Proof. Arguing as in the proof of [Reference Birkar7, Proposition 3.4], we see that

(3.1) $$ \begin{align} \alpha(S)=\mathrm{lct}(S,D) \end{align} $$

for some effective $\mathbb {Q}$ -divisor $D\sim _{\mathbb {Q}} -K_S$ on the surface S. One can deduce (3.1) without using the proof of [Reference Birkar7, Proposition 3.4], since it follows from the Kollár–Shokurov connectedness theorem [Reference Kollár and Mori50, Corollary 5.49] that

$$ \begin{align*}\alpha(S)=\mathrm{inf}\Big\{\frac{1}{\tau(C)}\ \big\vert\ C\ \text{is an irreducible curve in}\ S\Big\}, \end{align*} $$

where $\tau (C)=\mathrm {sup} \big \{u\in \mathbb {R}_{\geqslant 0}\ \big \vert \ \text {the divisor}\ -K_S-uC\ \text {is pseudo-effective}\big \}$ .

The log pair $(S,\alpha (S)D)$ has log canonical singularities, but it is not klt (Kawamata log terminal), hence the locus $\mathrm {Nklt}(S,\alpha (S)D)$ is not empty and it is connected by Kollár–Shokurov connectedness. Since $S(\Bbbk )=\varnothing $ , we conclude that $\mathrm {Nklt}(S,\alpha (S)D)$ is a connected union of irreducible curves and, in particular, the surface S contains a ${\Bbbk }$ -irreducible curve C with $ D=\tau C+\Delta $ , where $\tau =\frac {1}{\alpha (S)}$ and $\Delta $ is an effective $\mathbb {Q}$ -divisor whose support does not contain C. This gives

$$ \begin{align*}\tau=\mathrm{sup}\big\{u\in\mathbb{R}_{\geqslant 0}\ \big\vert\ \text{the divisor}\ -K_S-uC\ \text{is pseudo-effective}\big\}. \end{align*} $$

If C is not a minimal log canonical center of the log pair $(S,\alpha (S)D)=(S,C+\alpha (S)\Delta )$ , then using the Kawamata–Shokurov trick [Reference Cheltsov and Shramov20, Lemma 2.4.10], also known as tie breaking, we can replace the divisor D by an effective $\mathbb {Q}$ -divisor

$$ \begin{align*}D^\prime\sim_{\mathbb{Q}}(1+\epsilon)\big(-K_S\big) \end{align*} $$

for some sufficiently small $\epsilon \in \mathbb {Q}_{>0}$ such that the log pair $(S,\alpha (S)D^\prime )$ has log canonical singularities but $\mathrm {Nklt}(S,\alpha (S)D^\prime )$ consists of finitely many points. Now, using Kollár–Shokurov connectedness, we obtain a contradiction with the assumption that $S(\Bbbk )=\varnothing $ . Hence, the curve C is a minimal log canonical center of the log pair $(S,C+\alpha (S)\Delta )$ , which implies that C is smooth [Reference Kawamata45].

Now, using properties of log canonical centers [Reference Kawamata44, Reference Kawamata45], we conclude that $\mathrm {Nklt}(S,\alpha (S)D)=C$ , which implies that the curve C is connected and the log pair $(S,\alpha (S)D)$ is purely log terminal. Hence, the curve C is geometrically irreducible. Finally, using Kawamata’s subadjunction theorem, we see that the curve C is geometrically rational.

Proof. Suppose that $\alpha (S)<\frac {1}{2}$ . By Lemma 3.1, there is a geometrically irreducible curve $C\subset S$ such that

$$ \begin{align*}-K_S\sim_{\mathbb{Q}}\frac{1}{\alpha(S)}C+\Delta, \end{align*} $$

where $\Delta $ is an effective $\mathbb {Q}$ -divisor on S. Without loss of generality, we may assume that $C\cdot C_1>0$ .

Observe that $|nC_1|$ is base point free for $n\gg 0$ . Moreover, replacing $\Bbbk $ by its algebraic closure, we may assume that $|nC_1|$ gives a morphism $\pi \colon S\to \mathbb {P}^1$ such that its general fiber $F\simeq \mathbb {P}^1$ . Then

$$ \begin{align*}2=-K_S\cdot F=\frac{1}{\alpha(S)} C\cdot F+\Delta\cdot F\geqslant \frac{1}{\alpha(S)} C\cdot F\geqslant \frac{1}{\alpha(S)}>2, \end{align*} $$

which is absurd.

Proof. All required assertions follow from [Reference Fujita32, Lemma 9.7].

Proof. If $\widetilde {C}^2<0$ , then it follows from the adjunction formula that $\widetilde {C}^2=-1$ , and $\widetilde {C}$ is a $\Bbbk $ -form of $\mathbb {P}^1$ . Since $-\widetilde {C}\vert _{\widetilde {C}}$ is a line bundle of degree $1$ , we have $\widetilde {C}\cong \mathbb {P}^1$ which implies that $\widetilde {S}(\Bbbk )\ne \varnothing $ .

Proof. Suppose that $\tau \leqslant 2$ . Then $C^2=0$ and $C\cap \mathrm {Sing}(S)=\varnothing $ by Lemma 3.3 and Corollary 3.5. Moreover, it follows from Lemma 3.3 that $\tau =2$ and $ -K_S\sim _{\mathbb {Q}}2C+aZ $ , for some positive rational number a and an irreducible curve $Z\subset S$ with $Z^2=0$ . By the Riemann–Roch formula, the linear system $|C|$ is a pencil that gives a conic bundle $S\to \mathbb {P}^1$ . Since S is a Mori Dream Space, the linear system $|nZ|$ is base point free for some positive integer n, and it also gives a conic bundle $S\to C_2$ where $C_2$ is a conic defined over $\Bbbk $ . If $C_2(\Bbbk )\ne \varnothing $ , then $C_2(\Bbbk )\cong \mathbb {P}^1$ , and we can replace Z by a general fiber of the conic bundle $S\to C_2$ . Similarly, if $C_2(\Bbbk )=\varnothing $ , we may assume that Z is an irreducible geometrically reducible curve that is a preimage of a general irreducible zero-dimensional subscheme of the conic $C_2$ of length $2$ . In both cases, we have $Z\cap \mathrm {Sing}(S)=\varnothing $ . Using the adjunction formula, we get

$$ \begin{align*}2C\cdot Z=\big(2C+aZ\big)\cdot Z=-K_S\cdot Z= \left\{\begin{array}{l} 2\ \text{if } C_2(\Bbbk)\ne\varnothing, \\ 4\ \text{if } C_2(\Bbbk)=\varnothing. \end{array} \right. \end{align*} $$

But $C\cdot Z\ne 1$ , because $S(\Bbbk )=\varnothing $ . Thus, we see that $C_2$ is a pointless conic and $C\cdot Z=2$ . Now, taking the product of the morphisms $S\to \mathbb {P}^1$ and $S\to C_2$ , we obtain the isomorphism $S\to \mathbb {P}^1\times C_2$ .

Proof. Set $\tau =\frac {1}{\alpha (S)}$ and suppose that $S(\mathbb {R})=\varnothing $ . Then it follows from Lemma 3.1 that S contains a geometrically irreducible curve C with $ -K_{S}\sim _{\mathbb {Q}}\tau C+\Delta $ for some effective $\mathbb {Q}$ -divisor $\Delta $ on S.

