A new example of an algebraic surface with canonical map of degree 16

TL;DR

This paper constructs a minimal algebraic surface of general type with specific invariants, featuring a canonical map that is an abelian cover of degree 16 of P^1 x P^1, illustrating a new example in algebraic geometry.

Contribution

It presents a novel example of a minimal surface with a canonical map of degree 16, expanding known classifications of algebraic surfaces.

Findings

Constructed a minimal surface with p_g=4, K^2=32, q=1

Canonical map is an abelian cover of degree 16 of P^1 x P^1

Provides a new example of algebraic surface with these properties

Abstract

In this note, we construct a minimal surface of general type with geometric genus p g = 4, self-intersection of the canonical divisor K^2 = 32 and irregularity q = 1 such that its canonical map is an abelian cover of degree 16 of P^1 x P^1.

Equations

$$16\left(p_g-2\right)\le 16\mathrm{deg}\left(\mathrm{im}\left({\phi }_{|K_X|}\right)\right)\le K_X^2\le 9\chi \left({\mathcal{O}}_X\right)=9p_g-9q+9.$$

$$K_X^2=32,p_g\left(X\right)=4,q\left(X\right)=1$$

$${\chi }_{j_1{j}_2{j}_3{j}_4}\left(a_1,a_2,a_3,a_4\right):=e^{\left(\pi a_1{j}_1\right)i}{e}^{\left(\pi a_2{j}_2\right)i}{e}^{\left(\pi a_3{j}_3\right)i}{e}^{\left(\pi a_4{j}_4\right)i}$$

$$2L_{\chi }\equiv \sum _{\chi \left(\sigma \right)=-1}{D}_{\sigma }$$

$$2L_{1000}$$

$$2K_X$$

$$K_X^2$$

$$p_g\left(X\right)$$

$$\chi \left({\mathcal{O}}_X\right)$$

$$L_{0001}\equiv$$

$$2K_X$$

$$K_X^2=32,p_g\left(X\right)=4,q\left(X\right)=1,\chi \left({\mathcal{O}}_X\right)=4.$$

$$H^0\left(X,K_X\right)=H^0\left({\mathbb{P}}^1\times {\mathbb{P}}^1,K_{{\mathbb{P}}^1\times {\mathbb{P}}^1}\right)\oplus \underset{\chi \ne {\chi }_{0000}}{⨁}{H}^0\left({\mathbb{P}}^1\times {\mathbb{P}}^1,K_{{\mathbb{P}}^1\times {\mathbb{P}}^1}+L_{\chi }\right)$$

$$h^0\left(K_{{\mathbb{P}}^1\times {\mathbb{P}}^1}+L_{\chi }\right)=0$$

$$B_1:$$

$$\equiv 2L_{1100}$$

$$\equiv 6F+6G.$$

$$K_{X_1}$$

$$⟨\left(1,0,0,0\right),\left(0,1,0,0\right),\left(0,0,0,1\right)⟩$$

Full text

**Mathematics Subject Classification (2010):**14J29.

Keywords: Surfaces of general type, Canonical maps, Abelian covers

1 Introduction

Let $X$ be a minimal smooth complex surface of general type (see [4] or [2]) and denote by \textstyle{\varphi_{\left|K_{X}\right|}:X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}^{p_{g}\left(X\right)-1}} the canonical map of $X$, where $p_{g}\left(X\right)=\dim\left(H^{0}\left(X,K_{X}\right)\right)$ is the geometric genus and $\left|K_{X}\right|$ is the canonical system of $X$. A classical result of A. Beauville [3, Proposition 4.1 and its proof] says that if the image of $\varphi_{\left|K_{X}\right|}$ is a surface, the degree $d$ of the canonical map of $X$ is less than or equal to $36$ and that for large invariants the degree of the canonical map is less than or equal to $9$. Later, G. Xiao improved this result by showing that if the geometric genus of $X$ is bigger than $132$, the degree of the canonical map is less than or equal to $8$ (see [14, Theorem 3]). Only few surfaces with $d$ greater than $8$ have been known so far, such as: S. L. Tan’s example [13] with $d=9$, U. Persson’s example [9] with $d=16$ and C. Rito’s examples [10], [12], [11] with $d=12,16,24$. There is a recent preprint [6] of C. Gleissner, R. Pignatelli and C. Rito constructing surfaces with $d=24$, $q=1$ and $d=32$, $q=0$, where $q=\dim\left(H^{1}\left(X,K_{X}\right)\right)$ is the irregularity of $X$.

