Nonrational del Pezzo fibrations admitting an action of the Klein simple group

Abstract

We present a series of del Pezzo fibrations of degree 2 admitting an action of the Klein simple group and prove their nonrationality by the reduction modulo p method of Kollár. This is relevant to the embedding of the Klein simple group into the Cremona group of rank 3.

1 Introduction

The Klein simple group $G_\mathrm{K}$ is a finite simple group $G_\mathrm{K} \cong \mathrm {PSL}_2 (\mathbb {F}_7)$ of order 168. It is well known that $G_\mathrm{K}$ is the automorphism group of the Klein quartic curve which is defined in $\mathbb {P}^2$ by the equation . Let $S_\mathrm{K}$ be the double cover of $\mathbb {P}^2$ ramified along the Klein quartic curve. Then $S_\mathrm{K}$ is a nonsingular del Pezzo surface of degree 2 admitting a faithful action of $G_\mathrm{K}$. Belousov [3] proved that $\mathbb {P}^2$ and $S_\mathrm{K}$ are the only del Pezzo surfaces admitting a faithful action of $G_\mathrm{K}$. In [1], Ahmadinezhad presented a series of $G_\mathrm{K}$-Mori fiber spaces over $\mathbb {P}^1$ whose general fibers are isomorphic to $S_\mathrm{K}$ for $n \geqslant 0$. A G-Mori fiber space, where G is a group, is a G-equivariant version of Mori fiber space (see Definition 2.9). Among the above series of varieties, is a del Pezzo fibration for $n \geqslant 1$ while is not (see Sect. 2 for details). We have the following conjectures concerning these varieties.

Conjecture 1.1

(Cheltsov–Shramov [3, Conjecture 1.4]) The fibrations and , for $n \geqslant 0$, are the only $G_\mathrm{K}$-Mori fiber spaces over $\mathbb {P}^1$ in dimension 3.

Conjecture 1.2

(Ahmadinezhad [1, Conjecture 3.5]) The varieties $X_n$ are non-rational for $n \geqslant 2$.

Note that $X_0$ and $X_1$ are both rational. The main result of this paper is the following theorem which supports Conjecture 1.2.

Theorem 1.3

For $n \geqslant 5$, a very general $X_n$ is not rational.

We refer the reader to Sect. 3 for the meaning of very generality. Note that is a del Pezzo fibration of degree 2 and it satisfies the so-called K-condition (or $K^2$-condition) for $n \geqslant 2$. Thus, by the results of Pukhlikov [12] and Grinenko [5–7] on nonsingular del Pezzo fibrations of degree 2, if $X_n$ were nonsingular, then it would be birationally rigid for $n \geqslant 2$, which would imply nonrationality in a strong sense. Unfortunately, $X_n$ is singular and we cannot apply the above results directly. Instead, we apply the Kollár’s reduction modulo p method introduced in [8] (see also [9]) to prove nonrationality of $X_n$.

This is relevant to the study of embeddings of the Klein simple group $G_\mathrm{K} = \mathrm {PSL}_2 (\mathbb {F}_7)$ into the Cremona group ${\text {Cr}}_3 (\mathbb {C})$ of rank 3. If we are given a finite simple subgroup G of ${\text {Cr}}_3 (\mathbb {C}^3)$, then there is a rational G-Mori fiber space X / S such that the embedding $G \subset {\text {Cr}}_3 (\mathbb {C})$ is given by $G \subset \mathrm{Aut}(X) \subset \mathrm{Bir}(X) \cong {\text {Cr}}_3 (\mathbb {C})$ (see [11, Section 4.2]). Such a G-Mori fiber space X / S is called a Mori regularization of $G \subset {\text {Cr}}_3 (\mathbb {C})$. Moreover, two embeddings $G_1$ and $G_2$ into ${\text {Cr}}_3 (\mathbb {C})$ of a finite simple subgroup G are conjugate if and only if there is a G-equivariant birational map between Mori regularizations $X_1/S_1$ and $X_2/S_2$ of $G_1 \subset {\text {Cr}}_3 (\mathbb {C})$ and $G_2 \subset {\text {Cr}}_3 (\mathbb {C})$, respectively. In [4], Cheltsov and Shramov proved that there are at least three non-conjugate embeddings of $\mathrm {PSL}_2 (\mathbb {F}_7)$ into ${\text {Cr}}_3 (\mathbb {C})$ and each of them comes from rational ($G_\mathrm{K}$-) Fano threefolds. Theorem 1.3 implies that, for $n \geqslant 5$, a very general cannot be a Mori regularization of any subgroup of ${\text {Cr}}_3 (\mathbb {C})$ isomorphic to $G_\mathrm{K}$. If Conjectures 1.1 and 1.2 are both true, then it follows that there is no embedding of $\mathrm {PSL}_2 (\mathbb {F}_7)$ into ${\text {Cr}}_3 (\mathbb {C})$ coming from a $G_\mathrm{K}$-Mori fiber space over $\mathbb {P}^1$ other than , and . Note that, by [1, Theorem 3.4], there is a $G_\mathrm{K}$-equivariant birational map between $X_1$ and .