Let $f\colon \widetilde {S}\to S$ be the minimal resolution of the del Pezzo surface S and denote by $\widetilde {C}$ and $\widetilde {\Delta }$ , respectively, the strict transforms on $\widetilde {S}$ of the curve C and the divisor $\Delta $ . Then $\widetilde {S}(\mathbb {R})=\varnothing $ and $ -K_{\widetilde {S}}\sim _{\mathbb {Q}}\tau \widetilde {C}+\widetilde {\Delta }+\widetilde {B} $ , where $\widetilde {B}$ is an effective $\mathbb {Q}$ -divisor on the surface $\widetilde {S}$ whose support consists of f-exceptional curves. Now, applying the Minimal Model Program to $\tilde {S}$ over $\mathbb {R}$ , we obtain a birational morphism $h\colon \widetilde {S}\to \overline {S}$ defined over $\mathbb {R}$ such that one of the following two cases holds:

1. (1) $\overline {S}$ is a real smooth del Pezzo surface of Picard rank $1$ ;

2. (2) $\overline {S}$ is a real smooth surface of Picard rank $2$ , and there is a (standard) conic bundle $\pi \colon \overline {S}\to C_2$ defined over $\mathbb {R}$ , where $C_2$ is a geometrically irreducible real conic in $\mathbb {P}^2$ .

In both cases, we have $\overline {S}(\mathbb {R})=\varnothing $ by Lemma 2.1. Note that $\widetilde {C}$ is not h-exceptional, because $\widetilde {C}(\mathbb {R})=\varnothing $ . Set $\overline {C}=h(\widetilde {C})$ and let $\overline {\Delta }$ and $\overline {B}$ be the strict transforms on $\overline {S}$ of the divisors $\widetilde {\Delta }$ and $\widetilde {B}$ , respectively. Then

(3.2) $$ \begin{align} -K_{\overline{S}}\sim_{\mathbb{Q}}\tau\overline{C}+\overline{\Delta}+\overline{B}. \end{align} $$

Hence, if $\overline {S}$ is a smooth del Pezzo surface of Picard rank $1$ , then it follows from (3.2) and $\tau>2$ that $\overline {S}$ is a Severi–Brauer surface and $\overline {C}$ is a twisted line on it [Reference Kollár48], which implies that $\overline {S}\cong \mathbb {P}^2$ , which is a contradiction since $\overline {S}(\mathbb {R})=\varnothing $ .

Thus, there is a conic bundle $\pi \colon \overline {S}\to C_2$ . Now, using (3.2) and intersecting $\tau \overline {C}+\overline {\Delta }+\overline {B}$ with a general fiber of the conic bundle $\pi $ , we see that $\overline {C}$ is a fiber of $\pi $ , because $\tau>2$ . Then $\pi (\overline {C})$ is a real point in $C_2$ , so that $C_2\simeq \mathbb {P}^1$ . Now, using $\rho (\overline {S})=2$ and $\overline {S}(\mathbb {R})=\varnothing $ , we see that $\overline {S}$ is a form of $\mathbb {F}_n$ for some $n\in \mathbb {Z}_{\geqslant 0}$ , see [Reference Kollár49, Reference Mangolte58]. Then (3.2) and $\tau>2$ give $n\ne 0$ , so the surface $\overline {S}$ contains the unique geometrically irreducible curve $\overline {Z}$ with $\overline {Z}^2=-n$ . Since $\overline {Z}\cdot \overline {C}=1$ , we see that $\overline {Z}\cap \overline {C}$ consists of a single point in $\overline {S}(\mathbb {R})$ , which is a contradiction.

Proof. Suppose that $S_{\mathbb {C}}$ is not K-polystable. Then it follows from [Reference Fujita33, Reference Li54, Reference Zhuang81] that S contains a geometrically irreducible curve C with $\beta (C)\leqslant 0$ . Moreover, using Corollary 3.6, we see that $-K_{S}\sim _{\mathbb {Q}}\tau C+\Delta $ for some rational number $\tau>2$ and an effective $\mathbb {Q}$ -divisor $\Delta $ on the surface S. This gives $\alpha (S)\leqslant \frac {1}{\tau }<\frac {1}{2}$ , which contradicts Lemma 3.8.

Proof. Set $\tau =\frac {1}{\alpha (S)}$ and suppose that $S(\Bbbk )=\varnothing $ . By Lemma 3.1, there exists a geometrically irreducible smooth and geometrically rational curve $C\subset S$ with $ -K_{S}\sim _{\mathbb {Q}}\tau C, $ and the log pair $(S,C)$ has purely log terminal singularities. Let us seek a contradiction.

Let $f\colon \widetilde {S}\to S$ be the minimal resolution of S and let $\widetilde {C}$ be the strict transform on $\widetilde {S}$ of the curve C. Then $\widetilde {S}(\Bbbk )=\varnothing $ and $ -K_{\widetilde {S}}\sim _{\mathbb {Q}}\tau \widetilde {C}+\widetilde {B} $ , where $\widetilde {B}$ is an effective $\mathbb {Q}$ -divisor on the surface $\widetilde {S}$ whose support consists of f-exceptional curves. Now, applying the Minimal Model Program to $\widetilde {S}$ over $\Bbbk $ , we obtain a birational morphism $h\colon \widetilde {S}\to \overline {S}$ defined over $\Bbbk $ such that one of the following two cases holds:

• • $\overline {S}$ is a smooth del Pezzo surface defined over $\Bbbk $ of Picard rank $1$ ;

• • $\overline {S}$ is a smooth surface defined over $\Bbbk $ of Picard rank $2$ and there is a (standard) conic bundle $\pi \colon \overline {S}\to C_2$ defined over $\Bbbk $ , where $C_2$ is a geometrically irreducible conic in $\mathbb {P}^2$ defined over $\Bbbk $ .

Moreover, arguing as in the proof of Lemma 3.8, we see that the former case is impossible, which implies that $\overline {S}$ is a smooth surface of Picard rank $2$ and there exists a conic bundle $\pi \colon \overline {S}\to C_2$ . Note that $\overline {S}(\Bbbk )=\varnothing $ by Lemma 2.1.

Set $\overline {C}=h(\widetilde {C})$ , and let $\overline {B}$ be the strict transform of the divisor $\widetilde {B}$ on the surface $\overline {S}$ . Then

(3.3) $$ \begin{align} -K_{\overline{S}}\sim_{\mathbb{Q}}\tau\overline{C}+\overline{B}. \end{align} $$

Arguing as in the proof of Lemma 3.8, we see that $C_2\simeq \mathbb {P}^1$ and $\overline {C}$ is a fiber of the conic bundle $\pi $ . Then $\widetilde {C}^2\leqslant \overline {C}^2=0$ , which implies that $\widetilde {C}^2=0$ by Lemma 3.4, because we know that $\widetilde {S}(\Bbbk )=\varnothing $ . Hence, we see that h is an isomorphism in a neighborhood of the curve $\widetilde {C}$ , and the complete linear system $|\widetilde {C}|$ gives the composition morphism $\pi \circ h\colon \widetilde {S}\to C_2$ .

