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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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