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A special configuration of 12 conics and generalized Kummer surfaces

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Abstract

A generalized Kummer surface X obtained as the quotient of an abelian surface by a symplectic automorphism of order 3 contains a \(9\mathbf{A}_{2}\)-configuration of \((-2)\)-curves. Such a configuration plays the role of the \(16\mathbf{A}_{1}\)-configurations for usual Kummer surfaces. In this paper we construct 9 other such \(9\mathbf{A}_{2}\)-configurations on the generalized Kummer surface associated to the double cover of the plane branched over the sextic dual curve of a cubic curve. The new \(9\mathbf{A}_{2}\)-configurations are obtained by taking the pullback of a certain configuration of 12 conics which are in special position with respect to the branch curve, plus some singular quartic curves. We then construct some automorphisms of the K3 surface sending one configuration to another. We also give various models of X and of the generic fiber of its natural elliptic pencil.

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Acknowledgements

The authors wish to thank Carlos Rito for sharing his program LinSys, Remke Kloosterman for pointing out the reference [10], and also Cédric Bonnafé, Igor Dolgachev, Antonio Laface, Ulf Persson, Piotr Pokora, Giancarlo Urzúa. Part of the computations were done using Magma software [8]. The second author thanks the Max-Planck Institute for Mathematics of Bonn for its hospitality and support. The third author is partially supported by the ANR project No. ANR-20-CE40-0026-01 (SMAGP).

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Authors and Affiliations

  1. CNRS, I2M UMR 7373, Aix-Marseille Université, Marseille, France

    David Kohel & Xavier Roulleau

  2. Laboratoire de Mathématiques et Applications, UMR CNRS 7348, Université de Poitiers, TSA 61125, 11 bd Marie et Pierre Curie, 86073, POITIERS Cedex 9, France

    Alessandra Sarti

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  1. David Kohel
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  2. Xavier Roulleau
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  3. Alessandra Sarti
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Correspondence to Xavier Roulleau.

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Appendix

Appendix

Let \(D_{2}\) be the pull-back of a line by the double cover map \(\eta :X_{\lambda }\rightarrow \mathbb {P}^{1}\). Let us define the following classes in the \(\mathbb {Q}\)-basis \(\mathcal {B}_{0}=(D_{2},A_{1},A_{1}',\dots ,A_{9},A_{9}')\):

$$\begin{aligned} B_{1}=2D_{2}-\tfrac{1}{3}\left( {\textstyle \sum _{j=1}^{9}}2A_{j}+A_{j}'\right) ,\,\,B_{2}=2D_{2}-\tfrac{1}{3}\left( {\textstyle \sum _{j=1}^{9}}A_{j}+2A_{j}'\right) . \end{aligned}$$

We remark that \(B_{1}^{2}=B_{2}^{2}=2\), \(B_{1}B_{2}=5\) and \(B_{1}+B_{2}=D_{14}\). We have \(D_{14}A_{j}=D_{14}A_{j}'=1\), therefore \(B_{i}A_{j}\in \{0,1\}\), \(B_{i}A_{j}'\in \{0,1\}\). Using algorithms described in [21], we find that for \(j\in \{1,\dots ,9\}\), the classes of the curves \(\gamma _{j},\gamma _{j}'\) above the quartic \(Q_{j}\) are

$$\begin{aligned} \gamma _{j}=B_{1}-\left( A_{j}+A_{j}'\right) ,\,\gamma _{j}'=B_{2}-\left( A_{j}+A_{j}'\right) . \end{aligned}$$

It is easy to check that \(\gamma _{j}^{2}=\gamma _{j}'^{2}=-2\), \(\gamma _{j}\gamma _{j}'=1\), and for \(1\le i\ne j\le 9\), we have \(\gamma _{i}\gamma _{j}=\gamma _{i}'\gamma _{j}'=0\) and \(\gamma _{i}\gamma _{j}'=3\). In fact, using that the image in \(\mathbb {P}^{2}\) of \(\gamma _{j},\gamma _{j}'\) is a quartic curve that goes through the points in \(\mathcal {P}_{9}\) with a multiplicity 3 at \(p_{j}\), one gets

$$\begin{aligned} 4D_{2}\equiv \gamma _{j}+\gamma _{j}'+2\left( A_{j}+A_{j}'\right) +{\textstyle \sum _{j=1}^{9}}\left( A_{j}+A_{j}'\right) . \end{aligned}$$