Let $S_{\mathbb {C}}$ and $\widetilde {S}_{\mathbb {C}}$ be the models of the surfaces S and $\widetilde {S}$ over the algebraic closure $\mathbb {C}$ of the field $\Bbbk $ , let $C_{\mathbb {C}}$ and $\widetilde {C}_{\mathbb {C}}$ be the curves in $S_{\mathbb {C}}$ and $\widetilde {S}_{\mathbb {C}}$ that correspond to C and $\widetilde {C}$ , respectively. Then $C_{\mathbb {C}}\simeq \widetilde {C}_{\mathbb {C}}\simeq \mathbb {P}^1$ . Note that $\widetilde {C}_{\mathbb {C}}^2=0<C_{\mathbb {C}}^2$ , which implies that $C_{\mathbb {C}}\cap \mathrm {Sing}\big (S_{\mathbb {C}}\big )\ne \varnothing $ . Since $(S_{\mathbb {C}},C_{\mathbb {C}})$ has purely log terminal singularities, it follows from [Reference Kawamata45, Reference Prokhorov69] that $C_{\mathbb {C}}$ contains at most three singular points of the surface $S_{\mathbb {C}}$ , and all these singular points are cyclic quotient singularities. Thus, since $C(\Bbbk )=\varnothing $ and $C_{\mathbb {C}}\simeq \mathbb {P}^1$ , the curve $C_{\mathbb {C}}$ contains two singular points of the surface $S_{\mathbb {C}}$ , which are swapped by the action of $\mathrm {Gal}(\mathbb {C}/\Bbbk )$ .

Let $P_1$ and $P_2$ be the singular points of the surface $S_{\mathbb {C}}$ contained in $C_{\mathbb {C}}$ . Then the exceptional curves of the minimal resolution $\widetilde {S}_{\mathbb {C}}\to S_{\mathbb {C}}$ that are mapped to the singular points $P_1$ and $P_2$ form two disjoint Hirzebruch–Jung strings [Reference Hirzebruch37], that is, exceptional loci of minimal resolutions of quotient singularities, which are swapped by the action of the group $\mathrm {Gal}(\mathbb {C}/\Bbbk )$ . Since $(S_{\mathbb {C}},C_{\mathbb {C}})$ is purely log terminal, the curve $\widetilde {C}_{\mathbb {C}}$ intersects the first (or the last) curves of these strings, which we denote by $E_1$ and $E_2$ , respectively. Set $E=E_1+E_2$ . Then E is defined over $\Bbbk $ , so we consider it as a curve in $\widetilde {S}$ . Then E is the only f-exceptional curve that intersects the curve $\widetilde {C}$ . Hence, there exists the following commutative diagram:

where g is the contraction of all f-exceptional curves except for the curve E, and q is a partial resolution of singularities of the surface S that contracts the strict transform of the curve E.

Let $\widehat {E}=g(E)$ and $\widehat {C}=g(\widetilde {C})$ , Then $-K_{\widehat {S}}\sim _{\mathbb {Q}}\tau \widehat {C}+a\widehat {E}$ for some $a\in \mathbb {Q}_{>0}$ . On the other hand, we have $\widehat {C}^2=0$ and $\widehat {C}\cdot \widehat {E}=2$ , because g is an isomorphism in a neighborhood of the curve $\widetilde {C}$ . Furthermore, we have $-K_{\widehat {S}}\cdot \widehat {C}=2$ by the adjunction formula. This gives $a=1$ , because

$$ \begin{align*}2=-K_{\widehat{S}}\cdot \widehat{C}=\big(\tau\widehat{C}+a\widehat{E}\big)\cdot \widehat{C}=a\widehat{E}\cdot \widehat{C}=2a. \end{align*} $$

Hence, since $\widehat {E}$ is smooth, the subadjunction formula applied to $\widehat {E}$ gives

$$ \begin{align*}-4=\mathrm{deg}\big(K_{\widehat{E}}\big)\leqslant\big(K_{\widehat{S}}+\widehat{E}\big)\cdot \widehat{E}=-\tau\widehat{C}\cdot\widehat{E}=-2\tau<-4, \end{align*} $$

which is a contradiction.

Proof. Suppose that $S_{\mathbb {C}}$ is not K-polystable. Then it follows from [Reference Li54, Reference Fujita33, Reference Zhuang81] that S contains a geometrically irreducible curve C such that $\beta (C)\leqslant 0$ . Then $-K_{S}\sim _{\mathbb {Q}}\tau C$ for some rational number $\tau \geqslant 3$ , which implies that $\alpha (S)\leqslant \frac {1}{\tau }\leqslant \frac {1}{3}<\frac {1}{2}$ , but this contradicts Lemma 3.10.

Proof. Let us prove the assertion by induction on $\rho (S)$ . The case $\rho (S)=1$ is done by Lemma 3.10. Suppose that $\rho (S)\geqslant 2$ , and the assertion holds for del Pezzo surfaces with smaller Picard rank. We have to show that $S(\Bbbk )\ne \varnothing $ . Suppose that $S(\Bbbk )=\varnothing $ . Let us seek a contradiction.

If there exists a nonbiregular birational morphism $S\to S^\prime $ such that $S^\prime $ is a normal surface, then $S^\prime $ is a del Pezzo surface with quotient singularities, and it follows from Corollary 3.7 that $\alpha (S^\prime )\leqslant \alpha (S)<\frac {1}{2}$ and $S^\prime (\Bbbk )=\varnothing $ , which contradicts the induction hypotheses. Hence, we see that S does not admit any nonbiregular birational morphism to a normal surface. This is only possible when $\rho (S)=2$ , and $\mathrm {NE}(S)$ is generated by irreducible curves $C_1$ and $C_2$ such that $C_1^2=C_2^2=0$ and $C_1\cdot C_2>0$ . But in this case we have $\alpha (S)\geqslant \frac {1}{2}$ by Corollary 3.2.

Proof. By Mori–Mukai [Reference Mori and Mukai61], we have the following commutative diagram

where f is the blowup of a smooth curve C of degree $7$ and genus $5$ , $\pi $ is a standard conic bundle with discriminant curve $\Delta \subset \mathbb {P}^2$ of degree $5$ , and the dashed arrow is given by the two-dimensional linear system of all cubic surfaces that contain the curve C. Moreover, it follows from Lemma 2.5 that this diagram can be defined over $\Bbbk $ with $X_{\mathbb {C}}$ replaced by X, $\mathbb {P}^3$ replaced by a $\Bbbk $ -form U of $\mathbb {P}^3$ and $\mathbb {P}^2$ replaced by a $\Bbbk $ -form V of $\mathbb {P}^2$ . Applying Lemma 2.2 to $\Delta $ and V, we conclude that $V\simeq \mathbb {P}^2$ . Let L be a line in V and let $S=f_\ast \big {(} \pi ^\ast (L) \big {)}$ . Then $S_{\mathbb {C}}$ is a cubic surface, so applying Lemma 2.2 to S and U, we conclude that $U\simeq \mathbb {P}^3$ . In particular, we see that $X({\Bbbk })\ne \varnothing $ .

