birational morphism to the rational del Pezzo surface |$Y$| with |$\textrm{rk} \, \textrm{NS}(Y) = 1$|⁠, which yields the claim.

Let |$X$| be a del Pezzo surface of degree |$5$| such that |$ \rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})) = \langle (123),(12),(45) \rangle \simeq \operatorname{Sym}_{3} \times \mathbb{Z}/2\mathbb{Z} $| in |$ \textrm{Sym}_{5} $| as indicated in Figure 2l. Then the following holds:

• 1.

  There exist a quadratic extension |$ L/\textbf{k} $| and a birational morphism |$ \eta : X \longrightarrow \mathcal{Q}^{L} $| contracting an irreducible curve onto a point |$ p=\lbrace p_{1},p_{2},p_{3} \rbrace $| of degree |$3$| whose geometric components are contained in pairwise distinct rulings of |$ \mathcal{Q}^{L}_{L} $|⁠, and with splitting field |$ F $| such that |$ \operatorname{Gal}(F/\textbf{k}) \simeq \operatorname{Gal}(FL/L) \simeq \operatorname{Sym}_{3} $|⁠, where |$FL$| is a splitting field of |$p$| over |$L$|⁠.

• 2.

  Any two such del Pezzo surfaces are isomorphic if and only if the corresponding field extensions of degree two and six are |$\textbf{k}$|-isomorphic.

• 3.

  |$ \operatorname{Aut}_{\textbf{k}}(X)=\langle \widehat{\Phi _{q}} \rangle \simeq \mathbb{Z}/2\mathbb{Z} $|⁠, where |$\widehat{\Phi _{q}}$| is the lift of a quadratic birational involution of |$\mathbb{P}^{2}_{L}$| with base-point |$q$| of degree |$3$|⁠.

• 4.

  |$ \textrm{rk} \, \textrm{NS}(X)^{\operatorname{Aut}_{\textbf{k}}(X)} = 2 $|⁠, so in particular |$ X \rightarrow \ast $| is not an |$ \operatorname{Aut}_{\textbf{k}}(X) $|-Mori fibre space, and the |$\operatorname{Aut}_{\textbf{k}}(X)$|-minimal models of |$X$| are the quadric |$\mathcal{Q}^{L}$| and the rational del Pezzo surface |$Y$| of degree |$6$| such that |$Y_{L}$| is isomorphic to the blow-up of |$\mathbb{P}^{2}_{L}$| in a point |$q$| of degree |$3$| with splitting field |$E$| such that |$\operatorname{Gal}(E/L) \simeq \operatorname{Sym}_{3}$|⁠.

The proof of points (1), (2), and (4) is analogous to that of Proposition 3.4.1. (3) In this case, we have |$\operatorname{Aut}(\Pi _{\overline{\textbf{k}},\rho }) \simeq C_{\operatorname{Sym}_{5}}(\langle (123),(12),(45) \rangle ) = \langle (45) \rangle $|⁠, and from Proposition 3.4.1(3) there exists a birational quadratic involution |$\phi _{q} \in \textrm{Bir}_{L}(\mathbb{P}^{2})$| that lifts to an automorphism |$\widehat{\Phi _{q}}$| of |$X$| defined over |$\textbf{k}$| and that acts as |$(45)$| on the |$(-1)$|-curves, so |$\operatorname{Aut}_{\textbf{k}}(X) = \langle \widehat{\Phi _{q}} \rangle \simeq \mathbb{Z}/2\mathbb{Z}$|⁠.

Let |$X$| be a del Pezzo surface of degree |$5$| such that |$ \rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})) = \langle (123),(12)(45) \rangle \simeq \operatorname{Sym}_{3} $| in |$ \operatorname{Sym}_{5} $| as indicated in Figure 2n. Then the following holds:

• 1.

  There exist a quadratic extension |$ L/\textbf{k} $| and a birational morphism |$ \eta : X \longrightarrow \mathcal{Q}^{L} $| contracting an irreducible curve onto a point |$ p=\lbrace p_{1},p_{2},p_{3} \rbrace $| of degree |$3$| whose geometric components are contained in pairwise distinct rulings of |$ \mathcal{Q}^{L}_{L} $|⁠, and with splitting field |$F$| such that |$ \operatorname{Gal}(F/\textbf{k}) \simeq \operatorname{Sym}_{3} $| and |$ \operatorname{Gal}(FL/L) \simeq \mathbb{Z}/3\mathbb{Z} $|⁠, where |$FL$| is a splitting field of |$p$| over |$L$|⁠.