When the canonical map has degree $16$, by the Bogomolov-Miyaoka-Yau inequality, the geometric genus is less than or equal to $5$ and the irregularity is at most $2$. Indeed, we have

[TABLE]

So $p_{g}\leq 5$. In addition, since $p_{g}\geq 3$, we get $q\leq 2$.

In 1977, U. Persson gave an example with $d=16$, $K^{2}=16$, $p_{g}=3$ and $q=0$ [9]. And in 2017, C. Rito constructed a surface with $d=16$, $K^{2}=16$, $p_{g}=3$ and $q=2$ [12]. As far as we know, these are all the known examples of surfaces with $d=16$. In this note, we construct a minimal surface of general type with $d=16$, $K^{2}=32$, $p_{g}=4$ and $q=1$. It is worth mentioning that the canonical map of the surface constructed by U. Persson is an abelian cover of $\mathbb{P}^{2}$. In [5], R. Du and Y. Gao showed that if the canonical map is an abelian cover of $\mathbb{P}^{2}$ of degree $d>8$, then $d=9$ or $16$.

In our (very simple) construction, the canonical map is an abelian cover of $\mathbb{P}^{1}\times\mathbb{P}^{1}$. The following theorem is the result of this note:

Theorem 1**.**

There exists a minimal surface of general type $X$ satisfying

[TABLE]

such that the canonical map $\varphi_{\left|K_{X}\right|}$ is a $\mathbb{Z}_{2}^{4}-$cover of $\mathbb{P}^{1}\times\mathbb{P}^{1}$.

We construct this surface by taking a $\mathbb{Z}_{2}^{4}-$cover of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ branched in some fibres of the two rulings of $\mathbb{P}^{1}\times\mathbb{P}^{1}$.

Notation and conventions: All surfaces are projective algebraic over the complex numbers. Linear equivalence of divisors is denoted by $\equiv$. The rest of the notation is standard in algebraic geometry.

2 $\mathbb{Z}_{2}^{4}-$covers

The construction of abelian covers was studied by R. Pardini in [7]. For details about the building data of abelian covers we refer the reader to Section 1 and Section 2 of R. Pardini’s work ([7]).

We will denote by $\chi_{j_{1}j_{2}j_{3}j_{4}}$ the character of $\mathbb{Z}_{2}^{4}$ defined by

[TABLE]

for all $j_{1},j_{2},j_{3},j_{4},a_{1},a_{2},a_{3},a_{4}\in\mathbb{Z}_{2}$.

We can define $\mathbb{Z}_{2}^{4}-$covers as follows (see [7, Theorem 2.1] and [1, Remark 1.4] ):

Proposition 1**.**

Given $Y$ a smooth projective surface, let $L_{\chi}$ be divisors of $Y$ such that $L_{\chi}\not\equiv\mathcal{O}_{Y}$ for all nontrivial characters $\chi$ of $\mathbb{Z}_{2}^{4}$ and let $D_{\sigma}$ be effective divisors of $Y$ for all $\sigma\in\mathbb{Z}_{2}^{4}\setminus\left\{\left(0,0,0,0\right)\right\}$ such that the branch divisor $B:=\sum\limits_{\sigma\neq 0}{D_{\sigma}}$ is reduced. Then $\left\{L_{\chi},D_{\sigma}\right\}_{\chi,\sigma}$ is the building data of a normal $\mathbb{Z}_{2}^{4}-$cover \textstyle{f:X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y} if and only if

[TABLE]

for all nontrivial characters $\chi$ of $\mathbb{Z}_{2}^{4}$ and for all $\sigma\in\mathbb{Z}_{2}^{4}\setminus\left\{\left(0,0,0,0\right)\right\}$.

For the reader’s convenience, we leave here the relations of the reduced building data of $\mathbb{Z}_{2}^{4}-$ covers:

[TABLE]

By [7, Theorem 3.1] if each $D_{\sigma}$ is smooth and $B$ is a normal crossings divisor, then $X$ is a smooth surface.

From [7, Lemma 4.2, Proposition 4.2] one has:

Proposition 2**.**

If  $Y$ is a smooth surface and \textstyle{f:X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y} is a smooth $\mathbb{Z}_{2}^{4}-$cover with building data $\left\{L_{\chi},D_{\sigma}\right\}_{\chi,\sigma}$, the surface $X$ satisfies the following:

[TABLE]