The paper is organized as follows. In Sect. 2, we give an explicit construction of varieties $X_n$. They are constructed as hypersurfaces of suitable weighted projective space bundle over $\mathbb {P}^1$. Then we show that is indeed a del Pezzo fibration for $n \geqslant 1$. In Sect. 3, we prove the main theorem. The proof will be done by the Kollár’s reduction modulo p method, which we briefly recall in Sect. 3.2. The very first reduction step is done in Sect. 3.1. In Sect. 3.3, we work over a field of characteristic 2 and construct a specific big line bundle on some nonsingular model of $X_n$ by making use of the purely inseparable double covering structure. This will complete the proof in view of the non-ruledness criterion given in Lemma 3.2.

2 Construction of del Pezzo fibrations

We construct del Pezzo fibrations as hypersurfaces in suitable weighted projective space bundles over $\mathbb {P}^1$. We refer the reader to [2] for Cox rings (which are also known as homogeneous coordinate rings) of toric varieties. In this section we work over $\mathbb {C}$.

Throughout this paper, we define . We see that f is the defining polynomial of the Klein quartic curve whose automorphism group is the Klein simple group. Let $P_n$ be the projective simplicial toric variety with the Cox ring

which is $\mathbb {Z}^2$-graded as

and with the irrelevant ideal $I = (w_0,w_1) \cap (x_0,x_1,x_2,y)$, that is, $P_n$ is the geometric quotient

where the action of $(\mathbb {C}^*)^2$ on $\mathbb {A}^6 = {\text {Spec}}\mathrm{Cox}(P_n)$ is given by the above matrix. Note that the Weil divisor class group $\mathrm{Cl}(P_n)$ is isomorphic to $\mathbb {Z}^2$. There is a natural morphism $\mathrm{\Pi } :P \rightarrow \mathbb {P}^1$ defined as the projection to coordinates $w_0,w_1$, and this realizes P as a weighted projective space bundle over $\mathbb {P}^1$ whose fibers are . For a nonnegative integer n and homogeneous polynomials and of degree respectively and 4, define

and let . Throughout this paper, we assume that a does not have a multiple component.

Remark 2.1

Let us note that constructed as above coincides with the one given in [1, Section 3]. Indeed, choose and fix any pair of homogeneous polynomials of degree n such that and define

Let $\pi ' :\mathscr {X}'_n \rightarrow \mathbb {P}^1$ be the projection to the coordinates $w_0,w_1$. Then, $(c = t = f = 0) \subset \mathscr {X}'_n$ is a disjoint union of n-curves $C'_1,\ldots ,C'_n$ and $\mathscr {X}'_n$ has a singularity of type along each $C'_i$. Blowing up $\mathscr {X}'_n$ along these curves and then contracting the strict transforms of the $\pi '$-fibers containing $C'_i$, we obtain a birational map $\mathscr {X}'_n \dashrightarrow \mathscr {X}_n$ to the del Pezzo fibration $\mathscr {X}_n \rightarrow \mathbb {P}^1$ constructed in [1].

Now we have a birational map defined by the correspondence . It is easy to see that $\mathrm{\Psi }$ restricts to a birational map $\psi :X_n \dashrightarrow \mathscr {X}'_n$. Moreover, it is straightforward to see that $\psi ^{-1} :\mathscr {X}'_n \dashrightarrow X_n$ is obtained by blowing up $\mathscr {X}'_n$ along $C'_1,\ldots ,C'_n$ and then contracting the strict transforms of the fibers containing $C'_i$. This shows .

Remark 2.2

Let us explain that both $X_0$ and $X_1$ are rational. If $n = 0$, then we have , where is a (nonsingular) del Pezzo surface of degree 2, and thus $X_0$ is clearly rational. Suppose $n = 1$. Then, as explained in Remark 2.1, $X_1$ is birational to , where are homogeneous polynomials of degree 1 such that . It is clear that the projection is birational. Hence $\mathscr {X}'_1$ and $X_1$ are rational.

In the rest of this section, we show that $\pi :X_n \rightarrow \mathbb {P}^1$ is indeed a del Pezzo fibration for $n \geqslant 1$.