• 2.

  Any two such del Pezzo surfaces |$ X,X^{\prime} $| are isomorphic if and only if the corresponding field extensions |$ L,L^{\prime} $| and |$ F,F^{\prime} $| are |$\textbf{k}$|-isomorphic.

• 3.

  |$ \operatorname{Aut}_{\textbf{k}}(X)=\langle \widehat{\Phi _{q}} \rangle \simeq \mathbb{Z}/2\mathbb{Z} $|⁠, where |$\widehat{\Phi _{q}}$| is the lift of a quadratic birational involution of |$\mathbb{P}^{2}_{L}$| with base-point |$q$| of degree |$3$|⁠.

• 4.

  |$ \textrm{rk} \, \textrm{NS}(X)^{\operatorname{Aut}_{\textbf{k}}(X)} = 2 $|⁠, so in particular |$ X \rightarrow \ast $| is not an |$ \operatorname{Aut}_{\textbf{k}}(X) $|-Mori fibre space, and the |$\operatorname{Aut}_{\textbf{k}}(X)$|-minimal models of |$X$| are the quadric |$\mathcal{Q}^{L}$| and the rational del Pezzo surface |$Y$| of degree |$6$| such that |$Y_{L}$| is isomorphic to the blow-up of |$\mathbb{P}^{2}_{L}$| in a point |$q$| of degree |$3$| with splitting field |$E$| such that |$\operatorname{Gal}(E/L) \simeq \mathbb{Z}/3\mathbb{Z}$|⁠.

(1) There is only one |$(-1)$|-curve defined over |$\textbf{k}$| on |$X_{\overline{\textbf{k}}}$| namely |$ E_{4} $|⁠. Its contraction yields a birational morphism |$ \varepsilon : X \rightarrow Y $| onto a rational del Pezzo surface of degree |$6$| as in Figure 9, where the |$\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})$|-action on the hexagon of |$Y_{\overline{\textbf{k}}}$| is indicated by arrows. On the other hand, the contraction of the irreducible curve denoted |$D$| with geometric components |$D_{34},D_{24},D_{14}$| yields a birational morphism |$ \eta : X \rightarrow Z $| onto a rational del Pezzo surface of degree |$8$|⁠. It contracts |$D$| onto a point |$p = \lbrace p_{1},p_{2},p_{3} \rbrace $| of degree |$3$| and it conjugates the |$ \operatorname{Gal}(\overline{\textbf{k}}/\textbf{k}) $|-action on |$X_{\overline{\textbf{k}}}$| to an action that exchanges the fibrations of |$ Z_{\overline{\textbf{k}}} $| (see Figure 9). Thus, |$ Z \simeq \mathcal{Q}^{L} $| for some quadratic extension |$ L/\textbf{k} $| [21, Lemma 3.2(1)]. Then |$ \eta $| conjugates the |$ \operatorname{Gal}(\overline{\textbf{k}}/L) $|-action on |$ \mathcal{Q}^{L}_{\overline{\textbf{k}}} $| to an action on |$ \Pi _{\overline{\textbf{k}}} $| with |$ \rho (\operatorname{Gal}(\overline{\textbf{k}}/L)) = \mathbb{Z}/3\mathbb{Z} $|⁠. Let |$ F/\textbf{k} $| be any splitting field of |$ p \in \mathcal{Q}^{L} $|⁠. Let us show that |$\operatorname{Gal}(F/\textbf{k}) \simeq \operatorname{Sym}_{3}$|⁠. Let |$ M/\textbf{k} $| be a minimal normal extension containing |$F$| such that |$ \operatorname{Gal}(\overline{\textbf{k}}/M) $| acts trivially on |$ \Pi _{\overline{\textbf{k}}} $|⁠. The |$ \operatorname{Gal}(\overline{\textbf{k}}/\textbf{k}) $|-action on |$ \Pi _{\overline{\textbf{k}}} $| induces a short exact sequence |$ 1 \rightarrow \operatorname{Gal}(\overline{\textbf{k}}/M) \rightarrow \operatorname{Gal}(\overline{\textbf{k}}/\textbf{k}) \rightarrow \operatorname{Sym}_{3} \rightarrow 1 $|⁠, which implies that |$ \operatorname{Gal}(M/\textbf{k}) \simeq \operatorname{Gal}(\overline{\textbf{k}}/\textbf{k}) / \operatorname{Gal}(\overline{\textbf{k}}/M) \simeq \textrm{Sym}_{3} $|⁠. Then |$F$| is an intermediate field between |$M$| and |$\textbf{k}$|⁠, and since |$ F/\textbf{k} $| is normal, we get |$ \operatorname{Sym}_{3} \subseteq \operatorname{Gal}(F/\textbf{k}) \subseteq \operatorname{Gal}(M/\textbf{k}) \simeq \operatorname{Sym}_{3} $|⁠. This implies |$ \operatorname{Gal}(F/\textbf{k}) \simeq \operatorname{Sym}_{3} $|⁠. Then |$ FL/L $| is a splitting field of |$p$| over |$L$| as well, |$ FL/L $| is Galois and |$ \operatorname{Gal}(FL/L) \simeq \operatorname{Gal}(F/F \cap L) $| (see [17, Theorem 5.5]). Since |$ \rho (\operatorname{Gal}(\overline{\textbf{k}}/L)) = \langle (123)\rangle = \mathbb{Z}/3\mathbb{Z} $|⁠, the action of |$ \operatorname{Gal}(FL/L) $| on |$ p = \lbrace p_{1},p_{2},p_{3} \rbrace $| induces an exact sequence |$ 1 \rightarrow H \rightarrow \operatorname{Gal}(FL/L) \rightarrow \mathbb{Z}/3\mathbb{Z} \rightarrow 1 $|⁠, and |$H=\lbrace \operatorname{id} \rbrace $| using the same argument as in Proposition 3.3.7(1) (see [17, Corollary 2.10]). Then we get |$\operatorname{Gal}(FL/L) \simeq \mathbb{Z}/3\mathbb{Z}$|⁠.

