On the multicanonical systems of quasi-elliptic surfaces in characteristic 3

TL;DR

This paper investigates the properties of multicanonical systems on quasi-elliptic surfaces in characteristic 3, establishing that for m ≥ 5, these systems define a quasi-elliptic fiber space, with 5 being the minimal such value.

Contribution

It proves that for quasi-elliptic surfaces in characteristic 3, the multicanonical system |mK_S| induces a fiber space structure for all m ≥ 5, and 5 is the optimal bound.

Findings

|mK_S| defines a fiber space for m ≥ 5

The bound m=5 is optimal for such surfaces

Characterization of multicanonical systems in characteristic 3

Abstract

We consider the multicanonical systems $|mK_S|$ of quasi-elliptic surfaces with Kodaira dimension $1$ in characteristic 3. We show that for any $m \geq 5$ $|mK_S|$ gives the structure of quasi-elliptic fiber space, and the number $5$ is best possible to give the structure for any such surfaces.

Equations

$$K_S\sim {\phi }^{\ast }(K_B-\mathbf{f})+\sum _{i=1}^{\lambda }{a}_i{F}_i,$$

$$\begin{array}[]{rccc}\varphi:&\quad S&\longrightarrow&B\\ &\psi\downarrow&&\downarrow\\ &{\rm Alb}(S)&\longrightarrow&{\rm J}(B)\\ &\quad\cup&&\cup\\ &\quad\psi(S)&&B.\end{array}$$

$$12\chi ({\mathcal{O}}_S)=c_1(S{)}^2+c_2(S)=2-4q(S)+b_2(S).$$

$$t^2(t-1{)}^{2}z+t^2(t-1{)}^2+x^3+t{z}^3=0$$

$$\begin{array}[]{rl}K_{S}&\sim\varphi^{*}(K_{{\bf P}^{1}}-{\bf f})+2F_{0}+2F_{1}\\ &\sim-F+2F_{0}+2F_{1}.\end{array}$$

$$(\ast )2g-2+\chi ({\mathcal{O}}_S)+t+\sum _{i=1}^{\lambda }(a_i/3)>0.$$

$$(\ast \ast )m(2g-2+\chi ({\mathcal{O}}_S)+t)+\sum _{i=1}^{\lambda }[m{a}_i/3]\ge 2g+1$$

Full text

Research of the author is partially supported by Grant-in-Aid for Scientific Research (B) No. 15H03614.

Dedicated to Professor Piotr Pragacz on the occasion of his sixtieth birthday

1. Introduction

Let k be an algebraically closed field in characteristic $p>0$, and let $\varphi:S\longrightarrow B$ be an elliptic surface with Kodaira dimension $\kappa(S)=1$ over $k$. Let $K_{S}$ be a canonical divisor of $S$. We consider the multicanonical system $|mK_{S}|$. In Katsura-Ueno [4] and Katsura [3], we considered the following question:

Question 1.1*.*

Is there a positive integer $M$ such that if $m\geq M$, then the multicanonical system $|mK_{S}|$ gives a structure of elliptic surface for any elliptic surface $S$ over $k$ with $\kappa(S)=1$?

For the complex analytic case, Iitaka showed that $M=86$ and 86 is best possible (cf. Iitaka [2]). Namely, if $m$ is smaller than 86, then there exists an elliptic surface $\varphi:S\longrightarrow B$ with $\kappa(S)=1$ such that $|mK_{S}|$ does not give the structure of elliptic surface. In the case of algebraic elliptic surfaces, if the characteristic $p=0$ or $p\geq 3$, we showed that $M=14$ and 14 is best possible (Katsura-Ueno [4] and Katsura [3]). If $p=2$, then we showed that $M=12$ and 12 is best possible (cf. Katsura [3]).

In this paper, we treat quasi-elliptic surfaces $\varphi:S\longrightarrow B$ and consider the following similar question.

Question 1.2*.*

Is there a positive integer $M$ such that if $m\geq M$, then the multicanonical system $|mK_{S}|$ gives the structure of quasi-elliptic surface for any quasi-elliptic surface $S$ over $k$ with $\kappa(S)=1$?

Note that quasi-elliptic surfaces exist only in characterisitics $p=2$ and $3$ (cf. Bombieri-Mumford [1]). We show that if $p=3$, then we have $M=5$ and the number $5$ is best possible. In characteristic $p=2$, we have still some difficulties to determine the best possible number.