Definition 2.3

Let $\pi :X \rightarrow \mathbb {P}^1$ be a surjective morphism with connected fibers from a normal projective 3-fold X. We say that $\pi :X \rightarrow \mathbb {P}^1$ is a del Pezzo fibration over $\mathbb {P}^1$ if the following conditions are satisfied:

• X is $\mathbb {Q}$-factorial and has only terminal singularities.

• is $\pi$-ample.

• $\rho (X) = 2$.

Remark 2.4

We explain the natural affine open subsets of $P_n$ and $X_n$. We refer the reader to [13] for details. Since we will work over an algebraically closed field of characteristic 2 in the next section, we assume in this remark that the ground field of $P_n$ and $X_n$ is an algebraically closed field $\Bbbk$ of arbitrary characteristic.

Denote by $U_{w_i,x_j}$ the open subset $(w_i \ne 0) \cap (x_j \ne 0) \subset P_n$ and by $U_{w_i,y}$ the open subset $(w_i \ne 0) \cap (y \ne 0) \subset P_n$. Then $P_n$ is covered by $U_{w_i,x_j}$ and $U_{w_i,y}$ for $i = 0,1$ and $j = 0,1,2$; and $U_{w_0,x_0}$ is the affine 4-space $\mathbb {A}^4$. The restrictions of $w_1, x_1,x_2,y$ on $U_{w_0,x_0}$ form affine coordinates of $U_{w_0,x_0}$. Indeed, if we denote by $\widetilde{w}_1 = w_1/w_0$, $\widetilde{x}_i = x_i/x_0$ for $i = 1,2$ and , then $U_{w_0,x_0}$ is an affine 4-space with affine coordinates $\widetilde{w}_0,\widetilde{x}_1,\widetilde{x}_2,\widetilde{y}$. The affine scheme $X_n \cap U_{w_0,x_0}$ is defined by the equation . The same description applies for the other $U_{w_i,x_j}$.

We see that $U_{w_0,y}$ is the quotient of $\mathbb {A}^4$ by the action of . Indeed, if we denote by $\widetilde{w}_1 = w_1/w_0$, for $i = 0,1,2$, then $U_{w_0,y}$ is the quotient of $\mathbb {A}^4$ with coordinates $\widetilde{w}_1, \widetilde{x}_0,\widetilde{x}_1,\widetilde{x}_2$ under the $\varvec{\mu }_2$-action given by

where . Here, the above operation defines a ring homomorphism , where , and . When $\Bbbk = \mathbb {C}$, we can replace $\varvec{\mu }_2$ with and the action is given simply by $\widetilde{w}_0 \mapsto \widetilde{w}_0$ and . The affine scheme $X_n \cap U_{w_0,y}$ is the quotient of the affine scheme defined by the $\varvec{\mu }_2$-action. The same description applies for $U_{w_1,y}$.

Sometimes we will abuse the notation and say that $U_{w_0,x_0}$ is the affine 4-space $\mathbb {A}^4$ with coordinates $w_1,x_1,x_2,y$ and $X_n \cap U_{w_0,x_0}$ is defined by .

Lemma 2.5

The variety $X_n$ is nonsingular outside $(x_0 = x_1 = x_2 = 0) \cap X_n$ and it has a singular point of type at each point of $(x_0 = x_1 = x_2 = 0) \cap X_n$.

Proof

Set $U = U_{w_0,x_0}$ which is an affine 4-space with affine coordinates $w_1,x_1,x_2,y$, then $X_n \cap U$ is defined by , where and $f_0 = f (1,x_1,x_2)$. It is straightforward to see that

where the last equality holds since $f_0 = f (1,x_1,x_2)$ defines a nonsingular curve in $\mathbb {A}^2$. By symmetry, we conclude that $X \cap U_{w_i,x_j}$ is nonsingular for $i = 0,1$ and $j = 0,1,2$. Since the open subsets $U_{w_i,x_j}$ for $i = 0,1$ and $j = 0,1,2$ cover , we see that $X_n$ is nonsingular outside $(x_0 = x_1 = x_2 = 0) \cap X_n$.