(2) Two del Pezzo surfaces |$ \mathcal{Q}^{L} $| and |$ \mathcal{Q}^{L^{\prime}} $| are isomorphic if and only if the corresponding quadratic extensions |$L$| and |$L^{\prime}$| are |$\textbf{k}$|-isomorphic [21, Lemma 3.2(3)]. Moreover, |$ \operatorname{Aut}_{\textbf{k}}(\mathcal{Q}^{L}) $| acts transitively on the set of points of degree |$3$| with |$\textbf{k}$|-isomorphic splitting fields and whose geometric components are in general position in |$\mathcal{Q}^{L}_{\overline{\textbf{k}}}$| by Lemma 2.2.4. This yields the claim.

(3&4) By Lemma 3.1.1, the action of |$ \operatorname{Aut}_{\textbf{k}}(X) $| on the set of |$ (-1) $|-curves of |$X$| gives an isomorphism |$ \Psi : \operatorname{Aut}_{\textbf{k}}(X) \overset{\sim }{\longrightarrow } \operatorname{Aut}(\Pi _{\overline{\textbf{k}},\rho }) $|⁠. As before, we have |$\operatorname{Aut}(\Pi _{\overline{\textbf{k}},\rho }) \simeq C_{\operatorname{Sym}_{5}}(\langle (123),(12)(45) \rangle ) = \langle (45) \rangle $|⁠, and we now construct the corresponding geometric action. From part (1) we know that |$Y_{L}$| is isomorphic to the blow-up |$ \pi : Y_{L} \rightarrow \mathbb{P}^{2}_{L} $| of a point |$q=\lbrace q_{1},q_{2},q_{3} \rbrace $| of degree |$3$|⁠, so that |$X_{L}$| is isomorphic to the del Pezzo surface of degree |$5$| from Proposition 3.3.2. From the proof of Proposition 3.4.1(3) there exists |$ \Phi _{q} \in \operatorname{Aut}_{\textbf{k}}(Y) $| that induces a rotation of order two on the hexagon of |$ Y $| and that lifts by |$ \varepsilon $| to an automorphism |$ \widehat{\Phi _{q}} $| of |$X$| defined over |$ \textbf{k} $| that acts exactly as |$(45)$| on |$ \Pi _{\overline{\textbf{k}}} $|⁠. In conclusion, |$ \operatorname{Aut}_{\textbf{k}}(X)=\langle \widehat{\Phi _{q}} \rangle \simeq \mathbb{Z}/2\mathbb{Z} $|⁠. To finish, note that from (1) we have a contraction |$\eta : X \rightarrow \mathcal{Q}^{L}$| onto a point of degree |$3$|⁠, so thanks to Lemma 2.2.4, we get |$\textrm{rk} \, \textrm{NS}(X)^{\operatorname{Aut}_{\textbf{k}}(X)} = 2$|⁠. Now, note that the curve |$E_{4}$| is stabilized by |$\operatorname{Aut}_{\textbf{k}}(X)$|⁠. It can then be contracted equivariantly onto the rational del Pezzo surface |$Y$| of degree |$6$| in Figure 9 with |$\textrm{rk} \, \textrm{NS}(Y) = 1$|⁠, which gives the last assertion.