Proof. Arguing as in the proof of Lemma 4.1, we see that there exists the following commutative diagram

where Y is a form of a smooth cubic hypersurface $Y_{\mathbb {C}}\subset \mathbb {P}^4$ , f is the blowup of a curve $C\subset Y$ such that $C_{\mathbb {C}}$ is a line in the cubic hypersurface $Y_{\mathbb {C}}$ , $\pi $ is a conic bundle. Then Y is a cubic hypersurface in $\mathbb {P}^4$ by Corollary 2.3, so C is a line in it, which gives $C(\Bbbk )\ne \varnothing $ . In particular, $Y(\Bbbk )\ne \varnothing $ , and Lemma 2.1 says that $X({\Bbbk })\ne \varnothing $ as well.

Proof. By Mori–Mukai [Reference Mori and Mukai61], $X_{\mathbb {C}}$ can be obtained by blowing up the smooth quintic del Pezzo 3-fold $V_5$ along an elliptic curve. Thus, it follows from Lemma 2.5 that X can be obtained by blowing up a $\Bbbk $ -form of $V_5$ . By [Reference Kuznetsov and Prokhorov51, Theorem 1.1], any $\Bbbk $ -form of $V_5$ is $\Bbbk $ -rational, hence so is X, in particular $X(\Bbbk )\ne \varnothing $ .

Proof. It follows from Mori–Mukai [Reference Mori and Mukai61] and Lemma 2.5 that there exists a birational morphism $\pi \colon X\to Q$ such that Q is a $\Bbbk $ -form of a smooth quadric 3-fold in $\mathbb {P}^4$ , and $\pi $ is the blowup of a smooth elliptic curve C such that $-K_Q\cdot C=15$ . Then Q is a quadric in $\mathbb {P}^4$ by Corollary 2.3, so C is a curve of degree $5$ in it. Now, taking a hyperplane section of C, we obtain a zero-cycle in Q of degree $5$ defined over $\Bbbk $ , which implies that Q has a $\Bbbk $ -point, so X also has a $\Bbbk $ -point by Lemma 2.1.

Proof. By Mori–Mukai [Reference Mori and Mukai61], $X_{\mathbb {C}}$ can be obtained by blowing up a smooth quintic del Pezzo 3-fold $V_5$ along a twisted cubic curve. Now, arguing as in the proof of Lemma 4.3, we conclude that X is rational over $\Bbbk $ and, in particular, it has a $\Bbbk $ -point.

Proof. By Mori–Mukai [Reference Mori and Mukai61], $X_{\mathbb {C}}$ can be obtained by blowing up a smooth quintic del Pezzo 3-fold $V_5$ along a conic. Now, arguing as in the proof of Lemma 4.3, we conclude that X is rational over $\Bbbk $ and, in particular, it has a $\Bbbk $ -point.

Proof. Note that $X_{\mathbb {C}}\subset \mathbb {F}_1\times \mathbb {P}^2$ is a divisor in the linear system $|(\varsigma \circ \mathrm {pr}_1)^*(\mathcal {O}_{\mathbb {P}^2}(1))\otimes \mathrm {pr}_2^*(\mathcal {O}_{\mathbb {P}^2}(2))|$ , where $\mathrm {pr}_1\colon \mathbb {F}_1\times \mathbb {P}^2\to \mathbb {F}_1$ and $\mathrm {pr}_2\colon \mathbb {F}_1\times \mathbb {P}^2\to \mathbb {P}^2$ are projections to the first and the second factors, respectively, and $\varsigma \colon \mathbb {F}_1\to \mathbb {P}^2$ is the blowup of a point. Combining $\varsigma \circ \mathrm {pr}_1$ and $\mathrm {pr}_2$ , we obtain a morphism $\sigma \colon X_{\mathbb {C}}\to Y$ such that Y is a smooth divisor of degree $(1,2)$ in $\mathbb {P}^2\times \mathbb {P}^2$ . Let $\pi _1\colon Y\to \mathbb {P}^2$ and $\pi _2\colon Y\to \mathbb {P}^2$ be projections to the first and the second factors, respectively. Then $\sigma $ is a blowup of a smooth curve $\mathcal {C}$ that is a fiber of the morphism $\pi _1$ . Let $p=\pi _1(\mathcal {C})$ . Then $\varsigma $ is a blowup of the point p with commutative diagram

where $\vartheta $ is a natural projection, $\theta $ is a fibration into del Pezzo surfaces of degree $5$ . Moreover, combining morphisms $\theta $ and $\mathrm {pr}_2\vert _{X_{\mathbb {C}}}$ , we obtain a birational morphism $\upsilon \colon X_{\mathbb {C}}\to \mathbb {P}^1\times \mathbb {P}^2$ that is a blowup of a smooth curve of degree $(4,2)$ . This shows that the Mori cone $\mathrm {NE}(X_{\mathbb {C}})$ is simplicial and is generated by the following extremal rays:

1. 1. the ray generated by the curves contracted by $\sigma \colon X_{\mathbb {C}}\to Y$ ,

2. 2. the ray generated by the curves contracted by $\upsilon \colon X_{\mathbb {C}}\to \mathbb {P}^1\times \mathbb {P}^2$ ,

3. 3. the ray generated by the curves contracted by $\mathrm {pr}_1\vert _{X_{\mathbb {C}}}\colon X_{\mathbb {C}}\to \mathbb {F}_1$ .

Now, arguing as in the proof of Lemma 2.5, we see that the conic bundle $\mathrm {pr}_1\vert _{X_{\mathbb {C}}}\colon X_{\mathbb {C}}\to \mathbb {F}_1$ descends to a conic bundle $X\to \mathbb {F}_1$ defined over $\Bbbk $ , because $\mathbb {F}_1$ does not have nontrivial forms over $\Bbbk $ (indeed, a form of $\mathbb {F}_1$ can be obtained by blowing up a $\Bbbk $ -form of $\mathbb {P}^2$ , so the image of the exceptional curve yields a ${\Bbbk }$ -rational point of this Severi–Brauer surface, and, therefore, it must be isomorphic to $\mathbb {P}^2$ ). Now, composing this conic bundle with the projection $\vartheta \colon \mathbb {F}_1\to \mathbb {P}^1$ , we see that the del Pezzo fibration $\vartheta \colon X_{\mathbb {C}}\to \mathbb {P}^1$ is also defined over $\Bbbk $ . Since del Pezzo surfaces of degree $5$ are rational over any field, we see that X is $\Bbbk $ -birational to $\mathbb {P}^2\times \mathbb {P}^1$ , so that it is $\Bbbk $ -rational and, in particular, it has a $\Bbbk $ -point.

Proof. Over $\mathbb {C}$ we have a commutative diagram

where $\vartheta $ is the blowup of a point $p\in \mathbb {P}^3$ , $\pi $ is the blowup of the strict transform of a smooth quartic elliptic curve C that passes through the point p, $\zeta $ is a birational contraction of the strict transform of the cubic cone in $\mathbb {P}^3$ with vertex at p that contains the elliptic curve C to a smooth curve in $\mathbb {P}^1\times \mathbb {P}^2$ of degree $(2,3)$ , $\varpi $ is the blowup of the curve C, $\theta $ is the blowup of the fiber of $\varpi $ over the point p, $\eta $ is a $\mathbb {P}^1$ -bundle, $\nu $ is a fibration into quadric surfaces, $\sigma $ is a conic bundle, the left dashed arrow is given by the pencil of quadric surfaces that contain C, the right dashed arrow is the linear projection from the point p, and $\mathrm {pr}_1$ and $\mathrm {pr}_2$ are projections to the first and the second factors, respectively. This shows that the Mori cone $\mathrm {NE}(X_{\mathbb {C}})$ is simplicial and is generated by the following extremal rays:

1. 1. the ray generated by the curves contracted by $\theta \colon X_{\mathbb {C}}\to Y$ ,

2. 2. the ray generated by the curves contracted by $\pi \colon X_{\mathbb {C}}\to V_7$ ,

3. 3. the ray generated by the curves contracted by $\zeta \colon X_{\mathbb {C}}\to \mathbb {P}^1\times \mathbb {P}^2$ .