In Section 2, we summarize basic facts on quasi-elliptic surfaces. In Section 3, we examine the multicanonical system of quasi-elliptic surfaces in characteristic 3 and show our main theorem.

2. Some lemmas for quasi-elliptic surfaces

Let k be an algebraically closed field in characteristic $p>0$ and let $\varphi:S\longrightarrow B$ be a quasi-elliptic surface defined over $k$. Throughout this paper, we assume that any exceptional curve of the first kind is not contained in fibers. Such a surface exists if and only if $p$ is equal to 2 or 3, and the multiplicities of multiple fibers are all equal to $p$ (cf. Bombieri-Mumford [1]). We denote by $g$ the genus of the curve $B$. In this section, we recall some facts on quasi-elliptic surfaces. We denote by $pF_{i}$ $(i=1,\ldots,\lambda)$ the multiple fibers. Let ${\mathcal{T}}$ be the torsion part of $R^{1}\varphi_{*}{\mathcal{O}}_{S}$. Then, there exists a Cartier divisor ${\bf f}$ on $B$ such that $R^{1}\varphi_{*}{\mathcal{O}}_{S}/{\mathcal{T}}\cong{\mathcal{O}}_{B}({\bf f})$. The canonical divisor formula of $S$ is given by

[TABLE]

where $-{\text{deg}}{\bf f}=\chi({{\mathcal{O}}_{S}})+t$ with $t=$ the length of the torsion part of $R^{1}\varphi_{*}{\mathcal{O}}_{S}$ and $0\leq a_{i}\leq p-1$. Here, $\sim$ means linear equivalence. If $pF_{i}$ is a tame multiple fiber, then we have $a_{i}=p-1$. For details, see Bombieri-Mumford [1].

Lemma 2.1**.**

The Albanese variety ${\rm Alb}(S)$ of $S$ is isomorphic to the Jacobian variety ${\rm J}(B)$ of $B$.

Proof.

Let $\psi:S\longrightarrow{\rm Alb}(S)$ be the Albanese mapping. If ${\rm Alb}(S)$ is a point, then by the universality of Albanese variety we see that the Jacobian variety ${\rm J}(B)$ of $B$ is also a point. Now, assume ${\rm Alb}(S)$ is not a point. Since the general fiber of $\varphi$ is a rational curve with one cusp, the fibers are contracted by $\psi$. Therefore, $\psi(S)$ is a curve. We have, by the universality of Albanese variety, a commutative diagram:

[TABLE]

By this diagram, we have a morphism $\psi(S)\longrightarrow B$. Therefore, by the Stein factorization theorem, we see that $\psi(S)$ is isomorphic to $B$. Therefore, by the universality of Jacobian variety, we conclude ${\rm Alb}(S)\cong{\rm J}(B)$ (see also Katsura-Ueno [4], Lemma 3.4).

We find the following lemma and corollary in Lang [5] and Raynaud [6]. We give here an easy proof for the lemma.

Lemma 2.2**.**

Let $\varphi:S\longrightarrow B$ be a quasi-elliptic surface over a non-singular complete curve $B$ with genus $g$. Then, we have the inequality $\chi({\mathcal{O}}_{S})\geq(1-g)/3$.

Proof.

By Noether’s formula and the self-intersection number $c_{1}(S)^{2}=0$ of the first Chern class of $S$, we have

[TABLE]

By Lemma 2.1, we have $q(S)=g$. Denoting by $\rho(S)$ the Picard number of $S$, we have also $b_{2}(S)\geq\rho(S)\geq 2$. Hence, we have $\chi({\mathcal{O}}_{S})\geq(1-g)/3$.

Corollary 2.3**.**

(i)* If $g=1$, then $\chi({\mathcal{O}}_{S})\geq 0$.*

(ii)* If $g=0$, then $\chi({\mathcal{O}}_{S})\geq 1$.*

3. Multicanonical systems

In this section, let $k$ be an algebraically closed field of characteristic 3. Let $\varphi:S\longrightarrow B$ be a quasi-elliptic surface defined over $k$.