The classes in the \(\mathbb {Q}\)-basis \(\mathcal {B}_{0}=(L,A_{1},A_{1}',\dots ,A_{9},A_{9}')\) of the 24 \(\mathrm{(-2)}\)-curves \(\theta _{i,\dots ,n},\theta '_{i,\dots ,n}\) above the 12 conics \(C_{i,\dots ,n}\) in \(\mathcal {C}_{12}\) are

$$\begin{aligned}&\theta _{123456}=\frac{1}{3}(3,-2,-1,-2,-1,-2,-1,-1,-2,-1,-2,-1,-2,0,0,0,0,0,0),\\&\theta '_{123456}=\frac{1}{3}(3,-1,-2,-1,-2,-1,-2,-2,-1,-2,-1,-2,-1,0,0,0,0,0,0),\\&\theta _{123789}=\frac{1}{3}(3,-2,-1,-2,-1,-2,-1,0,0,0,0,0,0,-1,-2,-1,-2,-1,-2),\\&\theta '_{123789}=\frac{1}{3}(3,-1,-2,-1,-2,-1,-2,0,0,0,0,0,0,-2,-1,-2,-1,-2,-1),\\&\theta _{124578}=\frac{1}{3}(3,-2,-1,-1,-2,0,0,-2,-1,-1,-2,0,0,-2,-1,-1,-2,0,0),\\&\theta '_{124578}=\frac{1}{3}(3,-1,-2,-2,-1,0,0,-1,-2,-2,-1,0,0,-1,-2,-2,-1,0,0),\\&\theta _{124689}=\frac{1}{3}(3,-2,-1,-1,-2,0,0,-1,-2,0,0,-2,-1,0,0,-2,-1,-1,-2),\\&\theta '_{124689}=\frac{1}{3}(3,-1,-2,-2,-1,0,0,-2,-1,0,0,-1,-2,0,0,-1,-2,-2,-1),\\&\theta _{125679}=\frac{1}{3}(3,-2,-1,-1,-2,0,0,0,0,-2,-1,-1,-2,-1,-2,0,0,-2,-1),\\&\theta '_{125679}=\frac{1}{3}(3,-1,-2,-2,-1,0,0,0,0,-1,-2,-2,-1,-2,-1,0,0,-1,-2),\\&\theta _{134589}=\frac{1}{3}(3,-2,-1,0,0,-1,-2,-1,-2,-2,-1,0,0,0,0,-1,-2,-2,-1),\\&\theta '_{134589}=\frac{1}{3}(3,-1,-2,0,0,-2,-1,-2,-1,-1,-2,0,0,0,0,-2,-1,-1,-2),\\&\theta _{134679}=\frac{1}{3}(3,-2,-1,0,0,-1,-2,-2,-1,0,0,-1,-2,-2,-1,0,0,-1,-2),\\&\theta '_{134679}=\frac{1}{3}(3,-1,-2,0,0,-2,-1,-1,-2,0,0,-2,-1,-1,-2,0,0,-2,-1),\\&\theta _{135678}=\frac{1}{3}(3,-2,-1,0,0,-1,-2,0,0,-1,-2,-2,-1,-1,-2,-2,-1,0,0),\\&\theta '_{135678}=\frac{1}{3}(3,-1,-2,0,0,-2,-1,0,0,-2,-1,-1,-2,-2,-1,-1,-2,0,0),\\&\theta {}_{234579}=\frac{1}{3}(3,0,0,-2,-1,-1,-2,-2,-1,-1,-2,0,0,-1,-2,0,0,-2,-1),\\&\theta '_{234579}=\frac{1}{3}(3,0,0,-1,-2,-2,-1,-1,-2,-2,-1,0,0,-2,-1,0,0,-1,-2),\\&\theta {}_{234678}=\frac{1}{3}(3,0,0,-2,-1,-1,-2,-1,-2,0,0,-2,-1,-2,-1,-1,-2,0,0),\\&\theta '_{234678}=\frac{1}{3}(3,0,0,-1,-2,-2,-1,-2,-1,0,0,-1,-2,-1,-2,-2,-1,0,0),\\&\theta {}_{235689}=\frac{1}{3}(3,0,0,-2,-1,-1,-2,0,0,-2,-1,-1,-2,0,0,-2,-1,-1,-2),\\&\theta '_{235689}=\frac{1}{3}(3,0,0,-1,-2,-2,-1,0,0,-1,-2,-2,-1,0,0,-1,-2,-2,-1),\\&\theta _{456789}=\frac{1}{3}(3,0,0,0,0,0,0,-1,-2,-1,-2,-1,-2,-2,-1,-2,-1,-2,-1),\\&\theta '_{456789}=\frac{1}{3}(3,0,0,0,0,0,0,-2,-1,-2,-1,-2,-1,-1,-2,-1,-2,-1,-2). \end{aligned}$$

We also denote by \(\varTheta _{j},\,j=1,\dots ,24\) these curves in the order of the above list.