Now, arguing as in the proof of Lemma 2.5, we see that these extremal rays are defined over $\Bbbk $ , so their contractions can also be defined over $\Bbbk $ . Since $V_7$ does not have nontrivial forms over $\Bbbk $ , see Lemma 4.12 below, we see that X is rational over $\Bbbk $ . In particular, we have $X(\Bbbk )\ne \varnothing $ .

Proof. By Mori–Mukai [Reference Mori and Mukai63], $X_{\mathbb {C}}$ can be obtained by blowing up a smooth quintic del Pezzo 3-fold $V_5$ along a line. Arguing as in the proof of Lemma 4.3, we conclude that X is rational over $\Bbbk $ and $X(\Bbbk )\ne \varnothing $ .

Proof. Over $\mathbb {C}$ , the 3-fold $X_{\mathbb {C}}$ can be obtained by blowing up $\mathbb {P}^3$ along a smooth plane curve of degree $3$ or $2$ . Applying Lemma 2.5 and Lemma 2.4, we see that X is also obtained by blowing up $\mathbb {P}^3$ along a smooth plane curve defined over $\Bbbk $ . This implies that X is rational over $\Bbbk $ , in particular $X(\Bbbk )\ne \varnothing $ .

Proof. By Mori–Mukai [Reference Mori and Mukai63], the base extension $X_{\mathbb {C}}$ can be obtained by blowing up a smooth quadric 3-fold $Q\subseteq \mathbb {P}^4$ along a line. Let E be the exceptional divisor of this blowup. Then, by Lemma 2.5, the surface E is defined over $\Bbbk $ . On the other hand, it is well known that $E_{\mathbb {C}}$ is isomorphic to $\mathbb {F}_1$ . Since, as we explained in the proof of Lemma 4.7, $\mathbb {F}_1$ does not have nontrivial forms over $\Bbbk $ , we conclude that $E\simeq \mathbb {F}_1$ , so $E(\Bbbk )\ne \varnothing $ . Hence, $X(\Bbbk )\ne \varnothing $ as well. This also implies that X is $\Bbbk $ -rational, since forms of smooth quadrics containing $\Bbbk $ -points are $\Bbbk $ -rational.

Proof. A variety in this family is often called $V_7$ . By Mori–Mukai [Reference Mori and Mukai63], $X_{\mathbb {C}}$ can be obtained by blowing up $\mathbb {P}^3$ at a point p. By Lemma 2.5, X can be obtained by blowing up a $\Bbbk $ -form of $\mathbb {P}^3$ , say $X\to U$ such that the image p of the exceptional divisor yields a ${\Bbbk }$ -rational point of a Severi–Brauer 3-fold U, in particular, U is isomorphic to $\mathbb {P}^3$ .

Proof. By Mori–Mukai [Reference Mori and Mukai63], the base extension $X_{\mathbb {C}}$ , which is isomorphic to $\mathbb {P}( {\mathcal O}_{\mathbb {P}^2} \oplus {\mathcal O}_{\mathbb {P}^2}(-2))$ , possesses two extremal contractions: a divisorial contraction $f\colon X_{\mathbb {C}} \to \mathbb {P}(1,1,1,2)$ and a $\mathbb {P}^1$ -bundle $\pi \colon X_{\mathbb {C}} \to \mathbb {P}^2$ . Since the action of $\mathrm {Gal}(\mathbb {C}/{\Bbbk })$ on the Mori cone $\mathrm {NE}(X_{\mathbb {C}})$ leaves the two rays invariant, both f and $\pi $ are defined over $\Bbbk $ . With a slight abuse of notation, denote by $\pi $ the descent $\pi \colon X \to U$ over $\Bbbk $ , where U is a Severi–Brauer surface. The exceptional divisor E of f is defined over $\Bbbk $ . Since f is the blowup of the vertex of the cone $\mathbb {P}(1,1,1,2)$ , $E_{\mathbb {C}}$ is a Veronese surface. Furthermore, $\pi $ induces an isomorphism $E\simeq U$ . Moreover, applying Lemma 2.2 to the divisor $E|_E$ and the Severi–Brauer surface E, we conclude that $E\cong \mathbb {P}^2$ . In particular, we have $E(\Bbbk )\ne \varnothing $ , so X has a $\Bbbk $ -point. Indeed, one can show that X is rational over $\Bbbk $ .

Proof. Let $\Pi $ be a plane in $\mathbb {P}^3$ , and let p be a point in $\mathbb {P}^3$ with $p\not \in \Pi $ , let $\phi \colon V_7\to \mathbb {P}^3$ be the blowup of this point, and let $\widetilde {\Pi }$ be the proper transform on $V_7$ of the plane $\Pi $ . Then there exists a birational morphism $\pi \colon X_{\mathbb {C}}\to V_7$ that is a blowup of a smooth elliptic curve $C\subset \widetilde {\Pi }$ . Set $\mathscr {C}=\phi (C)$ . Then $\mathscr {C}$ is a smooth plane cubic curve in $\mathbb {P}^3$ . Let $E_C$ be the $\pi $ -exceptional surface, and let $E_P$ , $H_C$ , F be the proper transforms on the 3-fold $X_{\mathbb {C}}$ of the $\phi $ -exceptional surface, the plane $\Pi $ , and the cubic cone in $\mathbb {P}^3$ over the curve $\mathscr {C}$ with vertex p, respectively. Then we have the following commutative diagram:

where $\varpi $ is the blowup of the curve $\mathscr {C}$ , $\varphi $ is the contraction of the surface $E_P$ , $\sigma $ and $\psi $ are the contractions of the surfaces $H_C$ and F, respectively, $\varsigma $ is the contraction of the surface $\varphi (H_C)$ , Y is a Fano 3-fold that has a singular point of type $\frac {1}{2}(1,1,1)$ , the morphism $\widehat {Y}\to Y$ is the blowup of a smooth point of the 3-fold Y, both $V_7\to \mathbb {P}^2$ and $\mathbb {P}(\mathcal {O}_{\mathbb {P}^2}\oplus \mathcal {O}_{\mathbb {P}^2}(2))\to \mathbb {P}^2$ are $\mathbb {P}^1$ -bundles, the morphism $\mathbb {P}(\mathcal {O}_{\mathbb {P}^2}\oplus \mathcal {O}_{\mathbb {P}^2}(2))\to \mathbb {P}(1,1,1,2)$ is the contraction of the surface $\psi (H_C)$ , and $\widehat {Y}\to \mathbb {P}(1,1,1,2)$ is the contraction of $\sigma (F)$ . This shows that the Mori cone $\mathrm {NE}(X_{\mathbb {C}})$ is generated by the extremal rays that are spanned by the curves contracted by $\psi $ , $\varphi $ , $\pi $ , $\sigma $ . Since the Galois group $\mathrm {Gal}({\mathbb {C}}/{\Bbbk })$ cannot permute any of these rays, we see that the commutative diagram above descends to $\Bbbk $ . Since $V_7$ does not have nontrivial forms over $\Bbbk $ by Lemma 4.12, we see that X is $\Bbbk $ -rational and, in particular, has a $\Bbbk $ -point.

