A special configuration of 12 conics and generalized Kummer surfaces
In analogy with classical Kummer surfaces, a \textit{generalized Kummer surface} is the minimal desingularization of the quotient of an abelian surface \(A\) by a symplectic group \(G\subseteq \mathrm{Aut}(A)\). In this article the authors focus on the case \(G\cong \mathbb{Z}/3\mathbb{Z}\): the minimal desingularization \(\mathrm{Kum}_3(A)\) of a quotient \(A/G\) is a \(K3\) surface with nine disjoint \(A_2\)-configurations (i.e. pairs of smooth rational curves \(C,C'\) with \(C\cdot C'=1\)), arising over the nine cusps of \(A/G\). We call this a \textit{\(9A_2\)-configuration}. Conversely, any \(9A_2\) configuration on a \(K3\) surface \(X\) leads to a generalized Kummer structure on \(X\), i.e. an abelian surface \(A\) such that \(X\cong \mathrm{Kum}_3(A)\). The goal of the article under review is to investigate the geometry of a particular family of generalized Kummer surfaces, previously studied by \textit{C. Birkenhake} and \textit{H. Lange} [Manuscr. Math. 85, No. 3--4, 393--407 (1994; Zbl 0821.14027)]. For generic \(\lambda\), the dual of the cubic curve \(E_\lambda: x^3+y^3+z^3-3\lambda xyz=0\) is a sextic curve \(C_\lambda\) with nine \(A_2\)-singularities. Thus the double cover \(X_\lambda\) of \(\mathbb{P}^2\), branched along \(C_\lambda\) is a \(K3\) surface with a generalized Kummer structure. The authors show that there are precisely \(12\) conics in \(\mathbb{P}^2\) passing through \(6\) of the \(9\) cusps of \(C_\lambda\). Their pullbacks on \(X_\lambda\) are \(24\) smooth rational curves, and the authors use these curves to produce nine new \(9A_2\)-configurations on \(X_\lambda\). Moreover they prove that these new \(9A_2\)-configurations are in the \(\mathrm{Aut}(X_\lambda)\)-orbit of the original \(9A_2\)-configuration. Finally, the authors realize the surfaces \(X_\lambda\) as degree \(8\) non-complete intersections in \(\mathbb{P}^5\), and they obtain a Weierstrass model for them.
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