For \(k=1,\ldots ,9\), the quartic curves \(Q_{k}\) through \(\mathcal {P}_{9}\) that have a multiplicity 3 singular point at \(p_{k}\) are:

$$\begin{aligned}&Q_{1}:\,\,x^{4}-2\lambda x^{3}y\!+\!3\lambda ^{2}x^{2}y^{2}\!-\!(\lambda ^{3}+1)xy^{3}\!+\!\lambda y^{4}-2\lambda x^{3}z+(-\lambda ^{3}+1)xy^{2}z-2\lambda y^{3}z\\&\quad +\,3\lambda ^{2}x^{2}z^{2}+(-\lambda ^{3}+1)xyz^{2}+(\lambda ^{4}+2\lambda )y^{2}z^{2}-(\lambda ^{3}+1)xz^{3}-2\lambda yz^{3}+\lambda z^{4}=0,\\&Q_{2}:\,\,x^{4}\!-\!(\lambda ^{3}\!+\!1)/\lambda x^{3}y\!+\!3\lambda x^{2}y^{2}\!-\!2xy^{3}\!+\!1/\lambda y^{4}\!-\!2x^{3}z\!+\!(-\lambda ^{3}+1)/\lambda x^{2}yz-2y^{3}z\\&\quad +\,(\lambda ^{3}+2)x^{2}z^{2}+(1-\lambda ^{3})/\lambda xyz^{2}+3\lambda y^{2}z^{2}-2xz^{3}-(\lambda ^{3}+1)/\lambda yz^{3}+z^{4}=0,\\&Q_{3}:\,\,x^{4}-2x^{3}y+(\lambda ^{3}+2)x^{2}y^{2}-2xy^{3}+y^{4}-(\lambda ^{3}+1)/\lambda x^{3}z+(1-\lambda ^{3})/\lambda x^{2}yz\\&\quad +\,(1-\lambda ^{3})/\lambda xy^{2}z-(\lambda ^{3}+1)/\lambda y^{3}z+3\lambda x^{2}z^{2}+3\lambda y^{2}z^{2}\!-\!2xz^{3}-2yz^{3}+1/\lambda z^{4}\!=\!0,\\&Q_{4}:\,\,x^{4}+(2\omega +2)\lambda x^{3}y+3\omega \lambda ^{2}x^{2}y^{2}-(\lambda ^{3}+1)xy^{3}-(\omega +1)\lambda y^{4}-2\omega \lambda x^{3}z\\&\quad -\,(\omega ^{2}\lambda ^{3}+\omega +1)xy^{2}z-2\omega \lambda y^{3}z-(3\omega +3)\lambda ^{2}x^{2}z^{2}+(\omega -\omega \lambda ^{3})xyz^{2}\\&\quad +\,(\lambda ^{4}+2\lambda )y^{2}z^{2}-(\lambda ^{3}+1)xz^{3}+(2\omega +2)\lambda yz^{3}+\omega \lambda z^{4}=0,\\&Q_{5}:\,\,x^{4}-(\omega ^{2}\lambda ^{3}+\omega ^{2})/\lambda x^{3}y+3\omega \lambda x^{2}y^{2}-2xy^{3}-(\omega +1)/\lambda y^{4}-2\omega x^{3}z\\&\quad +\,(1-\lambda ^{3})/\lambda x^{2}yz-2\omega y^{3}z-((\omega +1)\lambda ^{3}+2\omega +2)x^{2}z^{2}+(\omega -\omega \lambda ^{3})/\lambda xyz^{2}\\&\quad +\,3\lambda y^{2}z^{2}-2xz^{3}-\omega ^{2}(\lambda ^{3}+1)/\lambda yz^{3}+\omega z^{4}=0,\\&Q_{6}:\,\,x^{4}+(2\omega +2)x^{3}y+(\omega \lambda ^{3}+2\omega )x^{2}y^{2}-2xy^{3}+\omega ^{2}y^{4}-(\omega \lambda ^{3}+\omega )/\lambda x^{3}z\\&\quad +\,\,(1-\lambda ^{3})/\lambda x^{2}yz-(\omega ^{2}\lambda ^{3}-\omega ^{2})/\lambda xy^{2}z-(\omega \lambda ^{3}+\omega )/\lambda y^{3}z\\&\quad -\,(3\omega +3)\lambda x^{2}z^{2}+3\lambda y^{2}z^{2}-2xz^{3}+(2\omega +2)yz^{3}+\omega /\lambda z^{4}=0,\\&Q_{7}:\,\,x^{4}-2\omega \lambda x^{3}y-(3\omega +3)\lambda ^{2}x^{2}y^{2}-(\lambda ^{3}+1)xy^{3}+\omega \lambda y^{4}+(2\omega +2)\lambda x^{3}z\\&\quad +\,(\omega -\omega \lambda ^{3})xy^{2}z+(2\omega +2)\lambda y^{3}z+3\omega \lambda ^{2}x^{2}z^{2}-(\omega ^{2}\lambda ^{3}-\omega ^{2})xyz^{2}\\&\quad +\,(\lambda ^{4}+2\lambda )y^{2}z^{2}-(\lambda ^{3}+1)xz^{3}-2\omega \lambda yz^{3}+\omega ^{2}\lambda z^{4}=0,\\&Q_{8}:\,\,x^{4}-(\omega \lambda ^{3}+\omega )/\lambda x^{3}y-(3\omega +3)\lambda x^{2}y^{2}-2xy^{3}+\omega /\lambda y^{4}+(2\omega +2)x^{3}z\\&\quad +\,(1-\lambda ^{3})/\lambda x^{2}yz+(2\omega +2)y^{3}z+(\omega \lambda ^{3}+2\omega )x^{2}z^{2}-(\omega ^{2}\lambda ^{3}-\omega ^{2})/\lambda xyz^{2}\\&\quad +\,3\lambda y^{2}z^{2}-2xz^{3}-(\omega \lambda ^{3}+\omega )/\lambda yz^{3}+\omega ^{2}z^{4}=0,\\&Q_{9}:\,\,x^{4}-2\omega x^{3}y+(\omega ^{2}\lambda ^{3}-2\omega -2)x^{2}y^{2}-2xy^{3}+\omega y^{4}-(\omega ^{2}\lambda ^{3}+\omega ^{2})/\lambda x^{3}z\\&\quad +\,(1-\lambda ^{3})/\lambda x^{2}yz+(-\omega \lambda ^{3}+\omega )/\lambda xy^{2}z-(\omega ^{2}\lambda ^{3}+\omega ^{2})/\lambda y^{3}z\\&\quad +\,3\omega \lambda x^{2}z^{2}+3\lambda y^{2}z^{2}-2xz^{3}-2\omega yz^{3}+\omega ^{2}/\lambda z^{4}=0, \end{aligned}$$

