On equations of double planes with $p_g=q=1$

This paper describes how to compute equations of plane models of minimal Du Val double planes of general type with $p_g=q=1$ and $K^2=2,...,8.$ A double plane with $K^2=8$ having bicanonical map not composed with the associated involution is also constructed. The computations are done using the algebra system Magma.


Introduction
There are two typical methods to construct examples of surfaces: Campedellidouble covers ramified over curves, usually highly singular -and Godeauxquotients by group actions (cf. [Re]). In this paper the first method is used to obtain new examples of double planes of general type with p g = q = 1. Often the singularities impose too many conditions on the linear parameters of systems of curves, and this implies hard calculations. The Computational Algebra System MAGMA ( [BCP]) is used to perform the computations.
For such a smooth minimal surface S, one has 2 ≤ K 2 ≤ 9 and K 2 = 9 only if the bicanonical map of S (given by |2K|) is birational (from [CM] and [Xi1,Théorème 2.2]). Since q = 1, the Albanese map of S is a connected fibration onto an elliptic curve. We denote by g the genus of a general Albanese fibre of S. Du Val ([Du]) classified the regular surfaces S of general type with p g ≥ 3 whose general canonical curve is smooth and hyperelliptic. For these surfaces the bicanonical map is composed with an involution i such that S/i is rational. The families of surfaces exhibited by Du Val are nowadays called the Du Val examples. The standard case of non-birationality of the bicanonical morphism φ 2 is the case where S has a genus 2 fibration.
For the non-standard case with φ 2 of degree 2 and bicanonical image a ruled surface, Xiao Gang ([Xi2, Theorem 2]) extended Du Val's list to p g (S) ≥ 1 and added two extra families (this result is still valid assuming only that φ 2 is composed with an involution such that the quotient surface is a ruled surface). Later G. Borrelli ([Br]) excluded these two families, confirming that the only possibilities for this case are the Du Val examples. This way, if the bicanonical map φ 2 of S is composed with i, S/i is ruled and S presents the non-standard case, then S is a Du Val double plane (see Definition 3). But, to my knowledge, the existence of these double planes with p g = q = 1 and K 2 = 4, . . . , 7 has not been shown yet, and no equation has been given for K 2 = 3 or 8.
In this paper we compute equations of plane models of minimal Du Val double planes of general type with p g = q = 1 and K 2 = 2, . . . , 8. We also construct a double plane with K 2 = 8 having bicanonical map not composed with the associated involution.
Notice that we give the first example of a minimal surface of general type with p g = q = 1 and K 2 = 7. We also give the first example with K 2 = 5 having Albanese fibration of genus g = 2.
The first example with K 2 = 6 has been given by the author in [Ri], as a double cover of a K3 surface. Polizzi ([Po2]) also gives examples with K 2 = 6, 4 and g = 2. These surfaces contain 8 − K 2 (−2)-curves, all in the same Albanese fibre. One can verify that, in the double plane examples with K 2 = 4 or 6, there is no Albanese fibre containing 8 − K 2 (−2)-curves. Therefore our surfaces are different from the above examples.
In [Po4], Polizzi classifies surfaces of general type with p g = q = 1, K 2 = 8 and bicanonical map of degree 2. He constructs examples using quotients under the action of a group and shows that these surfaces are Du Val double planes, describing the branch locus of the respective plane models. In sections 3.1 and 3.4 we show how to obtain equations for such branch loci.
Section 2 contains a structure description of the Albanese fibration of S and the definition of Du Val double plane. In Section 3 we describe the principal steps of the constructions. To obtain the equations of the branch loci we impose conditions to the linear parameters of systems of plane curves. Since the ramification curves are contained in Albanese fibres, in some cases it is easier to start by constructing non-reduced Albanese fibres, which simplify the computations. These are done using the algebra system Magma in Appendix A.