Let $P \in (x_0 = x_1 = x_2 = 0) \cap X_n$. Then $a (P) = 0$ and we may assume that $w_1$ vanishes at P after replacing $w_0,w_1$. We work on . We see that $X_n \cap U$ is the quotient of by the $\varvec{\mu }_2$-action and P corresponds to the origin. Since a vanishes at P and it does not have a multiple component, we have , so that $x_0,x_1,x_2$ form local coordinates of V at the origin. Thus the point P is of type . $\square$

For $n \geqslant 1$, we construct a birational morphism $\theta :X_n \rightarrow V_n$ as follows. Set $\xi _0 = w_0^n$, $, \ldots , \xi _n = w_1^n$ and let

be the toric morphism defined by the correspondence . Then the image of $\mathrm{\Theta }$, which we denote by $T_n$, is defined by $h_1 = \cdots = h_N = 0$, where $h_1, \ldots , h_N$ are the homogeneous polynomials in $y_0,\ldots ,y_n$ defining the image of the n-ple Veronese embedding $\mathbb {P}^1 \hookrightarrow \mathbb {P}^n_{y_0,\ldots ,y_n}$. We see that $\mathrm{\Theta } :P_n \rightarrow T_n$ is a birational morphism contracting the divisor to the plane $\mathrm{\Delta } = (y_0 = \cdots = y_n = 0) \subset T_n$. It follows that $T_n$ is a projective simplicial toric variety with Picard number 1. The image of $X_n$ under $\mathrm{\Theta }$ is a hypersurface $V_n$ in $T_n$ defined by $q + f = 0$, where $q = q (y_0,\ldots ,y_n)$ is a quadratic polynomial such that . The morphism is a birational morphism contracting the divisor to the curve $\mathrm{\Delta } \cap V_n \cong C$, where C is the plane curve defined by $f = 0$.

Lemma 2.6

If $n \geqslant 1$, then $V_n$ is a normal projective $\mathbb {Q}$-factorial 3-fold with Picard number 1.

Proof

Note that $X_n$ is $\mathbb {Q}$-factorial since it has only quotient singularities. It follows that $V_n$ is $\mathbb {Q}$-factorial since $\theta$ is an extremal contraction (which is not necessarily $K_{X_n}$-negative). We see that the singularity of $T_n$ along the plane $\mathrm{\Delta }$ is of type and $V_n$ intersects $\mathrm{\Delta }$ transversally. Moreover, outside the curve $\mathrm{\Delta } \cap V_n$, singular points of $V_n$ are of type . This implies that $V_n$ is a V-submanifold of $T_n$ and thus, by [2, Proposition 3.5], $V_n$ is quasi-smooth in $T_n$. Here, we refer the reader to [2, Section 3] for the definitions of V-submanifold and quasi-smoothness. It then follows, from [14, Proposition 4], that $\rho (V_n) = \rho (T_n) = 1$ since $V_n$ is quasi-smooth hypersurface defined by a global section of an ample divisor on $T_n$. $\square$

Lemma 2.7

For $n \geqslant 1$, the fibration $\pi :X_n \rightarrow \mathbb {P}^1$ is a del Pezzo fibration.

Proof

Assume that $n \geqslant 1$. We see that X has only terminal singularities of type and it is $\mathbb {Q}$-factorial. By Lemma 2.6, we have $\rho (X_n) = \rho (V_n) + 1 = 2$ since $\theta :X_n \rightarrow V_n$ is a birational morphism contracting a prime divisor. This shows that $\pi$ is an extremal contraction and thus is indeed a del Pezzo fibration. $\square$

Remark 2.8

The above arguments apply to more general cases without any change. Let be a homogeneous polynomial of degree 4 such that the plane curve in $\mathbb {P}^2$ defined by g is nonsingular. Then the hypersurface , where is a homogeneous polynomial of degree which does not have a multiple component, together with the projection $\pi :X_n \rightarrow \mathbb {P}^1$ is a del Pezzo fibration provided that $n \geqslant 1$.

The variety $V_n$ and the birational map $\theta :X_n \dashrightarrow V_n$ are constructed in order to prove Lemma 2.7 and we will not use them in what follows. We give a definition of G-Mori fiber space.

Definition 2.9

Let G be a group. A G-Mori fiber space is a normal projective variety X, where G acts faithfully on X, together with a G-equivariant morphism $\pi :X \rightarrow S$ onto a normal projective variety S with the following properties:

• X is $G \mathbb {Q}$-factorial, that is, every G-invariant Weil divisor on X is $\mathbb {Q}$-Cartier, and X has only terminal singularities.

• is $\pi$-ample.

• $\dim S < \dim X$ and $\pi$ has connected fibers.

• ${\text {rank}}\, {\text {Pic}}^G (X) - {\text {rank}}\, {\text {Pic}}^G (S) = 1$.

Note that the Klein simple group $G = \mathrm {PSL}_2 (\mathbb {F}_7)$ acts on along the fibers, so that is a G-Mori fiber space for $n \geqslant 1$. For $n = 0$, is not a del Pezzo fibration. Nevertheless, we have $\rho ^G (X_0) = 1$, so that is a G-Mori fiber space as well.