Proof. Let $\mathscr {C}$ be a twisted cubic curve in the space $\mathbb {P}^3$ , let p be a point in the curve $\mathscr {C}$ , let $\phi \colon V_7\to \mathbb {P}^3$ be the blowup of this point, and let C be the proper transform of the cubic curve $\mathscr {C}$ on the 3-fold $V_7$ . Then $X_{\mathbb {C}}$ can be obtained as the blowup $\pi \colon X_{\mathbb {C}}\to V_7$ along the curve C. One can see that $X_{\mathbb {C}}$ fits into the commutative diagram

where W is a smooth divisor of degree $(1,1)$ in $\mathbb {P}^2\times \mathbb {P}^2$ , both $p_1$ and $p_2$ are $\mathbb {P}^1$ -bundles, the morphism $\varpi $ is the blowup of $\mathbb {P}^3$ along $\mathscr {C}$ , the morphism $\widetilde {\mathbb {P}}^3\to \mathbb {P}^2$ is a $\mathbb {P}^1$ -bundle whose fibers are proper transforms of the secant lines in $\mathbb {P}^3$ of the twisted cubic curve $\mathscr {C}$ , the morphism $V_7\to \mathbb {P}^2$ is the $\mathbb {P}^1$ -bundle whose fibers are proper transforms of the lines in the space $\mathbb {P}^3$ that pass through p, and $\varphi $ is the blowup of the fiber of $\varpi $ over p. Observe that the Mori cone $\mathrm {NE}(X_{\mathbb {C}})$ is simplicial and is generated by the extremal rays spanned by the curves contracted by $\psi $ , $\varphi $ , $\pi $ . Since the Galois group $\mathrm {Gal}({\mathbb {C}}/{\Bbbk })$ cannot permute any of these rays, we see that the commutative diagram above descends to $\Bbbk $ . Now, as in the proof of Lemma 4.14, we see that X is rational over $\Bbbk $ . In particular, we have $X(\Bbbk )\ne \varnothing $ .

Proof. The Fano 3-fold $X_{\mathbb {C}}$ can be obtained as a blowup $\pi \colon X_{\mathbb {C}}\to \mathbb {P}^3$ along a disjoint union of a smooth conic C and a line L. There is a commutative diagram

where $\vartheta $ is the blowup of the line L, the morphism $\varphi $ is the blowup of the conic C, the morphisms $\theta $ and $\phi $ are blowups of the proper transforms of the curves L and C, respectively, Q is a smooth quadric in $\mathbb {P}^4$ , the morphism $\eta $ is the blowup of a point in Q, the morphism $\widetilde {Q}\to Q$ is the blowup of a conic (the proper transform of the line L), the morphism $Y\to \mathbb {P}^1$ is a $\mathbb {P}^2$ -bundle, the morphism $\widetilde {Q}\to \mathbb {P}^1$ is a fibration into quadric surfaces, and $\psi $ is the contraction of the proper transform of the plane in $\mathbb {P}^3$ containing C.

This shows that the Mori cone $\mathrm {NE}(X_{\mathbb {C}})$ is simplicial and is generated by the extremal rays spanned by the curves contracted by $\theta $ , $\phi $ , $\psi $ . Since the Galois group $\mathrm {Gal}({\mathbb {C}}/{\Bbbk })$ cannot permute any of these rays, we see that the commutative diagram above descends to $\Bbbk $ . In particular, we can obtain X as the blowup of a Severi–Brauer 3-fold U along a disjoint union of a twisted line and a twisted conic. Now, applying Lemma 2.5 to U and the twisted conic, we see that $U\simeq \mathbb {P}^3$ . Hence, we see that X is rational over $\Bbbk $ .

Proof. Over $\mathbb {C}$ , there exists a blowup $\pi \colon X_{\mathbb {C}}\to \mathbb {P}^1\times \mathbb {P}^2$ of a smooth curve C of degree $(2,1)$ . Let S be the proper transform on X of the surface in $\mathbb {P}^1\times \mathbb {P}^2$ of degree $(0,1)$ that passes through the curve C, let $\ell _1$ and $\ell _2$ be the rulings of the surface $S\cong \mathbb {P}^1\times \mathbb {P}^1$ such that the curves $\pi (\ell _1)$ and $\pi (\ell _2)$ are of degree $(1,0)$ and $(0,1)$ in $\mathbb {P}^1\times \mathbb {P}^2$ , respectively. Let E be the $\pi $ -exceptional surface, and let $\ell _3$ be a fiber of the natural projection $E\to C$ . Then the curves $\ell _1$ , $\ell _2$ , $\ell _3$ generate the Mori cone $\overline {\mathrm {NE}}(X)$ , and the extremal rays $\mathbb {R}_{\geqslant 0}[\ell _1]$ and $\mathbb {R}_{\geqslant 0}[\ell _2]$ give birational contractions $X\to U_1$ and $X\to U_2$ , respectively. Moreover, it follows from the proof of [Reference Cheltsov and Shramov18, Lemma 8.22] that there is a commutative diagram

where the morphism $U_1\to \mathbb {P}^1$ is a quadric bundle, the morphism $U_2\to \mathbb {P}^2$ is a $\mathbb {P}^1$ -bundle, the map $U_1\dashrightarrow U_2$ is a flop, and V is a Fano 3-fold in Family № 1.15 with one isolated ordinary double point singularity. For details, we refer the reader to the case (2.3.2) in [Reference Takeuchi76, Theorem 2.3].

Since the Galois group $\mathrm {Gal}({\mathbb {C}}/{\Bbbk })$ cannot nontrivially permute extremal rays $\mathbb {R}_{\geqslant 0}[\ell _1]$ , $\mathbb {R}_{\geqslant 0}[\ell _2]$ , $\mathbb {R}_{\geqslant 0}[\ell _3]$ , we see that the diagram above is defined over $\Bbbk $ with $X_{\mathbb {C}}$ replaced by X, $\mathbb {P}^1$ replaced by a (possibly pointless) conic $C_2$ , and $\mathbb {P}^2$ is replaced by its $\Bbbk $ -form U. Then we may assume that C is a curve in $C_2\times U$ defined over $\Bbbk $ , and $\pi $ is the blowup of the product $C_2\times U$ along this curve. Then the image of the curve C in U via the natural projection $\mathrm {pr}_2\colon C_2\times U\to U$ is a twisted line in the Severi–Brauer surface U, so it follows from Lemma 2.2 that $U\simeq \mathbb {P}^2$ and $\mathrm {pr}_2(C)$ is a line. Moreover, since $\mathrm {pr}_2|_C: C \to \mathrm {pr}_2(C)$ is an isomorphism, we see that C is isomorphic to $\mathbb {P}^1$ , which implies in turn that $C_2\simeq \mathbb {P}^1$ via the projection $\mathrm {pr}_1\colon \colon C_2\times U\to C_2$ in consideration of Lüroth’s Theorem. Therefore X is birational to $\mathbb {P}^1\times \mathbb {P}^2$ over $\Bbbk $ , so X is rational over $\Bbbk $ . In particular, we have $X ({\Bbbk })\ne \varnothing $ .