where \(\omega ^{2}+\omega +1=0\).

The matrices in basis \(\mathcal {B}\) of the generators of the group \(G_{432}\simeq AGL_{2}(\mathbb {F}_{3})\) preserving the natural \(9\mathbf{A}_{2}\)-configuration \(A_{1},A_{1}',\dots ,A_{9},A_{9}'\) are

$$\begin{aligned}&{g_{1}=\left( \begin{array}{ccccccccccccccccccc} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} -2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} -1 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} -1 &{} 1 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1 &{} 0 &{} 0 &{} 1 &{} 0\\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 2 &{} -2 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1 &{} -1 &{} 0 &{} 1 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 4 &{} -2 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1 &{} -1 &{} -1 &{} 2 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 2 &{} -1 &{} 1 &{} -1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} -1 &{} 0 &{} -1 &{} 1 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} -1 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -2 &{} 2 &{} -1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} -1 &{} 0\\ 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} -1 &{} 0 &{} 0 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} -2 &{} 0 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} -1 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 2 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} -1 &{} -1 &{} 1 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -3 &{} 2 &{} -1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} -2 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -2 &{} 1 &{} -1 &{} 2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} -2 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} -1 &{} 2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} -1 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 3 &{} -3 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1 &{} -1 &{} -1 &{} 1 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -4 &{} 2 &{} -2 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 1 &{} -2 &{} 1\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 2 &{} -1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1 &{} -1 &{} 0 &{} 1 &{} 0 \end{array}\right) , }\\&{g_{2}=\left( \begin{array}{ccccccccccccccccccc} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} -2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 2 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} -1 &{} 1 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1 &{} 0 &{} 0 &{} -1 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 2 &{} -2 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -2 &{} 0 &{} 0 &{} -1 &{} 0 &{} 0\\ 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 4 &{} -2 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -4 &{} 0 &{} 0 &{} -2 &{} 1 &{} 0\\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 2 &{} -1 &{} 1 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} -2 &{} 0 &{} 0 &{} -1 &{} 1 &{} 0\\ 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} -1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} -2 &{} 2 &{} -1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 2 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1 &{} 0 &{} 0 &{} -1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -2 &{} 0 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 2 &{} 0 &{} 0 &{} 1 &{} -1 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 2 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} -2 &{} 0 &{} 0 &{} -1 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -3 &{} 2 &{} -1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 3 &{} 0 &{} 0 &{} 2 &{} -1 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -2 &{} 1 &{} -1 &{} 2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 3 &{} 0 &{} -1 &{} 2 &{} -1 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} -1 &{} 2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} -1 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 3 &{} -3 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -3 &{} 0 &{} 0 &{} -2 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -4 &{} 2 &{} -2 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 3 &{} 0 &{} 0 &{} 2 &{} -1 &{} 1\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 2 &{} -1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} -3 &{} 0 &{} 0 &{} -1 &{} 0 &{} 0 \end{array}\right) , } \end{aligned}$$

The automorphism f of Theorem 39 acts on the Néron–Severi lattice by

$$\begin{aligned} {f_{\mathcal {B}}=\left( \begin{array}{ccccccccccccccccccc} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1\\ 0 &{} 1 &{} -1 &{} 0 &{} 0 &{} -1 &{} 0 &{} 0 &{} 0 &{} -1 &{} 0 &{} 1 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1 &{} 0 &{} 0 &{} 0 &{} -1 &{} 0 &{} 0 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} -1 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0\\ 0 &{} -1 &{} 0 &{} 0 &{} 0 &{} 2 &{} -1 &{} 0 &{} 0 &{} 1 &{} -1 &{} -2 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1\\ 0 &{} -3 &{} 2 &{} 0 &{} 0 &{} 2 &{} -1 &{} 1 &{} -1 &{} 2 &{} -2 &{} -3 &{} 2 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} -1\\ 0 &{} -1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} -1 &{} -2 &{} 1 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} -1 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0\\ 1 &{} 1 &{} 0 &{} 0 &{} 0 &{} -1 &{} 0 &{} 0 &{} 0 &{} -1 &{} 2 &{} 2 &{} -1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1\\ 0 &{} 1 &{} -1 &{} 0 &{} 0 &{} -1 &{} 1 &{} -1 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1\\ 0 &{} 1 &{} -1 &{} 0 &{} 0 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} 1 &{} 2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1\\ 0 &{} -1 &{} 1 &{} 0 &{} 0 &{} 1 &{} -1 &{} 0 &{} 0 &{} 1 &{} -1 &{} -1 &{} 1 &{} -1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0\\ 1 &{} 2 &{} -1 &{} 0 &{} 0 &{} -2 &{} 1 &{} 0 &{} 0 &{} -1 &{} 2 &{} 2 &{} -1 &{} 1 &{} 0 &{} 1 &{} 1 &{} 0 &{} 1\\ 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} -1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0\\ 1 &{} 1 &{} -1 &{} 0 &{} 1 &{} -1 &{} 0 &{} 0 &{} 0 &{} -1 &{} 2 &{} 2 &{} -1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0\\ 1 &{} -1 &{} 1 &{} 1 &{} 0 &{} 0 &{} -1 &{} 1 &{} -1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0\\ -1 &{} -2 &{} 1 &{} 0 &{} 0 &{} 2 &{} -1 &{} 0 &{} 0 &{} 1 &{} -2 &{} -2 &{} 2 &{} -1 &{} 0 &{} -1 &{} 0 &{} 0 &{} -1\\ 0 &{} 2 &{} -1 &{} 0 &{} 0 &{} -2 &{} 1 &{} -1 &{} 0 &{} -2 &{} 2 &{} 3 &{} -1 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1\\ 0 &{} -2 &{} 1 &{} 0 &{} 0 &{} 2 &{} -1 &{} 1 &{} -1 &{} 1 &{} -1 &{} -2 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} -2 \end{array}\right) . } \end{aligned}$$