Notation and conventions
We work over the complex numbers; all varieties are assumed to be projective algebraic. For a projective smooth surface S, the canonical class is denoted by K, the geometric genus by p g := h 0 (S, O S (K)), the irregularity by q := h 1 (S, O S (K)) and the Euler characteristic by χ = χ(O S ) = 1 + p g − q.
An (−n)-curve C on a surface is a curve isomorphic to P 1 such that C 2 = −n. We say that a curve singularity is negligible if it is either a double point or a triple point which resolves to at most a double point after one blow-up. An (m 1 , m 2 , . . .)-point, or point of order (m 1 , m 2 , . . .), is a point of multiplicity m 1 , which resolves to a point of multiplicity m 2 after one blow-up, etc.
An involution of a surface S is an automorphism of S of order 2. We say that a map is composed with an involution i of S if it factors through the double cover S → S/i. The rest of the notation is standard in algebraic geometry.

Preliminaries
We say that a smooth surface S is a double plane if S has an involution i such that S/i is a rational surface. A plane model of S is a double cover X → P 2 such that X is a normal surface and there exists a commutative diagram such that the horizontal arrows denote birational maps.

Double planes with q = 1
Let S be a smooth minimal surface of general type with an involution i. Since S is minimal of general type, this involution is biregular. The fixed locus of i is the union of a smooth curve R ′′ (possibly empty) and of t ≥ 0 isolated points P 1 , . . . , P t . Let S/i be the quotient of S by i and p : S → S/i be the projection onto the quotient. The surface S/i has nodes at the points Q i := p(P i ), i = 1, . . . , t, and is smooth elsewhere. If R ′′ = ∅, the image via p of R ′′ is a smooth curve B ′′ not containing the singular points Q i , i = 1, . . . , t. Let now h : V → S be the blow-up of S at P 1 , . . . , P t and set R ′ = h * (R ′′ ). The involution i induces a biregular involution i on V, whose fixed locus is R := R ′ + t 1 h −1 (P i ), and the quotient W := V / i is smooth.
Suppose now that q(S) = 1 and S/i is a rational surface. Since q = 1, the Albanese variety of S is an elliptic curve E and the Albanese map is a connected fibration (see e.g. [Be] or [BPV]). This fibration is preserved by i and so we have a commutative diagram Denote by f A : W → P 1 the fibration induced by the Albanese fibration of S. The double cover E → P 1 is ramified over 4 points p j of P 1 , thus the branch locus B of π : V → W is contained in 4 fibres F j A := f * A (p j ), j = 1, ..., 4, of the fibration f A . By Zariski's Lemma (see e.g. [BPV]) the irreducible components B i of B satisfy B 2 i ≤ 0. Since π * (F j A ) has even multiplicity, each component of F j A which is not a component of the branch locus must be of even multiplicity.
In the following sections W can always be contracted to P 2 . We keep the same notation for the image of f A and F j A on P 2 .

Du Val double planes
Notation 1 Given distinct points p 0 , . . . , p j , . . . , p j+s ∈ P 2 , let T i denote the line through p 0 and p i , i = 1, . . . , j. We say that a plane curve is of type d m, (n, n) j T , r s if it is of degree d and if it has: an m-uple point at p 0 , an (n, n)-point at p 1 , . . . , p j , an r-uple point at p j+1 , . . . , p j+s and no other non-negligible singularities. The index T is used if T i is tangent to the (n, n)-point at p i . An obvious generalization is used if there are other singularities.
Definition 2 Let C 0 and F denote, respectively, the negative section and a rational fibre of the Hirzebruch surface F 2 .
The Bombieri-Du Val surface is the minimal model of the double cover of F 2 with branch locus a smooth curve in C 0 + |7C 0 + 14F |. Surfaces of type B, D or D n are called Du Val's ancestors. The Bombieri surface has K 2 = 9, p g = 6 and q = 0; a Du Val ancestor of type D has K 2 = 2, p g = 3 and q = 0; those of type D n , n ≥ 0, have K 2 = 8, p g = 6 − n and q = 0, except possibly in the case n = 6, where p g = q = 1 if the 6 singular points are contained in a conic.
Notice that the imposition of a 4-uple point to the branch locus decreases K 2 by 2 and the Euler characteristic χ by 1, while a (3, 3)-point decreases both K 2 and χ by 1. Negligible singularities in the branch locus do not change these invariants.
The imposition of 6 − n 4-uple or (3, 3)-points to the branch locus of a Du Val surface of type D n gives p g = 0 if the singular points p 1 , . . . , p 6 are not contained in a conic and p g = q = 1 otherwise.
Notice that, in the case D n , the branch locus contains the lines T 1 , . . . , T n . Each of these lines corresponds to two (−2)-curves in W, which contract to two nodes of S/i. A (3, 3)-point in the branch locus also corresponds to a node of S/i.
The bicanonical map of a Du Val surface factors through a map of degree 2 onto a rational surface. For more information on Du Val surfaces see [Du] or [Ci].