3 Proof of Theorem 1.3

3.1 Reduction modulo 2

In the following, we drop the subscript n and write $P = P_n$, $X = X_n$. In the previous section, the toric variety P was defined over $\mathbb {C}$. We can define P over an arbitrary field or more generally an arbitrary ring. For a field or a ring K, we denote by $P_K$ the toric variety over ${\text {Spec}}K$ defined by the same fan as that of P. Then, since is defined over $\mathbb {Z}$, we can define the subscheme for a homogeneous polynomial of degree .

Let , $\alpha _i \in \mathbb {C}$. Assume that $\alpha _0,\dots ,\alpha _{2 n}$ are very general so that they are algebraically independent over $\mathbb {Z}$. Then, the ring is isomorphic to a polynomial ring of variables over $\mathbb {Z}$ and the ideal (2) is a prime ideal. Define

which is a DVR whose residue field is of characteristic 2. Then we can define , which is a scheme over ${\text {Spec}}R$ and whose geometric generic fiber is isomorphic to $X_{\mathbb {C}}$.

Lemma 3.1

Let $\Bbbk$ be an algebraically closed field which is uncountable. If is not ruled for a very general , then is not ruled for a very general .

Proof

Let $X'$ be the geometric special fiber of $X_R \rightarrow {\text {Spec}}R$ defined over $\Bbbk$. We can write $X' = (a' y^2 + f = 0) \subset P_{\Bbbk }$ for some and $a'$ corresponds to a very general element. By the Matsusaka theorem [9, V.1.6 Theorem], if $X'$ is not ruled, then X is not ruled. This completes the proof. $\square$

3.2 Kollár’s technique

In this subsection, we briefly recall Kollár’s argument of proving non-ruledness of suitable covering spaces in positive characteristic. We apply the following non-ruledness criterion which is a slight generalization of [9, V.5.1 Lemma].

Lemma 3.2

Let Y be a smooth proper variety defined over an algebraically closed field and $\mathscr {M}$ a big line bundle on Y. If there is an injection $\mathscr {M}\hookrightarrow (\mathrm{\Omega }_Y^i)^{\otimes m}$ for some $i > 0$ and $m > 0$, then Y is not separably uniruled.

Proof

Suppose that Y is separably uniruled. Then, there exists a separable dominant map , where V is a normal projective variety. After shrinking V, we may assume that $\varphi$ is a morphism and V is smooth. The homomorphism is an isomorphism on a non-empty open subset since $\varphi$ is separable. This induces an injection for any $k \geqslant 1$. The invertible sheaf $\mathscr {M}$ is big so that the global sections of separate points on a nonempty open subset of for a sufficiently large k. This is a contradiction since the global sections of $(\mathrm{\Omega }^i_{V \times \mathbb {P}^1})^{\otimes m k}$ do not separate points in a fiber. $\square$

Remark 3.3

Our aim is to prove that the variety $X_n$ defined over an algebraically closed field of characteristic 2 is not ruled. In view of Lemma 3.2, it is enough to construct a resolution $r :Y \rightarrow X_n$ and a big line bundle $\mathscr {M}$ which is a subsheaf of $(\mathrm{\Omega }^i_Y)^{\otimes m}$ for some $m > 0$. As we will see in Sect. 3.3, there is a purely inseparable cover $X_n \rightarrow Z_n$ of degree 2 for some normal projective variety $Z_n$. In the following we explain the Kollár’s construction of a big line bundle on a nonsingular model of a suitable cyclic covering space in a general setting.

Let Z be a variety of dimension n defined over an algebraically closed field $\Bbbk$ of characteristic $p > 0$, $\mathscr {L}$ a line bundle on Z and $s \in H^0 (Z, \mathscr {L}^{\otimes m})$ a global section of $\mathscr {L}^m$ for some $m > 0$. Let $U = {\text {Spec}}\bigoplus _{i \geqslant 0} \mathscr {L}^{- i}$ be the total space of the line bundle $\mathscr {L}$ and let $\rho _U :U \rightarrow Z$ be the natural morphism. We denote by the zero section and define

We say that is the cyclic covering of Z obtained by taking mth roots of s. Set and let $\rho = \rho _U{|}_X :X \rightarrow Z$ be the cyclic covering.

From now on we assume that Z is nonsingular and m is divisible by p. We have a natural differential whose construction is given below. Let $\tau$ be a local generator of $\mathscr {L}$ and a local section. Let $x_1,\ldots ,x_n$ be local coordinates of of Z. Then define

This is independent of the choices of local coordinates and the local generator $\tau$, and thus defines d. For the section $s \in H^0 (Z, \mathscr {L}^m)$, we can view d (s) as a sheaf homomorphism . By taking the tensor product with $\mathscr {L}^{-m}$, we obtain $d s :\mathscr {L}^{- m} \rightarrow \mathrm{\Omega }^1_Z$.