Let us define

$$\begin{aligned} S={\textstyle \sum _{j=1}^{9}}A_{i},\,\,S'={\textstyle \sum _{j=1}^{9}}A_{i}' \end{aligned}$$

The classes of the \(\mathrm{(-2)}\)-curves \((R_{i})\) defined in Sect. 4.3 are

$$\begin{aligned} \left( R_{i}\right) =2L-\tfrac{1}{3}\left( S+2S'+3A_{i}+3A'_{i}\right) , \end{aligned}$$

where L is the pullback of a line by the double cover map \(X_{\lambda }\rightarrow \mathbb {P}^{2}\). The translation automorphism \(\tau \) defined in Sect. 4.3 sends L to the class

$$\begin{aligned} L'=7L-\tfrac{4}{3}(S+2S'). \end{aligned}$$

The three polynomials A, B, D in \(\mathbb {Q}(\omega )(t)\) of Sect. 4.4 are defined as follows:

$$\begin{aligned}&A=\left( \lambda ^{3}t^{2}+3\lambda ^{3}-4t^{2}\right) \left( \lambda ^{3}t^{2}+3\lambda ^{3}+(6\omega +6)\lambda ^{2}t^{2}+(-6\omega -6)\lambda ^{2}-4t^{2}\right) \\&\quad \cdot \left( \lambda ^{3}t^{2}+3\lambda ^{3}-6\lambda ^{2}t^{2}+6\lambda ^{2}-4t^{2}\right) \left( \lambda ^{3}t^{2}+3\lambda ^{3}-6\omega \lambda ^{2}t^{2}+6\omega \lambda ^{2}-4t^{2}\right) ,\\&B=(\lambda ^{6}t^{4}+6\lambda ^{6}t^{2}+9\lambda ^{6}+6\lambda ^{5}t^{4}+12\lambda ^{5}t^{2}-18\lambda ^{5}-18\lambda ^{4}t^{4}+36\lambda ^{4}t^{2}-18\lambda ^{4}\\&\quad -\,8\lambda ^{3}t^{4}-24\lambda ^{3}t^{2}-24\lambda ^{2}t^{4}+24\lambda ^{2}t^{2}+16t^{4})\\&\quad \cdot (\lambda ^{6}t^{4}\!+\!6\lambda ^{6}t^{2}\!+\!9\lambda ^{6}\!+\!(-6\omega -6)\lambda ^{5}t^{4}\!+\!(-12\omega -12)\lambda ^{5}t^{2}\!+\!(18\omega +18)\lambda ^{5}-18\omega \lambda ^{4}t^{4}\\&\quad +\,36\omega \lambda ^{4}t^{2}-18\omega \lambda ^{4}\!-\!8\lambda ^{3}t^{4}\!-\!24\lambda ^{3}t^{2}\!+\!(24\omega \!+\!24)\lambda ^{2}t^{4}\!+\!(-24\omega -24)\lambda ^{2}t^{2}\!+\!16t^{4})\\&\quad \cdot (\lambda ^{6}t^{4}+6\lambda ^{6}t^{2}+9\lambda ^{6}+6\omega \lambda ^{5}t^{4}+12\omega \lambda ^{5}t^{2}-18\omega \lambda ^{5}+(18\omega +18)\lambda ^{4}t^{4}\\&\quad +\,(-36\omega -36)\lambda ^{4}t^{2}+(18\omega +18)\lambda ^{4}-8\lambda ^{3}t^{4}\!-\!24\lambda ^{3}t^{2}\!-\!24\omega \lambda ^{2}t^{4}\!+\!24\omega \lambda ^{2}t^{2}\!+\!16t^{4}), \\&D=((\lambda +2)t-(2\omega +1)\lambda )((\lambda -2\omega -2)t-(2\omega +1)\lambda )((\lambda +2\omega )t-(2\omega +1)\lambda )\\&\quad \cdot (t^{2}-1)((\lambda +2)t\!+\!(2\omega +1)\lambda )((\lambda -2\omega -2)t\!+\!(2\omega +1)\lambda )((\lambda +2\omega )t\!+\!(2\omega +1)\lambda ). \end{aligned}$$

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Kohel, D., Roulleau, X. & Sarti, A. A special configuration of 12 conics and generalized Kummer surfaces. manuscripta math. 169, 369–399 (2022). https://doi.org/10.1007/s00229-021-01334-2

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