The Magma procedures LinSys and LinSys2
Magma has the function LinearSystem which calculates linear systems of plane curves with ordinary singularities, but we want to work with non-ordinary singularities. To achieve this, we define two procedures. The first one, LinSys, calculates the linear system L of plane curves of degree d, in an affine plane A, having singular points p i of order (m1 i , m2 i ) with tangent slope td i .
The other procedure, LinSys2, calculates the sub-system J, of a given linear system L of plane curves, of those sections which have a singularity, at a point q, of type m = (m 1 , . . . , m j ) with tangent slopes given by td = [td 1 , . . . , td j−1 ].
The code lines for these procedures are in Appendix A.1.
3 Equations of plane models with p g = q = 1 In this section we obtain equations of the branch locus B ⊂ P 2 = P(1, 1, 1) of minimal Du Val double planes S of type D n (see Definition 3) with p g = q = 1 and K 2 = 2, . . . , 8. Notice that if f = 0 is the defining equation of B, then the equation w 2 = f gives a plane model of S, in the weighted projective space P((deg f )/2, 1, 1, 1). In Section 3.5 we construct a double plane with K 2 = 8 having bicanonical map not composed with the associated involution.
Table (2) lists the type of each branch curve that we are going to obtain and the corresponding values of (K 2 , g), where g denotes the genus of a general Albanese fibre of S.
We keep Notation 1.
Type of branch curve (K 2 , g) First we find double planes with (K 2 , g) = (8, 5) and (K 2 , g) = (7, 5) such that the branch locus contains an element of the pencil f A which induces the Albanese fibration. From here constructions with (K 2 , g) = (6, 4), (5, 4), (4, 3), (3, 3), (2, 2) will follow easily. A construction with (K 2 , g) = (6, 3) and branch locus strictly contained in elements of f A is also given. Then we get surfaces with (K 2 , g) = (8, 4) or (8, 3). These three surfaces with K 2 = 8 are surfaces of type I, II, III described by F. Polizzi in [Po4]. Finally we construct a pencil l of plane curves such that the minimal double plane with branch locus a general element of l has K 2 = 8 and bicanonical map not composed with the corresponding involution.
In order to obtain q = 1, all the singularities, except the one at p 0 , are chosen to be contained in a conic.
A difficulty that arises in the computation of singular curves is that often the computer is not able to finish the calculations. Except for Section 3.2, our method is to try to figure out the configuration of the non-reduced fibres F i A of f A , described in Section 2.1. The problem of finding the support of F i A deals with curves of lower degree and with simpler singularities, hence with faster computations.
The next sections of this chapter describe the principal steps. The detailed calculations are done in Appendix A, using the Computational Algebra System Magma.
We keep Notation 1.