Example 3.1**.**

In characteristic 3, we consider the quasi-elliptic surface $\varphi:S\longrightarrow{\bf P}^{1}$ which is given by a non-singular complete model of the surface defined by

[TABLE]

Here, $t$ is a parameter of the base curve ${\bf P}^{1}$. By Lang [5] p.485, this surface has two tame multiple fibers at $t=0,1$, and we have $\chi({\mathcal{O}}_{S})=1$. We denote the two tame multiple fibers by $3F_{0}$ and $3F_{1}$. The canonical divisor $K_{S}$ is given by

[TABLE]

Here, ${\bf f}$ is a Cartier divisor on ${\bf P}^{1}$ with $-{\text{deg}}{\bf f}=\chi({\mathcal{O}}_{S})=1$ such that ${\mathcal{O}}_{{\bf P}^{1}}({\bf f})\cong R^{1}\varphi_{*}{\mathcal{O}}_{S}$, and $F$ is a general fiber of $\varphi:S\longrightarrow{\bf P}^{1}$. Since we have $F\sim 3F_{0}\sim 3F_{1}$, we see $4K_{S}\sim 2F_{0}+2F_{1}$. Therefore, we have $\dim{\rm H}^{0}(S,{\mathcal{O}}_{S}(4K_{S}))=1$, and $|4K_{S}|$ does not give the structure of quasi-elliptic surface. If $m\geq 5$, then we have $\dim{\rm H}^{0}(S,{\mathcal{O}}_{S}(mK_{S}))\geq 2$, and $|mK_{S}|$ gives the structure of quasi-elliptic surface.

We have the following theorem.

Theorem 3.2**.**

Assume that the characteristic $p=3$. Then, for any quasi-elliptic surface $f:S\to B$ with $\kappa(S)=1$ over $k$ and for any $m\geq 5$, the multicanonical system $|mK_{S}|$ gives the unique structure of quasi-elliptic surface, and the number 5 is best possible.

Proof.

The method of the proof is similar to the one in Iitaka [2], Katsura-Ueno [4] and Katsura [3]. Since the Kodaira dimension is equal to 1, the structure of quasi-elliptic surface is unique. The Kodaira dimension of $S$ is equal to 1 if and only if

[TABLE]

Therefore, we need to find the least integer $m$ such that

[TABLE]

holds under the condition $(*)$. Here, $[r]$ means the integral part of a real number $r$. We have the following 6 cases:

Case (I) $g\geq 2$

Case (II-1) $g=1,\chi({\mathcal{O}}_{S})+t\geq 1$

Case (II-2) $g=1,\chi({\mathcal{O}}_{S})=0,t=0$

Case (III-1) $g=0,\chi({\mathcal{O}}_{S})+t\geq 3$

Case (III-2) $g=0,\chi({\mathcal{O}}_{S})+t=2$

Case (III-3) $g=0,\chi({\mathcal{O}}_{S})=1,t=0$

We check $(**)$ under the condition $(*)$ for each case. In Case (I), by Lemma 2.2, we have $2g-2+\chi({\mathcal{O}}_{S})\geq 5(g-1)/3$. Hence, if $m\geq 3$, $(**)$ holds. In Case (II-1), if $m\geq 3$, $(**)$ holds. In Case (II-2), all multiple fibers are tame in this case, and we have at least one multiple fiber by $(*)$. Since $a_{i}=2$, $(**)$ holds for $m\geq 5$. In Case (III-1), $(**)$ holds for $m\geq 1$. In Case (III-2), since $\chi({\mathcal{O}}_{S})\geq 1$ by Corollary 2.3, we have $t\leq 1$. Therefore, the number of wild fibers is less than or equal to $1$. If there exists at least one tame multiple fiber, then $(**)$ holds for $m\geq 2$. If there exist no tame fibers and only one wild fiber, then by $(*)$ we have $a_{1}\geq 1$. Therefore, $(**)$ holds for $m\geq 3$. In Case (III-3), all multiple fibers are tame, and we have $\lambda\geq 2$ by $(*)$. Therefore, $(**)$ holds for $m\geq 5$. The result on the best possible number in characteristic 3 follows from Example 3.1.

In characteristic 2, we can also consider a similar question to the one in characteristic 3. We have still difficulties to decide the best possible number. For example we need to solve the following question.

Question 3.3*.*

Does there exist a quasi-elliptic surface over an elliptic curve with only one tame multiple fiber and with $\chi({\mathcal{O}}_{S})=0$ in characteristic 2?

If there don’t exist such quasi-elliptic surfaces, then we can show that in characteristic 2, $(**)$ holds for $m\geq 4$ and that the best possible number is equal to $4$. Namely, we have $M=4$ in characteristic 2.