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Construction of Nikulin configurations on some Kummer surfaces and applications. (English) Zbl 1411.14044
A Nikulin configuration \({\mathcal{C}}\) is the data of \(16\) disjoint smooth rational \((-2)\)-curves \(A_1,\cdots,A_{16}\) on a \(K3\) surface \(X\). Then \(X\) is a Kummer surface, \(X=\text{Km}(B)\) where \(B\) is an abelain surface determined by \({\mathcal{C}}\). Let \(B\) be a generic abelian surface having a polarization \(M\) with \(M^2=k(k+1)\) (for \(k>0\) an integer), and let \(X={\text{Km}}(B)\). Let \({\mathcal{C}}=A_1+\cdots+A_{16}\) be the natural Nikulin configuration on \(X=\text{Km}(B)\). The first result is the existence of another Nikulin configuration on \(X\).
Theorem 1. Let \(t\in\{1,\cdots,16\}\). There exists a \((-2)\)-curve \(A^{\prime}_i\) on \(\text{Km}(B)\) such that \(A_tA_t^{\prime}=4k+2\) and \({\mathcal{C}}_t=A^{\prime}_t+\sum_{j\neq i} A_j\) is another Nikulin configuration. The numerical class of \(A_t^{\prime}\) is \(2L-(2k+1)A_j\); the class \(L_t^{\prime}=(2k+1)L-2k(k+1)A_t\) generates the orthogonal complement of the \(16\) curves \(A_t^{\prime}\) and \(\{A_j|j\neq i\}\); moreover, \(L_t^{\prime,2}=L^2\).
Theorem 2. Suppose \(k\geq 2\). There is no automorphism of \(X\) sending the Nikulin configuration \({\mathcal{C}}=\sum_{j=1}^{16} A_j\) to the configuration \({\mathcal{C}}^{\prime}=A_t^{\prime} +\sum_{j\neq i} A_j\).
Theorem 2 gives the first explicit construction of two distinct Kummer structures on a Kummer surface, namely, \(X=\text{Km}(B)= \text{Km}(B^{\prime})\), but \(B\) and \(B^{\prime}\) are not isomorphic.
Then an infinite order automorphism on the Kummer surface is constructed.
Finally, given the two Nikulin configurations \({\mathcal{C}}\) and \({\mathcal{C}}^{\prime}\) on a \(K3\) surface \(X\), a bi-double cover \(S\to X\), which is a surface of general type, is constructed, and its properties are studied.
MSC:
14J28 \(K3\) surfaces and Enriques surfaces
14J50 Automorphisms of surfaces and higher-dimensional varieties
14J29 Surfaces of general type
14J10 Families, moduli, classification: algebraic theory
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