Proof. Let $\mathrm {pr}_1\colon \mathbb {P}^1\times \mathbb {P}^2\to \mathbb {P}^1$ and $\mathrm {pr}_2\colon \mathbb {P}^1\times \mathbb {P}^2\to \mathbb {P}^2$ be the projections to the first and the second factors, respectively, let $H_1$ be a fiber of the map $\mathrm {pr}_1$ , let $H_2=\mathrm {pr}_2^*(\mathcal {O}_{\mathbb {P}^2}(1))$ , and let C be a conic in $H_1\cong \mathbb {P}^2$ . Then there is a blowup $\psi \colon X_{\mathbb {C}}\to \mathbb {P}^1\times \mathbb {P}^2$ along the curve C.

Let $E_C$ be the $\psi $ -exceptional surface, let $\widetilde {H}_1$ be the proper transform of the surface $H_1$ on the 3-fold X, let F be the surface in $|H_2|$ that contains C, and let $\widetilde {F}$ be the proper transform of this surface on X. We have the following commutative diagram:

where $\pi $ and $\phi $ are the contraction of the surfaces $\widetilde {H}_1\cong \mathbb {P}^2$ and $\widetilde {F}\cong \mathbb {P}^1\times \mathbb {P}^1$ , respectively, the morphisms $\varpi $ and $\varphi $ are the contractions of the surfaces $\phi (\widetilde {H}_1)$ and $\pi (\widetilde {F})$ , respectively, the morphism $\sigma $ is a $\mathbb {P}^1$ -bundle, and $\eta $ is a fibration into del Pezzo surfaces such that all its fibers except $\pi (\widetilde {F})$ are isomorphic to $\mathbb {P}^2$ , while $\pi (\widetilde {F})\cong \mathbb {P}(1,1,4)$ . Note that the Mori cone $\overline {\mathrm {NE}}(X)$ is simplicial and it is generated by the extremal rays contracted by $\pi $ , $\phi $ and $\psi $ . As in the proof of Lemma 4.17, we see that the above diagram can be defined over $\Bbbk $ with $X_{\mathbb {C}}$ replaced by X, $\mathbb {P}^1$ replaced by a (possibly pointless) conic $C_2$ , and $\mathbb {P}^2$ is replaced by its $\Bbbk $ -form U. Then $\pi $ is a blowup of the product $C_2\times U$ along a curve C defined over $\Bbbk $ such that $\mathrm {pr}_1(C)$ is a point in $C_2$ , and $\mathrm {pr}_2(C)$ is a twisted conic in the Severi–Brauer surface U. Now, applying Lemma 2.2, we see that $C_2\cong \mathbb {P}^1$ and $U\cong \mathbb {P}^2$ , so X is rational over $\Bbbk $ , which gives $X ({\Bbbk })\ne \varnothing $ .

Proof. Let $\mathscr {C}$ be a smooth conic in $\mathbb {P}^3$ , let p be an arbitrary point in the conic $\mathscr {C}$ , let $\phi \colon V_7\to \mathbb {P}^3$ be the blowup of the point p, and let C be the proper transform on the 3-fold $V_7$ of the conic $\mathscr {C}$ . Then, over $\mathbb {C}$ , there exists a birational morphism $\pi \colon X_C\to V_7$ that is the blowup of the curve C. One can see that X fits into the commutative diagram

where Q is a smooth quadric 3-fold in $\mathbb {P}^4$ , the morphism $\varpi $ is the blowup of the conic $\mathscr {C}$ , the morphism $\widetilde {\mathbb {P}}^3\to Q$ is the contraction to a point of the proper transform of the plane in $\mathbb {P}^3$ that contains $\mathscr {C}$ , $\varphi $ is the blowup of the fiber of the morphism $\varpi $ over the point p, the morphism $\widehat {Q}\to Q$ is the blowup of a line in Q that passes through the latter point, and $\widehat {Q}\to \mathbb {P}^2$ is a $\mathbb {P}^1$ -bundle. Hence, the Mori cone $\mathrm {NE}(X_{\mathbb {C}})$ is simplicial and is generated by the extremal rays spanned by the curves contracted by $\psi $ , $\varphi $ , $\pi $ . Since the Galois group $\mathrm {Gal}({\mathbb {C}}/{\Bbbk })$ cannot permute any of these rays, the commutative diagram above descends to $\Bbbk $ . Since $V_7$ does not have nontrivial forms over $\Bbbk $ by Lemma 4.12, we see that X is $\Bbbk $ -rational.

Proof. Over $\mathbb {C}$ , there is a blowup $\phi \colon X_C\to \mathbb {P}^1\times \mathbb {P}^2$ of a smooth curve C of degree $(1,1)$ , and we have the following commutative diagram

where W is a divisor of degree $(1,1)$ on $\mathbb {P}^{2}\times \mathbb {P}^{2}$ , $\omega _{1}$ is a natural $\mathbb {P}^{1}$ -bundle, $\alpha $ contracts a smooth surface $E\cong \mathbb {P}^{1}\times \mathbb {P}^{1}$ to a fiber L of $\omega _{1}$ , $\gamma $ is the blowup of the point $\omega _1(L)$ , the morphism $\xi $ is a $\mathbb {P}^{1}$ -bundle, $\zeta $ is a $\mathbb {F}_{1}$ -bundle, $\mathrm {pr}_1$ and $\mathrm {pr}_2$ are projections to the first and the second factors, respectively. This commutative diagram shows that the Mori cone $\mathrm {NE}(X_{\mathbb {C}})$ is generated by the extremal rays spanned by the curves contracted by $\pi $ , $\alpha $ , $\phi $ . Since the Galois group $\mathrm {Gal}({\mathbb {C}}/{\Bbbk })$ cannot nontrivially permute these rays, we see that the commutative diagram above descends to $\Bbbk $ with $X_{\mathbb {C}}$ replaced by X, $\mathbb {P}^1$ replaced by a (possibly pointless) conic $C_2$ , and $\mathbb {P}^2$ is replaced by its $\Bbbk $ -form U. Then we may assume that C is a curve in $C_2\times U$ defined over $\Bbbk $ , and $\pi $ is the blowup of the product $C_2\times U$ along this curve. But $\mathrm {pr}_2(C)$ is a twisted line in U, which gives $U\simeq \mathbb {P}^2$ by Lemma 2.2. Moreover, since $\mathrm {pr}_2|_C\colon C \to \mathrm {pr}_2(C)$ is an isomorphism and $\mathrm {pr}_2(C)$ is a line, we see that $C\cong \mathbb {P}^1$ . Then $C_2\simeq \mathbb {P}^1$ as well, since $\mathrm {pr}_1|_C\colon C \to C_2$ is an isomorphism. Therefore, we see that X is birational to $\mathbb {P}^1\times \mathbb {P}^2$ over $\Bbbk $ . In particular, X has a $\Bbbk $ -point.

