Some surfaces with canonical maps of degrees 10, 11, and 14

In this note, we present examples of complex algebraic surfaces of general type with canonical maps of degrees 10, 11, and 14. They are constructed as quotients of a product of two Fermat septics using certain free actions of the group Z72$\mathbb {Z}_7^2$ .


INTRODUCTION
Beauville has shown in [5] that if the image of the canonical map Φ   of a surface  of a general type is a surface, then the following inequality holds: Here,  is the irregularity and   is the geometric genus of .In particular, 28 ≤  is only possible if  = 0 and   = 3. Motivated by this observation, the construction of surfaces with   = 3 and canonical map of degree  for every value 2 ≤  ≤ 36 is an interesting, but still widely open problem [12,Question 5.2].In particular for most values 10 ≤ , it is not known if a surface realising that degree exists.Indeed, for a long time, the only examples were the surfaces of Persson [16], with canonical map of degree 16 and Tan [22], with degree 12.In recent years, this problem attracted the attention of many authors, putting an increased effort in the construction of new examples.As a result, previously unknown surfaces with degrees  = 12, 16, 20, 24, 32, and 36 have been discovered, by Rito [17][18][19][20], Gleissner, Pignatelli and Rito [10] and Nguyen [13,14].In this work, we present surfaces with canonical maps of degrees  = 10, 11, and 14.According to our knowledge, they are the first surfaces for those degrees.They can be described using Pardini's theory of branched abelian covers [15], which is one of the standard techniques in this subject, cf.[12].However, we decided to present them in an elementary way using plane curves and basic algebraic geometry at graduate textbook level [21].Our construction is completely selfcontained, basically reference free and easily accessible.It can be sketched as follows: the surfaces , which all have   = 3, arise as quotients of a product of two Fermat septics none of them |  | is base-point free, that is, the canonical map Φ   ∶  ⤏ ℙ 2 is just a rational map.To compute its degree, we resolve the indeterminacy by a sequence of blowups and compute the degree of the resulting morphism via elementary intersection theory.We point out that our surfaces are particular examples of surfaces isogenous to a product, that is, quotients of a product of two curves modulo a free action of a finite group.This construction goes back to an exercise in Beauville's book on Complex Algebraic Surfaces [6, Exercises X. 13 4], where a free action of ℤ 2 5 on a product of two Fermat quintics is used to construct a surface with   =  = 0.This example served as an important inspiration for our work.By now, many more surfaces isogenous to a product have been constructed.Apart from other works, that mainly deal with irregular surfaces, we want to mention the complete classification of surfaces isogenous to a product with   =  = 0 [1] and the classification for   = 1 and  = 0 under the assumption that the action is diagonal [9].However, for higher values of   , a classification of regular surfaces isogenous to a product is much more involved and is not yet established.Recently, similar constructions involving non-free actions on a product of Fermat curves have been used to provide other interesting projective manifolds that helped us to understand some important geometric phenomena.Most notably are the rigid but not infinitesimally rigid manifolds [4] of Bauer and Pignatelli that gave a negative answer to a question of Kodaira and Morrow [11, p. 45] and, to a lesser degree, also the infinite series of -dimensional infinitesimally rigid manifolds of general type with non-contractible universal cover for each  ≥ 3, provided by Frapporti and the second author of this paper [8].