Definition 3.4

([9, V.5.8 Definition]) We define $\mathscr {Q}(\mathscr {L}, s) = ( \det \mathrm {Coker} (d s) )^{\vee \vee }$.

We have .

Lemma 3.5

([9, 5.5 Lemma]) There is an injection .

Remark 3.6

Let $x_1,\ldots ,x_n$ be local coordinates of Z at a point P and as before. Then, is generated by the form

See [9, V.5.9 Lemma] for details.

Let us show that if the singularity of X is mild, then we can lift to an invertible subsheaf of $\mathrm{\Omega }^{n-1}_Y$, where Y is a suitable nonsingular model of X. For simplicity of description, we assume that $p = 2$ and $n = \dim Z = 3$.

Definition 3.7

([9, V.5.6 Definition], see also [9, V.5.7 Exercise]) We say that $s \in H^0 (Z, \mathscr {L}^m)$ has a critical point at $P \in Z$ if vanishes at P. Denote by $\mathrm{Crit}(s) \subset Z$ the set of critical points of s. We say that s has an almost nondegenerate critical point at P if in suitable choice of local coordinates $x_1,x_2,x_3$ we can write

where $\alpha \in \Bbbk$, for a local generator $\tau$ of $\mathscr {L}$ at P, $h = h (x_1,x_2,x_3)$ consists of monomials of degree at least 3 and it does not involve $x_1^3$.

Lemma 3.8

([9, V.5.10 Proposition]) Suppose that s has only almost nondegenerate critical points. Then the singularities of X are isolated singularities and they can be resolved by blowing up each singular point of X. Moreover, if we denote by $r :Y \rightarrow X$ the blowup of each singular point of X, then ${r^*\rho ^* \mathscr {Q}(\mathscr {L},s) \hookrightarrow \mathrm{\Omega }^2_Y}$.

3.3 Construction of a big line bundle

Throughout this subsection, we work over an algebraically closed field $\Bbbk$ of characteristic 2 which is uncountable. We write $P = P_{\Bbbk }$ and $X = X_{\Bbbk }$. We do not assume $n \geqslant 5$ for the moment. Let and $X^{\circ }\! = X \cap P^{\circ }$. Note that $P^{\circ }$ is the nonsingular locus of P. Define

and set Z to be the hypersurface in Q defined by . Let $\rho :X \rightarrow Z$ be the morphism which is defined by the correspondence $z = y^2$, which is a purely inseparable finite morphism of degree 2.

Lemma 3.9

Let be a general homogeneous polynomial of degree . Then the set

consists of finitely many points and $\mathrm{Crit}(a) \cap (a = 0) = \varnothing$. Moreover, for each $P \in \mathrm{Crit}(a)$, we can choose a local coordinate w of $\mathbb {P}^1$ at P such that

for some $\alpha ,\beta \in \Bbbk$ with $\alpha \ne 0$.

Proof

The set $\mathrm{Crit}(a)$ is clearly a finite set of points. As a generality of a, we in particular require that a does not have a multiple component. It is then clear that $\mathrm{Crit}(a) \cap (a = 0) = \varnothing$. The last assertion follows by counting dimension. Let $P \in \mathbb {P}^1$ be a point and w a local coordinate of $\mathbb {P}^1$ at P. We can write , $\alpha _i \in \Bbbk$. We say that a has a bad critical point at P if $\alpha _1 = \alpha _3 = 0$. Two conditions $\alpha _1 = \alpha _3 = 0$ are imposed for a to have a bad critical point at a given $P \in \mathbb {P}^1$. Since P runs through $\mathbb {P}^1$, we see that homogeneous polynomials a which have a bad critical point at some point $P \in \mathbb {P}^1$ form at most $2 - 1 = 1$ codimensional subfamily in the space of all . Thus, a general a does not have a bad critical point at all and the proof is completed. $\square$

Lemma 3.10

The set

consists of finitely many closed points and $\mathrm{Crit}(f) \cap (f = 0) = \varnothing$. Moreover, for each $P \in \mathrm{Crit}(f)$, we can choose local coordinates $t_1, t_2$ of $\mathbb {P}^2$ at P such that

$$\begin{aligned} f = \gamma + t_1 t_2 + \text {higher order terms} \end{aligned}$$

for some $\gamma \ne 0$.