Let |$X$| be a del Pezzo surface of degree |$5$|⁠. Suppose that |$\operatorname{Aut}_{\textbf{k}}(X)=\langle \alpha \rangle \simeq \mathbb{Z}/5\mathbb{Z} $|⁠. Then over |$\overline{\textbf{k}}$|⁠, |$\alpha $| is the lift of a birational quadratic transformation of |$\mathbb{P}^{2}_{\overline{\textbf{k}}}$| of order |$5$|⁠, and up to conjugation, it is the lift of the order five Cremona transformation defined by |$ [x:y:z] {\scriptsize{\vdash}}\kern-5pt{\dashrightarrow} [x(z-y):z(x-y):xz] $|⁠.

Note that since |$\alpha $| is an automorphism of |$X_{\overline{\textbf{k}}}$| of order |$5$|⁠, it has two orbits of |$(-1)$|-curves of size five. Over |$\overline{\textbf{k}}$|⁠, one can contract the four disjoint |$(-1)$|-curves |$E_{1},E_{2},E_{3},E_{4}$| onto four points |$p_{1},p_{2},p_{3},p_{4}$| in general position in |$\mathbb{P}^{2}_{\overline{\textbf{k}}}$| and get a birational morphism |$ \pi : X_{\overline{\textbf{k}}} \longrightarrow \mathbb{P}^{2}_{\overline{\textbf{k}}} $|⁠, so that |$\phi := \pi \circ \alpha \circ \pi ^{-1}: \mathbb{P}^{2}_{\overline{\textbf{k}}} \dashrightarrow \mathbb{P}^{2}_{\overline{\textbf{k}}} $| is a birational transformation (see Figure 11).

Let us determine its degree |$d_{\phi }$|⁠. We let |$L \subseteq \mathbb{P}^{2}_{\overline{\textbf{k}}} $| be a general line, we denote |$ \phi ^{-1}(L) =: C $| its preimage by |$\phi $|⁠, and we compute the multiplicities |$ m_{p_{i}}(\phi ):= m_{p_{i}}(\phi ^{-1}(L)) = m_{p_{i}}(C) $| for |$ i=1,\dots ,4 $|⁠. If |$\alpha (E_{i}) = E_{j} $| for some |$j$|⁠, then |$ m_{p_{i}}(C) = \tilde{C} \cdot E_{i} = \alpha (\tilde{C}) \cdot \alpha (E_{i}) = \tilde{L} \cdot E_{j} = 0 $|⁠, and if |$ \alpha (E_{i}) \neq E_{j} $| for all |$j$|⁠, then |$\alpha (E_{i})=D_{jk} $| and |$ \tilde{C} \cdot E_{i} = \alpha (\tilde{C}) \cdot \alpha (E_{i}) = \tilde{L} \cdot D_{jk} = 1 $| because |$L$| is general. So |$ m_{p_{i}}(\phi ^{-1}(L)) \in \lbrace 0,1 \rbrace $|⁠. Applying now Noether’s relations (see for instance [14, Proposition 2.34]) to |$d_{\phi }$| and to the |$m_{p_{i}}(\phi )$|’s, we get |$d_{\phi } \in \lbrace 1,2 \rbrace $|⁠. If |$d_{\phi }=1$|⁠, then such an automorphism of |$\mathbb{P}^{2}_{\overline{\textbf{k}}}$| would permute the four points |$p_{1},\dots ,p_{4}$| via an element of |$\operatorname{Sym}_{4}$|⁠, which does not contain any element of order |$5$|⁠. It follows that |$\phi $| is a quadratic birational transformation of |$\mathbb{P}^{2}_{\overline{\textbf{k}}}$| of order |$5$| with |$3$| base-points among |$\lbrace p_{1},p_{2},p_{3},p_{4} \rbrace $| (see [14, Corollary 2.36]), and up to conjugation, there is only one such birational quadratic transformation which is defined by |$ \phi : [x:y:z] {\scriptsize{\vdash}}\kern-5pt{\dashrightarrow} [x(z-y):z(x-y):xz] $| ([1]; see also [8, Proposition 4.6.1] and [25, Example 4.19]). It has base-points |$p_{1}=[0:0:1], p_{2}=[1:0:0], p_{3}=[0:1:0]$| and it sends |$p_{4}=[1:1:1]$| onto |$p_{1}$|⁠.