THE SURFACES
Let  be the Fermat septic curve In this section, we construct a series of surfaces , as quotients of a product of two copies of , modulo a suitable diagonal action of the group ℤ 2 7 .For any surface , we determine the canonical map Φ   and compute its degree.On the first copy of , we define the action of ℤ ) .This action has 21 points with nontrivial stabilizer.They form three orbits of length 7. A representative of each orbit and a generator of the stabilizer is given by: Note that the automorphisms (, ) are precisely the deck transformations of the cover The cover has degree 49 and is branched along (0 ∶ 1), (1 ∶ 0), and (−1 ∶ 1).In particular, ∕ℤ 2 7 ≃ ℙ 1 and  is the quotient map.On the second copy of , for which we use the homogenous variables  = ( 0 ∶  1 ∶  2 ), the group acts by •, where  ∈ Aut(ℤ 2  7 ) is an automorphism depending on the specific example.The explicit choices for  are stated in the table below.Let  be the quotient  ∶= ( × )∕ℤ 2   7   modulo the diagonal action  × (•).
To write the canonical system of the corresponding unmixed quotient, we need to fix a suitable basis of the space  0 (, Ω 1  ) of global holomorphic 1-forms on .In affine coordinates, such a basis is given by where  ∶=  1  0 and  ∶=  2  0 .
Note that: I) The action of ℤ 2 7 on  0 (, Ω 1  ) under pullback with  is II) The space of canonical sections  0 (  ) is isomorphic to ) ℤ 2  7 , where the action on the tensor product is diagonal, that is, (, ) ∈ ℤ 2  7 acts via The observations I) and II) imply: ) .
We can now state and prove our main result: Theorem 1.For all  ∈ Aut(ℤ 2  7 ) in the table below, the diagonal action  × (•) of ℤ 2 7 on the product of two Fermat septics is free.The quotient is a regular smooth projective surface  of general type with   = 3.A basis of  0 (  ), the canonical map Φ   in projective coordinates and its degree are stated in the table: ( 5 4 6 5 3. ) ( 2 6 1 4 ( ) ( 2 (  Remark 1.The surfaces  in the table from above are examples of unmixed surfaces isogenous to a product of curves, that is, surfaces isomorphic to a quotient of a product of two curves  1 ×  2 of genus (  ) ≥ 2 modulo a free and diagonal action of a finite group .More precisely, they are Beauville surfaces, which are by definition the rigid surfaces isogenous to a product.Rigidity means that they do not admit nontrivial deformations.This is equivalent to the fact that the quotient curves   ∕ are isomorphic to ℙ 1 and the quotient maps   →   ∕ ≃ ℙ 1 are branched in three points.Such surfaces can be described purely in terms of group theory, cf.[3].Using the MAGMA [7] algorithm from the paper [10], one can classify all regular unmixed surfaces isogenous to a product of curves with   = 3 and abelian group .Among them are the unmixed Beauville surfaces with   = 3 and abelian group.The latter form seven biholomorphism classes, which are exactly the surfaces in the table of our theorem.
To compute the degree of the canonical map of each of our surfaces, we use the following lemma.Proof.We prove the lemma by induction on (, ) with  ≤ .Here, we are considering the lexicographic order ≤ defined on the lower diagonal Δ ≥ ∶= {(, ) ∶  ≥ } ⊆ ℕ × ℕ as follows: ( ′ ,  ′ ) ≤ (, ) if and only if  ′ <  or  ′ =  and  ′ ≤ .
Suppose that (, ) = 0.Then,  is not a base-point of || and so M =  2 =  2 − .Now suppose that the statement is true for ( ′ ,  ′ ) < (, ).Our aim is to prove it for (, ).We blow up the point , take the pullback of the divisors   and remove the fixed part, which is the exceptional divisor  of the blowup.In fact, the pullback of  3 contains  +  times  and  +  ≥ , thanks to the assumptions  +  ≥  or  ≥ : if  ≥ , then  +  ≥ , otherwise if  <  and  +  ≥ , then Restricted to the preimage of our neighborhood of , these divisors are: We need to distinguish two cases, when  = 1 or when  > 1.In the first case  −  =  < , so that (, ) < (, ).We can write  =  ′  +  ′ , with 0 ≤  ′ <  and define new coefficients  ′ ∶= ,  ′ ∶= ,  ′ ∶=  and  ′ ∶=  +  − .Then, they fulfill the inductive hypothesis, because: By induction, the self-intersection of the new linear system M is equal to In the case  > 1, then  ≤  −  and ( − , ) < (, ).We define new variables  ′ ∶=  − ,  ′ ∶= ,  ′ ∶=  +  − , and  ′ ∶= .Observe that  ′ =  −  = ( − 1) ′ +  and we can define  ′ ∶=  − 1.They satisfy the inductive hypothesis, because of the estimations: Hence, the self-intersection of the new linear system M is equal to Proof of Theorem 1. First, we show that the three diagonal actions  × (•) on  ×  are free.Indeed, as remarked above, the non-trivial stabilizers of the points on the first copy of  are generated by (1,0), ( . However, none of these elements have a fixed point on the second copy of  under the twisted actions •.Thus, the actions are free and the quotient surfaces  are smooth, projective, and of general type.The latter holds because the genus of the Fermat septic is () = 15 ≥ 2.Moreover, they are regular surfaces, because as ∕ℤ 2 7 is biholomorphic to ℙ 1 .The geometric genus of each  is therefore equal to Using Lemma 1, we compute a basis of  0 (  ) for each surface .Replacing the affine variables by Here, | M| is a base-point free linear system.The self-intersection M2 is positive if and only if Φ M is not composed with a pencil.In this case, Φ M is onto and it holds: For the computation of the resolution, it is convenient to write the divisors of the bi-quartics defining Φ   as linear combinations of the reduced curves   ∶= {  = 0} and   ∶= {  = 0} on .Note that   and   intersect transversally in only one point and (  ,   ) = (  ,   ) = 0, for all , .Thus, these curves can be illustrated as a grid of three vertical and three horizontal lines.
Consider the third surface in the table.Here, the divisors of the three bi-quartics spanning the canonical system |  | are: The degree of the canonical map is therefore given by The degree of the canonical map of all of the other surfaces can be computed in the same way, except for the second one that we treat separately.Here, the fixed part of |  | is  1 +  1 and the mobile part is spanned by three divisors 3 0 +  1 + 2 2 ,  0 +  1 +  2 + 2 1 +  2 , and 3 1 + 3 0 .

A C K N O W L E D G M E N T S
The authors would like to thank Roberto Pignatelli, Massimiliano Alessandro, and Stephen Coughlan for useful comments and discussions.A special thanks goes to the organizers of the conference Algabraic Geometry in Roma Tre, on the occasion of Sandro Verra's 70th birthday (June 2022), that enabled us to participate, present this work as a poster and provided financial support for our stay in Rome.The poster can be found here: http://ricerca.mat.uniroma3.it/users/moduli/verra70/Fallucca-Gleissner.pdf.

The degrees 5
and 7 of the first and second surfaces have also been realized by a different construction [12, Example 4.5].

Lemma 2 .
Let || be a two-dimensional linear system on a surface  spanned by  1 ,  2 , and  3 .Assume that  has at most isolated base-points.Given a point  of , suppose we can write the divisors   around  as  1 = ,  2 =  and  3 =  + , where  and  are reduced, smooth, and intersect transversally at  and , , ,  are non-negative integers,  ≤ .Assume that •  ≥  or •  ≠ 0 and  +  ≥ , where  =  +  with 0 ≤  < .Then after blowing up at most ()-times we obtain a new linear system | M| such that no infinitely near point of  is a base-point of | M|.Moreover, M2 =  2 − .
Ĥ + ( − ),  K and  Ĥ +  K + ( +  − ).Here, Ĥ and K are the strict transforms of  and .Let | M| be the linear system generated by these three divisors, then M2 =  2 −  2 .If  =  or  = 0, then | M| is base-point free and we are done.Otherwise, on the preimage, the linear system | M| has precisely one new base-point: the intersection point of K and .Locally near this point the three divisors spanning | M| are: ( − ),  K and  K + ( +  − ).