Proof. Let $V_7$ be the blowup of $\mathbb {P}^3$ at a point p, let L be a line in $\mathbb {P}^3$ not containing p, and let C be its strict transform on $V_7$ . Then $X_{\mathbb {C}}$ can be obtained by blowing up $V_7$ along the curve C, and it follows from [Reference Mori and Mukai61] and [Reference Mori and Mukai63] that $X_{\mathbb {C}}$ has exactly one divisorial contraction, the inverse of the blowing up $X_{\mathbb {C}} \to V_7$ of the curve C. Thus, the blowup $X_{\mathbb {C}} \to V_7$ descends to $\Bbbk $ , but $V_7$ does not have nontrivial $\Bbbk $ -forms by Lemma 4.12, which implies that X can be obtained by blowing up $V_7$ over $\Bbbk $ . In particular, $X({\Bbbk })\ne \varnothing $ .

Proof. As in the proof of Lemma 4.21, let $f\colon V_7\to \mathbb {P}^3$ be the blowup of a point p, let E be the f-exceptional surface, and let C be a line in $E\simeq \mathbb {P}^2$ . Then $X_{\mathbb {C}}$ can be obtained by blowing up $V_7$ along the curve C. Moreover, it follows from [Reference Mori and Mukai61] and [Reference Mori and Mukai63] that $X_{\mathbb {C}}$ has two extremal contractions, one of which is to $V_7$ and the other is to $\mathbb {P}(\mathcal {O}_{\mathbb {P}^2} \oplus \mathcal {O}_{\mathbb {P}^2}(2))$ . In particular, they are both defined over $\Bbbk $ . Recall from the proofs of Lemmas 4.12 and 4.13 that $V_7$ and $\mathbb {P}(\mathcal {O}_{\mathbb {P}^2} \oplus \mathcal {O}_{\mathbb {P}^2}(2))$ do not have nontrivial forms over $\Bbbk $ , so we obtain the required assertion as in the proof of Lemma 4.21.

Proof. As in the proof of Lemmas 4.21 and 4.22, let $f\colon V_7\to \mathbb {P}^3$ be the blowup of a point p. Then $V_7\cong \mathbb {P}( \mathcal {O}_{\mathbb {P}^2} \oplus \mathcal {O}_{\mathbb {P}^2}(1))$ . Let L be a fiber of the natural projection $V_7\to \mathbb {P}^2$ . Then $X_{\mathbb {C}}$ can be obtained by blowing up $V_7$ along L. Moreover, it follows from [Reference Mori and Mukai61] and [Reference Mori and Mukai63] that $X_{\mathbb {C}}$ has two extremal contractions: one of them is the birational morphism $X_{\mathbb {C}}\to V_7$ , and the other one is a birational contraction of $V_7\to \mathbb {P}(\mathcal {O}_{\mathbb {P}^1}^{\oplus 2}\oplus \mathcal {O}_{\mathbb {P}^1}(1))$ . In particular, both morphisms must be defined over $\Bbbk $ . Now, arguing as in the proof of Lemma 4.21, we see that X is $\Bbbk $ -rational and, in particular, it has a $\Bbbk $ -point.

Proof. Let C be a curve of degree $(2,1)$ in $\mathbb {P}^1\times \mathbb {P}^2$ , let L be a curve of degree $(1,0)$ in $\mathbb {P}^1\times \mathbb {P}^2$ that is disjoint from C. Let $f\colon Y\to \mathbb {P}^1\times \mathbb {P}^2$ be the blowup of the curve C, and let $\widetilde {L}$ be the strict transform of the curve L on the 3-fold Y. Then Y is a smooth Fano 3-fold in Family № 3.21, and it follows from [Reference Mori and Mukai61, Reference Mori and Mukai63] that the base extension $X_{\mathbb {C}}$ is isomorphic over $\mathbb {C}$ to the blowup of Y along the curve $\widetilde {L}$ . Moreover, analyzing the Mori cone of the 3-fold $X_{\mathbb {C}}$ , we see that the birational morphism $X_C\to Y$ descends to the $\Bbbk $ -birational morphism from X to a $\Bbbk $ -form of Y. From the proof of Lemma 4.17, we know that all $\Bbbk $ -forms of Y are $\Bbbk $ -rational, so X is also $\Bbbk $ -rational. In particular, X has a $\Bbbk $ -point.

Proof. Let $L_1$ and $L_2$ be two disjoint lines in $\mathbb {P}^3$ , let $f\colon Y\to \mathbb {P}^3$ be the blowup of the curves $L_1$ and $L_2$ , let $E_1$ and $E_2$ be the f-exceptional surfaces such that $f(E_1)=L_1$ and $f(E_2)=L_2$ , and let C be a fiber of the natural projection $E_1\to L_1$ . Then there exists a birational morphism $g\colon X_{\mathbb {C}}\to Y$ that blows up the curve C.

Let $E_C$ be the g-exceptional surface, let $\widetilde {E}_1$ and $\widetilde {E}_2$ be the strict transforms on $X_{\mathbb {C}}$ of the surfaces $E_1$ and $E_2$ , respectively. Then there exists a birational contraction $h\colon X_{\mathbb {C}}\to V_7$ of the surfaces $\widetilde {E}_1$ and $\widetilde {E}_2$ such that $V_7$ is the blowup of $\mathbb {P}^3$ at the point $f(C)$ , and $h(E_C)$ is the exceptional divisor of the morphism $V_7\to \mathbb {P}^3$ . To be precise, we have the following commutative diagram:

Moreover, it follows from [Reference Mori and Mukai61] and [Reference Mori and Mukai63] that this commutative diagram is defined over $\Bbbk $ , so that X is $\Bbbk $ -rational, since $V_7$ does not have nontrivial $\Bbbk $ -forms by Lemma 4.12. In particular, we see that X has a $\Bbbk $ -point.

Proof. Let $V=\mathbb {P}^1 \times \mathbb {F}_1$ , let S be a fiber of the natural projection $V\to \mathbb {P}^1$ , and let C be the $(-1)$ -curve in $S\cong \mathbb {F}_1$ . Then it follows from [Reference Mori and Mukai61, Reference Mori and Mukai63] that there exists a birational morphism $f\colon X_{\mathbb {C}} \to \mathbb {P}^1 \times \mathbb {F}_1$ that is the blowup of the curve C. Let E be the f-exceptional divisor. Then it follows from [Reference Matsuki59] that E is defined over $\Bbbk $ . Since $\mathbb {F}_1$ does not have nontrivial forms over $\Bbbk $ , the 3-fold X is the blowup of $\mathscr {C}\times \mathbb {F}_1$ , where $\mathscr {C}$ is a conic in $\mathbb {P}^2$ . However, the image of E in $\mathscr {C}\times \mathbb {F}_1$ is a curve that is contained in a fiber of the natural projection $\mathscr {C}\times \mathbb {F}_1\to \mathscr {C}$ , which implies that $\mathscr {C}$ has a $\Bbbk $ -point. Therefore, X is $\Bbbk $ -birational to $\mathbb {P}^1 \times \mathbb {F}_1$ and, in particular, it has a $\Bbbk $ -point.