Examples of surfaces with canonical map of degree 4

We give two examples of surfaces with canonical map of degree 4 onto a canonical surface.


Introduction
Let S be a smooth minimal surface of general type with geometric genus p g ≥ 3. Denote by φ : S P p g −1 the canonical map and let d := deg(φ). The following result of Beauville is well-known.

Moreover, in case (A) d ≤ 36 and in case (B) d ≤ 9.
The question of which pairs (d, p g ) can actually occur has been object of study for some authors. Several examples were given for case (A), but case (B) is still mysterious. It is known that if d > 3, then p g ≤ 12, but so far only the case (d, p g ) = (5, 4) has been shown to exist (independently by Tan [Tan92] and by Pardini [Par91b]). We refer the recent preprint by Mendes Lopes and Pardini [MLP21] for a more detailed account on the subject. They leave some open problems, this note is motivated by their last question.
Question. For what pairs (d, p g ), with d > 3, are there examples of surfaces in case (B) of Theorem 1.1?
Here we give examples for the cases (d, p g ) = (4, 5) and (4, 7), with canonical images a 40-nodal complete intersection surface in P 4 and a 48-nodal complete intersection surface in P 6 , respectively (Beauville also paid some attention to such nodal surfaces, see [Bea17]).
The strategy for the construction is the following. If X is a surface with nodes admitting a Galois covering Y → X ramified over the nodes and with Galois group G, a group with a "big" number of subgroups, then we have a "big" number of intermediate coverings of X. By computing the geometric genus p g of all involved surfaces, we may hope to find some ρ : W → Z with p g (W ) = p g (Z), hence such that the canonical map of W factors through ρ.
We work explicitely with the equations of a 40-nodal surface from [RRS19], all computations are implemented with Magma [BCP97].

Notation
As usual the holomorphic Euler characteristic of a surface S is denoted by χ(S), the geometric genus by p g (S), the irregularity by q(S), and a canonical divisor by K S . A (−m)-curve is a curve isomorphic to P 1 with self-intersection −m. A node of S is an ordinary double point of S. We say that a set of nodes of S is 2-divisible if the sum A i of the corresponding (−2)-curves in the smooth minimal model of S is 2-divisible in the Picard group.
Proposition 2.1. A normal finite G (Z/2) r -covering π : Y → X of a smooth variety X is completely determined by the datum of (1) reduced effective divisors D σ , for all σ ∈ G, with no common components; (2) divisor linear equivalence classes L χ 1 , . . . , L χ r , for χ 1 , . . . , χ r a basis of the group of characters G ∨ , such that D σ (with additive notation for the characters ).
Conversely, given (1) and (2), one obtains a normal scheme Y with a finite G (Z/2) r -covering Y → X, with branch curves the divisors D σ .
From now on, we assume that X and Y are surfaces. If each D σ is smooth and D σ has simple normal crossings, then Y is smooth and its invariants are (2.1) Let R σ be the support of π * (D σ ). The Hurwitz formula gives Now assume that the D σ are disjoint (−2)-curves. Then the R σ are disjoint (−1)-curves, the canonical map of Y factors through the covering Y → X if and only if p g (Y ) = p g (X), and one has a commutative diagram the surface X has nodes corresponding to the (−2)-curves of X, and Y → X is a (Z/2) r -covering ramified on those nodes. In this case Equation (2.1) becomes where m is the number of nodes of X .

Construction
Let X 40 be the surface in P 4 given by the equations It is the canonical model of a surface with invariants p g = 5, q = 0 and K 2 = 8. The above quartic I is classically known as the Igusa quartic; its singular set is the union of 15 lines. The quadric meets these lines transversally, and is tangent to I at 10 smooth points, thus the singular set of X 40 is the union of 40 nodes N 1 , . . . , N 40 (for more details see [RRS19]).
Let X 40 be the smooth minimal model of X 40 and denote by A i the (−2)-curves in X 40 corresponding to the nodes N i , i = 1, . . . , 40. Let a, b, c be the canonical generators of the group (Z/2) 3 and, for i, j, k ∈ Z/2, let χ ijk denote the character which takes the value i, j, k on a, b, c, respectively. We show in Section 4.1 that one can write abc is a sum of 4 (−2)-curves, each of D bc , D ac , D ab is a sum of 8 (−2)-curves, and such that there exist divisors L 100 , L 010 , L 001 satisfying: It follows from Proposition 2.1 that these data define a (Z/2) 3 -covering π : Y → X 40 branched on the (−2)-curves A i , equivalently a (Z/2) 3 -covering ψ : Y → X 40 branched on the nodes of X 40 (the surface Y is minimal because X 40 is minimal and ψ is étale in codimension 1). In particular there exist divisors L 111 , L 110 , L 101 , L 011 such that: Since ψ is ramified only on nodes, we have K Y ≡ ψ * (K X 40 ) and then K 2 Y = 8K 2 X 40 = 64. We show in Section 4.1 that h 0 X 40 , O X 40 K X 40 + L 111 = 2 and h 0 X 40 , O X 40 K X 40 + L ijk = 0 for ijk 111, thus p g (Y ) = p g (X 40 ) + 2 + 0 + · · · + 0 = 7. We get from (2.2) that χ(Y ) = 8(6 − 5) = 8, thus q(Y ) = 0. The covering ψ factors as with Y 48 and X 16 given by the quotients by the groups ab, ac and c , respectively (the subscript n means a surface with singular set the union of n nodes). All these surfaces are regular because q(Y ) = 0. It follows from (2.2) that χ(X 16 ) = 4(6 − 36/8) = 6, thus p g (X 16 ) = p g (X 40 ) = 5, and we conclude that the (Z/2) 2 -covering X 16 → X 40 is the canonical map of X 16 . Examples of surfaces with canonical map of degree 4 5 Analogously, p g (Y ) = p g (Y 48 ) = 7 and we claim that the (Z/2) 2 -covering Y → Y 48 is the canonical map of Y .
For this it suffices to show that Y 48 is a canonical surface. Since the canonical system of Y 48 contains the pullback of the canonical system of X 40 and since p g (Y 48 ) > p g (X 40 ), the canonical map of Y 48 must be birational. But we can be more precise. We follow Beauville [Bea17] and show that Y 48 can be embedded in P 6 as a complete intersection of 4 quadrics in the following way. The linear system L of quadrics through the branch locus of the covering Y 48 → X 40 (16 nodes) is of dimension 2. Using computer algebra it is not difficult to show that L contains quadrics B, C, D such that the surface X 40 is given by Q = 0, B 2 − CD = 0, where Q is the quadric from (3.1) (we write the quadrics as general elements of L, thus depending on some parameters; then we obtain a variety on these parameters by imposing that the hypersurfaces Q = 0 and B 2 − CD = 0 are tangent at the 24 nodes of X 40 which are disjoint from the 16 nodes of B 2 − CD = 0; finally we compute points in this variety).
Then Y 48 is given in P 6 (x, y, z, w, t, u, v) by equations We give these equations in Section 4.2 and verify that Y 48 is as stated.
Let us explain how we find 2-divisible sets of nodes in X 40 . The surface X 40 contains 40 tropes, which are hyperplane sections H i = 2T i with T i ⊂ X 40 a reduced curve through 12 nodes of X 40 , and smooth at these points. Thus in X 40 the pullback of such a trope can be written as Thus for each pair of tropes the sum of nodes contained in their union and not contained in their intersection is 2-divisible.
Using these 2-divisibilities, the strategy for finding configurations as in (3.2) is simple: we have used a computer algorithm to list and check possibilities.

The covering Y → X 40
We start by defining the surface X 40 and its singular set.