Let |$X$| be a del Pezzo surface of degree |$5$| with |$\textrm{rk} \, \textrm{NS}(X) = 1$|⁠. Then the following holds:

• 1.

  If |$ \rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k}))=\langle (12345)\rangle \simeq \mathbb{Z}/5\mathbb{Z} $|⁠, then |$\operatorname{Aut}_{\textbf{k}}(X)=\langle \widehat{\phi } \rangle \simeq \mathbb{Z}/5Z$|⁠, where |$\widehat{\phi }$| is the lift of a quadratic birational map of |$\mathbb{P}^{2}_{\overline{\textbf{k}}}$| of order |$5$|⁠.

• 2.

  If |$\rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k}))=\langle (12345),(25)(34)\rangle \simeq \operatorname{D}_{5}$|⁠, then |$\operatorname{Aut}_{\textbf{k}}(X)=\lbrace \operatorname{id} \rbrace $|⁠.

• 3.

  If |$\rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k}))=\langle (12345),(2354)\rangle \simeq \operatorname{GA}(1,5)$|⁠, then |$\operatorname{Aut}_{\textbf{k}}(X) = \lbrace \operatorname{id} \rbrace $|⁠.

• 4.

  If |$\rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k}))=\langle (12345),(123)\rangle \simeq \mathcal{A}_{5}$|⁠, then |$\operatorname{Aut}_{\textbf{k}}(X) = \lbrace \operatorname{id} \rbrace $|⁠.

• 5.

  If |$\rho (\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k}))=\langle (12345),(12)\rangle \simeq \operatorname{Sym}_{5}$|⁠, then |$\operatorname{Aut}_{\textbf{k}}(X)=\lbrace \operatorname{id} \rbrace $|⁠.