Proof

We have

By a straightforward computation, we have

where $\zeta \in \Bbbk$ is a primitive 7th root of unity. It is also straightforward to see that $f (P) \ne 0$ for $P \in \mathrm{Crit}(f)$. For the last assertion, we work on the affine open subset $U = (x_0 \ne 0) \subset \mathbb {P}^2$. Note that $\mathrm{Crit}(f) \subset U$. By setting $x_0 = 1$, we think of $x_1,x_2$ as affine coordinates of $U \cong \mathbb {A}^2$. We have on U. For the verification of the last assertion, it is enough to show that the Hessian of f at $P = (1\!:\!\zeta ^{3 i}\!:\!\zeta ^i) \in \mathrm{Crit}(f)$ is nonzero. We have $\partial ^2 f/\partial x_1^2 = 1$, $\partial ^2 f/\partial x_1 \partial x_2 = x_1^2$ and $\partial ^2 /\partial x_2^2 = 0$, so that we can compute the Hessian as

$$\begin{aligned} \left| \begin{matrix} 1 &{} \quad x_1^2 \\ x_1^2 &{} \quad 0 \end{matrix} \right| (P) = \zeta ^{12 i} \ne 0. \end{aligned}$$

Therefore, the last assertion is proved. $\square$

Set and $Z^{\circ }\! = Z \cap Q^{\circ }$.

Lemma 3.11

The quasi projective variety $Z^{\circ }$ is nonsingular.

Proof

We work on the open subset $U = U_{w_0,x_0} \subset Q$ which is an affine 4-space with coordinates $w_1,x_1,x_2,z$. We see that $Z \cap U$ is defined by , where and $f_0 = f (1,x_1,x_2)$. We have

$$\begin{aligned} \mathrm{Sing }(Z \cap U) = \biggl ( z\, \frac{a_0}{\partial w_1} = \frac{\partial f_0}{\partial x_1} = \frac{\partial f_0}{\partial x_1} = a_0 = f_0 = 0 \biggr ) = \varnothing , \end{aligned}$$

where the last equality follows since the curve $f_0 = 0$ in $\mathbb {A}^2$ is nonsingular. By symmetry, $Z \cap U_{w_i,x_j}$ is nonsingular for $i = 0,1$ and $j = 0,1,2$. Since $Z^{\circ }$ is covered by $U_{w_i,x_j}$ for $i = 0,1$ and $j = 0,1,2$, the proof is completed. $\square$

Let $H_Q$ and $F_Q$ be divisor classes on Q which correspond to the weight ${}^t (0 \ 1)$ and ${}^t (1 \ 0)$, respectively, that is, $F_Q$ is the fiber class of the projection $Q \rightarrow \mathbb {P}^1$ and $H_Q{|}_{F_Q} \in |\mathscr {O}_{\mathbb {P}(1,1,1,4)} (1)|$. We set $H_Z = H_Q{|}_Z$ and $F_Z = F_Q{|}_Z$. Define $\mathscr {L}$ to be the sheaf whose restriction on $Z^{\circ }$ is an invertible sheaf. Note that we have $z \in H^0 (Z,\mathscr {L}^2)$. It is clear that . In the following we choose and fix a general so that the assertions of Lemma 3.9 hold.

Lemma 3.12

The section $z \in H^0 (Z^{\circ }\!, \mathscr {L}^2)$ has only almost nondegenerate critical points on $Z^{\circ }$.

Proof

Let $\mathrm{Crit}(z) \subset Z^{\circ }$ be the set of critical points of z. Since

z can be chosen as a part of local coordinates at every point $P \in Z^{\circ }$ such that $a (P) = 0$. It follows that z does not have a critical point at any point $P \in X \cap (a = 0)$. We work on an open set $U \subset Z^{\circ }$ on which $a \ne 0$ and prove that has only almost nondegenerate critical points on U. Since on U and a is a unit on U, it is enough to show that has only almost nondegenerate critical points on U. Let $P \in U$ be a critical point of z. We have

for $i = 0,1$ and $j = 0,1,2$. Since $a (P) \ne 0$, we have $(\partial f/\partial x_j) (P) = 0$ for $j = 0,1,2$. By Lemma 3.10, we have $f (P) \ne 0$, which implies $(\partial a/\partial w_i) (P) = 0$ for $i = 0,1$. By Lemmas 3.9 and 3.10, we can choose local coordinates $w, t_1, t_2$ of Z at P such that

where $\alpha , \beta , \gamma \in \Bbbk$ with $\alpha ,\gamma \ne 0$, $h = h (w,t_1,t_2)$ consists of monomials of degree at least 3 and it does not involve $w^3$. This shows that z has only almost nondegenerate critical points on $Z^{\circ }$. $\square$

Define which is an invertible sheaf on $Z^{\circ }$. By Lemma 3.5, we have , where . By adjunction, we have , hence . Let $H_P$ and $F_P$ be the divisors on P which correspond to ${}^t (0 \ 1)$ and ${}^t (1 \ 0)$, respectively, so that $F_P$ is the fiber class of $\mathrm{\Pi } :P \rightarrow \mathbb {P}^1$ and . We set $H = H_P{|}_X$ and $F = F_P{|}_X$. We have $H = \rho ^*H_Z$ and $F = \rho ^*F_Z$, hence . Let $\iota :X^{\circ } \hookrightarrow X$ be the open immersion. The sheaf is a reflexive sheaf of rank 1 but is not invertible at each singular point of type . We define which is an invertible sheaf on X and we have an injection $\mathscr {M}\hookrightarrow ((\mathrm{\Omega }_X^2)^{\otimes 2})^{\vee \vee }$.