respectively.
To construct B ′ , we first find points p 0 , . . . , p 6 such that D 1 exists. Then we use the procedure LinSys (see Section 2.3) to verify the existence of D 2 and D 3 , also singular at p 0 , . . . , p 6 . The curve B ′ = D 4 is a general element of the pencil generated by D 1 , D 2 , D 3 . So let us look for D 1 . Briefly, the steps are as follows. Let A be an affine plane, C be a smooth conic not containing the origin p 0 of A and p 1 , . . . , p 4 be points in C. Denote by L the linear system of plane curves of type 6 2, (2, 2) 2 T , (2, 1) 2 T , with singularities at p 0 , . . . , p 4 , respectively. Let F be a general element of L and p 5 , p 6 be general points of A. We define a scheme Sch by imposing the following conditions: · p 5 , p 6 ∈ C F ; · p 5 , p 6 are double points of F ; · the singularities of F at p 5 , p 6 have one branch tangent to T 5 , T 6 ; · p 5 = p 6 and p 5 , p 6 ∈ {p 0 , . . . , p 4 }.
Now we compute the points of Sch with Magma, choosing one of the solutions for p 5 , p 6 , and we use the procedure LinSys to compute B ′ , a general element of the pencil of curves of type 16(8, (4, 4) 6 T ), with singularities at p 0 , . . . , p 6 . Finally we perform some verifications: that B ′ is reduced, the singularities are as expected, etc.
With this we find a minimal double plane with p g = q = 1 and K 2 = 8. The divisor 2D 1 + T 3 + · · · + T i is also a good candidate for one of the singular fibres F i A in the case where the branch locus is a curve of type [10 + 2n] 2n + 2, (5, 5) n T , 4 6−n , n = 5, 4, 3.
The corresponding Magma calculations are in Appendix A.2.1. There we use symmetry in order to obtain faster computations.
In this example the (3, 3)-point p 6 is infinitely near to the (5, 5)-point p 1 of B (i.e. there is a (5, 5, 3, 3)-point at p 1 ) and the line T 1 , through p 0 , p 1 , is tangent to the conic C defined by p 1 , . . . , p 5 . This implies q(S) = 1, because C − 6 1 E i is effective, where the curves E i are the exceptional divisors corresponding to the blow-ups at the points p i and C is the pullback of C.
To find B ′ = D 4 we proceed as follows. In an affine plane A, we fix a smooth conic C not containing the origin p 0 and we choose distinct points p 1 , . . . , p 5 ∈ C such that T 1 is tangent to C at p 1 . We compute the linear system L of plane curves of type 15(7, (4, 4) 5 T ) and we resolve the base point of L at p 1 , denoting the resulting linear system by L 1 . The defining polynomial of B ′ is given by (the blow-down of) a linear combination of elements of L 1 .
In order to obtain a (3, 3)-point p 6 (infinitely near to p 1 ), we impose conditions, given by the zeros of the derivatives up to order 2, to the elements of L 1 . Then we resolve this point and impose another triple (infinitely near) point. With all these conditions we define a matrix, which is denoted by M t in Appendix A.2.2. To have a solution it is necessary that M t has no maximal rank.
We define a scheme Sch by imposing the following conditions: · the vanishing of the maximal minors of M t; · the points are infinitely near.
Now we compute the points of Sch with Magma, choosing one of the solutions for p 6 , and we use the procedures of Section 2.3 to compute B ′ , with singularities at p 0 , . . . , p 6 . Finally we perform some verifications: that B ′ is reduced, the singularities are as expected, etc.
Notice that there is no need to verify the non-existence of other singularities: in the presence of another non-negligible singularity, the computation of the invariants of the corresponding double plane (using the formulas of [BPV,Section V. 22]) leads to a contradiction.
To find the pencil which induces the Albanese fibration, one uses again the procedures LinSys and LinSys2 to compute the pencil of curves of type 16(8, (4, 4) 5 T , (4, 4)), through p 0 , . . . , p 6 , and to verify that the elements F 1 A , . . . , F 4 A are as claimed.