3 Construction

Denote by $F=\left\{0\right\}\times\mathbb{P}^{1}$ and $G=\mathbb{P}^{1}\times\left\{0\right\}$ the generators of $\operatorname{Pic}\left(\mathbb{P}^{1}\times\mathbb{P}^{1}\right)$. Let $D_{1000}$, $D_{0101}$, $D_{0100}\in\left|2F\right|$, $D_{1001}\in\left|2G\right|$ and $D_{1011}$, $D_{1010}$, $D_{0111}$, $D_{0110}\in\left|G\right|$ be smooth distinct divisors of $\mathbb{P}^{1}\times\mathbb{P}^{1}$. Let \textstyle{f:X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}^{1}\times\mathbb{P}^{1}} be the $\mathbb{Z}^{4}_{2}-$cover with the branch locus $B=\sum\limits_{\sigma\neq 0}{D_{\sigma}}$, where $D_{\sigma}=0$ for the rest of $D_{\sigma}$. The building data is as follows:

[TABLE]

Since each $D_{\sigma}$ is smooth and $B$ is a normal crossings divisor, $X$ is smooth. From Proposition 2, we get

[TABLE]

This implies that $X$ is also minimal and of general type. Furthermore, from Proposition 2, $X$ has the following invariants:

[TABLE]

We show that the canonical map is an abelian cover of degree $16$. We have the $\mathbb{Z}_{2}^{4}-$equivariant decomposition

[TABLE]

where the group $\mathbb{Z}_{2}^{4}$ acts on $H^{0}\left(\mathbb{P}^{1}\times\mathbb{P}^{1},K_{\mathbb{P}^{1}\times\mathbb{P}^{1}}+L_{\chi}\right)$ via the character $\chi$ (see [7, Proposition 4.1c]).

We consider the subgroup $H:=\left\langle\left(0,0,0,1\right),\left(0,0,1,0\right),\left(1,1,0,0\right)\right\rangle$ of $\mathbb{Z}_{2}^{4}$. Let $H^{\perp}$ denote the kernel of the restriction map \textstyle{\left(\mathbb{Z}_{2}^{4}\right)^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{*}} , where $\left(\mathbb{Z}_{2}^{4}\right)^{*}$ and $H^{*}$ are the character groups of $\mathbb{Z}_{2}^{4}$ and $H$, respectively. We have $H^{\perp}=\left\langle\chi_{1100}\right\rangle$. Because

[TABLE]

for all $\chi\notin H^{\perp}$, the subgroup $H$ acts trivially on $H^{0}\left(X,K_{X}\right)$. So the canonical map $\varphi_{\left|K_{X}\right|}$ is the composition of the quotient map g:\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 7.53471pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-7.53471pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 31.53471pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 31.53471pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{X_{1}:=X/H}}}}}}}}\ignorespaces}}}}\ignorespaces with the canonical map $\varphi_{\left|K_{X_{1}}\right|}$ of $X_{1}$ (see e.g. [8, Example 2.1]).

The surface $X_{1}$ is the $\mathbb{Z}_{2}-$cover \textstyle{f_{1}:X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}^{1}\times\mathbb{P}^{1}} branched on

[TABLE]

So $X_{1}$ is a surface of general type with $p_{g}\left(X_{1}\right)=4$ whose only singularities are $36$ nodes. Since

[TABLE]

the linear system $\left|K_{X_{1}}\right|$ is base point free and the map \textstyle{f_{1}:X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}^{1}\times\mathbb{P}^{1}} is the canonical map of $X_{1}$.

As the quotient map g:\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 7.53471pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-7.53471pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 31.53471pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 31.53471pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{X_{1}}}}}}}}}\ignorespaces}}}}\ignorespaces ramifies only on nodes, the canonical class $K_{X}$ is the pullback of $K_{X_{1}}$ and so the linear system $\left|K_{X}\right|$ is base point free. Therefore the canonical map $\varphi_{\left|K_{X}\right|}$ coincides with the $\mathbb{Z}_{2}^{4}-$cover \textstyle{f:X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{P}^{1}\times\mathbb{P}^{1}} .

Remark 1**.**

The $\mathbb{Z}_{2}^{3}-$quotient of $X$ by the subgroup

[TABLE]

is the product $\mathbb{P}^{1}\times E$, where $E$ is an elliptic curve. Note that also the example with $p_{g}=3$, $d=24$ by C. Gleissner, R. Pignatelli and C. Rito [6] is obtained as $\mathbb{Z}_{2}^{3}-$cover of such a product. It is easy to check that the Albanese pencil of $X$ has genus $5$.

Acknowledgments

The author is supported by Fundação para a Ciência e Tecnologia (FCT), Portugal under the framework of the program Lisbon Mathematics PhD (LisMath), Programa de Doutoramento FCT. The author is deeply indebted to Margarida Mendes Lopes for all her help and thanks Carlos Rito for many interesting conversations and suggestions. Thanks are also due to the anonymous referee for his/her thorough reading of the paper and suggestions.