(1) In this case the |$\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})$|-action on the |$(-1)$|-curves of |$X_{\overline{\textbf{k}}}$| is given by |$ E_{4} \mapsto E_{1} \mapsto D_{34} \mapsto D_{14} \mapsto D_{12} \mapsto E_{4} $| and |$ D_{13} \mapsto E_{3} \mapsto D_{23} \mapsto E_{2} \mapsto D_{24} \mapsto D_{13} $|⁠, which gives the two orbits of size five on |$\Pi _{\overline{\textbf{k}}}$| represented in Figure 2o. The action of |$\operatorname{Aut}_{\textbf{k}}(X)$| on the set of |$(-1)$| curves of |$X$| yields an isomorphism |$ \Psi : \operatorname{Aut}_{\textbf{k}}(X) \overset{\sim }{\longrightarrow } \operatorname{Aut}(\Pi _{\overline{\textbf{k}},\rho }) $| by Lemma 3.1.1, where |$\operatorname{Aut}(\Pi _{\overline{\textbf{k}},\rho }) \simeq \lbrace \sigma \in \operatorname{Sym}_{5} \, \vert \, (12345) \circ \sigma = \sigma \circ (12345) \rbrace = \langle (12345) \rangle $|⁠. Now consider over |$\overline{\textbf{k}}$| the order five birational quadratic map |$ \phi : \mathbb{P}^{2}_{\overline{\textbf{k}}} \dashrightarrow \mathbb{P}^{2}_{\overline{\textbf{k}}}, \, [x:y:z] {\scriptsize{\vdash}}\kern-5pt{\dashrightarrow} [x(z-y):z(x-y):xz] $|⁠. Up to change of coordinates in |$\mathbb{P}^{2}_{\overline{\textbf{k}}}$| we can choose the four points |$p_{1}=[0:0:1], p_{2}=[1:0:0], p_{3}=[0:1:0]$| and |$p_{4}=[1:1:1]$|⁠. Then |$\phi $| has base-points |$p_{1}, p_{2}, p_{3}$|⁠, it sends |$p_{4}$| onto |$p_{1}$|⁠, it sends the lines |$L_{34}$| onto |$L_{14}$|⁠, |$L_{14}$| onto |$L_{12}$|⁠, |$L_{24}$| onto |$L_{13}$|⁠, and it contracts the lines |$L_{12}$| onto |$p_{4}$|⁠, |$L_{23}$| onto |$p_{2}$|⁠, |$L_{13}$| onto |$p_{3}$|⁠. Its birational inverse |$ \phi ^{-1}: [x:y:z] {\scriptsize{\vdash}}\kern-5pt{\dashrightarrow} [z(z-x):(z-y)(z-x):z(z-y)] $| has base-points |$p_{2}, p_{3}, p_{4}$|⁠, it sends |$p_{1}$| onto |$p_{4}$|⁠, and it contracts the lines |$L_{34}$| onto |$p_{1}$|⁠, |$L_{23}$| onto |$p_{3}$|⁠, |$L_{24}$| onto |$p_{2}$|⁠. Thus, by blowing up the points |$p_{1},p_{2},p_{3},p_{4}$|⁠, the map |$\phi $| lifts to an automorphism |$\widehat{\phi }$| of |$X_{\overline{\textbf{k}}}$| that acts exactly as |$(12345)$| on |$\Pi _{\overline{\textbf{k}}}$|⁠. If |$g \in \operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})$|⁠, then |$(g \widehat{\phi } g^{-1}) \widehat{\phi }^{-1} \in \operatorname{Aut}(X_{\overline{\textbf{k}}})$| preserves the curves |$E_{1},E_{2},E_{3},E_{4}$|⁠. It therefore descends to an element of |$\operatorname{Aut}(\mathbb{P}^{2}_{\overline{\textbf{k}}})$| fixing four points in general position and is hence equal to the identity. It follows that |$\widehat{\phi }$| is defined over |$\textbf{k}$| and so |$\operatorname{Aut}_{\textbf{k}}(X) = \langle \widehat{\phi } \rangle \simeq \mathbb{Z}/5\mathbb{Z}$|⁠.

(2) In this case the action of the Galois group |$\operatorname{Gal}(\overline{\textbf{k}}/\textbf{k})$| on the set of |$(-1)$| curves of |$X_{\overline{\textbf{k}}}$| is given by |$ E_{4} \mapsto E_{1} \mapsto D_{34} \mapsto D_{14} \mapsto D_{12} \mapsto E_{4} $|⁠, |$ D_{13} \mapsto E_{3} \mapsto D_{23} \mapsto E_{2} \mapsto D_{24} \mapsto D_{13} $| and |$ E_{1} \leftrightarrow D_{34}, D_{23} \leftrightarrow D_{24}, E_{3} \leftrightarrow D_{13}, E_{4} \leftrightarrow D_{14}, D_{12}^{\circlearrowleft }, E_{2}^{\circlearrowleft } $|⁠, which still gives two orbits of |$(-1)$|-curves of size five as indicated in Figure 2p. The action of |$\operatorname{Aut}_{\textbf{k}}(X)$| on the set of |$(-1)$|-curves of |$X$| yields an isomorphism |$\Psi : \operatorname{Aut}_{\textbf{k}}(X) \overset{\sim }{\longrightarrow } \operatorname{Aut}(\Pi _{\overline{\textbf{k}},\rho })$| by Lemma 3.1.1, where |$\operatorname{Aut}(\Pi _{\overline{\textbf{k}},\rho })=\lbrace \operatorname{id} \rbrace $| since there is no non-trivial element of |$\operatorname{Sym}_{5}$| that commutes with both |$(12345)$| and |$(25)(34)$|⁠.

The points (3), (4), and (5) are proven analogously to point (2).

1 comes from the description given in §3.5, as well as Lemma 3.5.1 and Proposition 3.5.2. 2 follows from Proposition 3.3.1. Cases 3a, 3b, 3c, 3d, 3j, 3k, and 3l are a consequence of Propositions 3.2.1, 3.2.3, 3.3.2, 3.3.4, 3.4.1, 3.4.3, and 3.4.5, respectively, and the cases {3e, 3f, 3g, 3h, 3i} come from Proposition 3.3.7. Finally, Case 4 comes from Proposition 3.2.5.