With this we find a minimal double plane with p g = q = 1, K 2 = 7 and g = 5. Using the above Magma procedures, one can verify that there exist also branch loci of type 3.3 K 2 = 6 and g = 3 In this section we impose two (3, 3)-points to the branch locus of a Du Val's ancestor of type D 4 , i.e. we construct a plane curve B ′ of type 14(6, (4, 4) 4 T , (3, 3) 2 ), singular at points p 0 , . . . , p 6 , such that B := B ′ + 4 1 T i is reduced. In this example the (3, 3)-points p 5 , p 6 are tangent to the conic C through p 1 , . . . , p 6 .
The double cover with branch locus B is a plane model of a Du Val double plane S with K 2 = 6, p g = q = 1 and g = 3.
To find D 1 , we follow the steps used in Section 3.1, but now with L the linear system of curves of type 6 2, (2, 2) 4 T and Sch defined by · p 5 , p 6 ∈ C F ; · F is smooth at p 5 , p 6 ; · F is tangent to C at p 5 , p 6 ; · p 5 = p 6 and p 5 , p 6 ∈ {p 0 , . . . , p 4 }.
The details can be found in Appendix A.2.3. Again we use symmetry in order to increase speed of calculations.
3.4 K 2 = 8 and g = 4 or 3 Given, in an affine plane A, a general plane cubic Q not containing the origin p 0 , there are 6 points p 1 , . . . , p 6 ∈ Q such that each line T i , defined by p 0 , p i , is tangent to Q. In this example we have where D 1 , D 2 , D 3 are curves of types 4 2, (1, 1) 6 T , 12 6, (3, 3) 6 T , 6 3, (2, 1) 6 T , respectively. Using the procedure LinSys (see Section 2.3), one can verify the existence of the curves D 1 , D 2 , D 3 , through p 0 , . . . , p 6 , and that p 1 , . . . , p 6 are contained in a conic. The double cover with branch locus D 1 + D 2 + 6 1 T i is a plane model of a Du Val double plane S with K 2 = 6, p g = q = 1 and g = 4. This is a surface of type II described in [Po4].
Finally the surface of type I described in [Po4]: Let C ⊂ P 2 be a smooth conic and p 0 ∈ C, p 1 , . . . , p 6 ∈ C be points such that the lines T 5 and T 6 , through p 0 , p 5 and p 0 , p 6 , are tangent to C. One can verify the existance of a smooth cubic D of type 3(1, (1, 1) 4 T , 1 2 ), through p 0 , . . . , p 6 . Let F 1 A = 2C + T 1 + · · · + T 4 , F 2 A = 2D + T 5 + T 6 and F 3 A , F 4 A be general elements of the pencil generated by F 1 A and F 2 A . The branch locus in this case is 3.5 K 2 = 8, g = 4, φ 2 not composed with i This section describes the steps to obtain an equation of a double plane S of general type with K 2 = 8, p g = q = 1 and Albanese fibres of genus g = 4, such that the bicanonical map φ 2 of S is not composed with the associated involution i of S.
In this example the pencil f A which induces the Albanese fibration has four fibres . . , D 4 are curves of types 10 4, (3, 3) 4 T , (3, 2) T , 3 1, 1, (1, 1) 4 T , 9 3, (3, 3), (3, 2) 3 T , (3, 3) , 21 9, (6, 6) 5 T , respectively. The surface S is the minimal double plane given by the branch locus From the double cover formulas ( [BPV,Section V. 22]), one obtains K 2 S = 8 (notice that the pullback of the lines T i contains 10 (−1)-curves), χ(O S ) = 1 and p g (S) is equal to the number of generators of the linear system of curves of type 10(6, (2, 3) 5 T ). The points p 0 , . . . , p 5 are chosen such that the line T 1 , through p 0 , p 1 , is tangent to the conic defined by p 1 , . . . , p 5 . One can verify that this implies the existence of the cubic D 2 . After finding D 1 , we verify the existence of D 2 and D 3 . The curve D 4 is a general element of the pencil generated by D 1 , D 2 , D 3 . One can use the procedure LinSys to compute the number of elements of the linear system of curves of type 10(6, (2, 3) 5 T ), showing that p g (S) = 1. The difficulty here is the computation of D 1 . This is done following the steps of Section 3.2. We omit the details.

A Appendix: Magma computations
In this appendix several computations are done using the Computational Algebra System MAGMA.