Note that X has two kinds of singularities both of which are isolated: one of them are the singular points on $X^{\circ }$ corresponding to the critical points of z and the other ones are singular points of type . Let $r :Y \rightarrow X$ be the blowup of X at each singular point. By Lemmas 3.8 and 3.12, Y is nonsingular and we have an injection on the open subset $Y^{\circ }\! = r^{-1} (X^{\circ })$. We will show that there is an injection .

Lemma 3.13

There is an injection .

Proof

Let P be a singular point of type . Since we know that on the open subset $Y^{\circ } = r^{-1} (X^{\circ })$, it is enough to show that locally around the exceptional divisor of $r :Y \rightarrow X$ over P. We can write for some local section $\eta$ of $((\mathrm{\Omega }_{X}^2)^{\otimes 2})^{\vee \vee }$ since $\mathscr {M}\subset ((\mathrm{\Omega }_{X}^2)^{\otimes 2})^{\vee \vee }$ is an invertible sheaf. We will show that

for some $g, h_1,h_2 \in \mathscr {O}_{X,P}$, and then we will show that $r^*\eta$ does not have a pole along the exceptional divisor over P.

After replacing $w_0,w_1$, we assume that $w_1$ vanishes at P (so that $w_0$ does not vanish at P). We work on an open subset U of $U_{w_0,x_0} \subset P$. Shrinking U, we assume $a \ne 0$ on U. Then . Let $\widetilde{w}_1 = w_1/w_0$, $\widetilde{x}_1 = x_1/x_0$, $\widetilde{x}_2 = x_2/x_0$ be the restrictions of $w_1,x_1,x_2$ to $U_{w_0,x_0}$. Then, in view of Remark 3.6, after further shrinking U, we see that is generated by

In particular,

where K(X) is the function field of X.

Set $\xi _i = x_i/y^{1/2}$ for $i = 0,1,2$. Then $\xi _0,\xi _1,\xi _2$ can be chosen as local coordinates of the orbifold chart of (X, P). Now we have $\widetilde{x}_i = \xi _1/\xi _0$ for $i = 1,2$, hence

Here, since the ground field is of characteristic 2 and $\xi _0^2 \in \mathscr {O}_{X,P}$, we have the equality

for $i = 1,2$. Thus , where . It follows that

for some rational function g. By [10, Lemma 5.3], we see that for some $h \in \mathscr {O}_{X,P}$.

Now, by shrinking X, we assume that $r :Y \rightarrow X$ is the blowup (more precisely, the weighted blowup with weight ) at P. Then the order of the pole of the rational 2-form

along the exceptional divisor E is at most 2 (in fact, an explicit computation shows that the above form does not have a pole along E but we do not need this strong estimate). It is clear that $r^*\xi _i^2$ vanishes along E to order 1 so that $r^* g$ vanishes along E to order at least 3. Therefore, $r^*\eta$ does not have a pole along E and we have an injection . $\square$

Lemma 3.14

If $n \geqslant 5$, then the invertible sheaf $\mathscr {M}$ is big.

Proof

Let m be a positive integer such that $m > n/(n-4)$. We show that the complete linear system of defines a birational map. Set and which are positive integers. Then

is a set of sections of $\mathscr {M}^m$ and they define a generically finite map. Indeed, the restriction of sections and for $0 \leqslant j_1, j_2 \leqslant 2$ on $X \cap U_{w_0,y}$ are $1, w_1$ and $x_i^2$ for $0 \leqslant j_1,j_2 \leqslant 2$ and they clearly define a generically finite map (in fact an isomorphism). It follows that the complete linear system of $\mathscr {M}^m$ defines a generically finite map and thus $\mathscr {M}$ is big. $\square$

Proof of Theorem 1.3

Assume that $n \geqslant 5$. By Lemmas 3.13, 3.14 and 3.2, a very general $X_n$ defined over $\Bbbk$ is not separably uniruled. In particular, it is not ruled. Then a very general $X_n$ defined over $\mathbb {C}$ is not ruled, by Lemma 3.1, and the proof is completed. $\square$