Construction of Nikulin configurations on some Kummer surfaces and applications

Autor: Alessandra Sarti, Xavier Roulleau
Přispěvatelé: Université d'Aix-Marseille, Laboratoire de Mathématiques et Applications (LMA-Poitiers), Université de Poitiers-Centre National de la Recherche Scientifique (CNRS), Aix Marseille Université (AMU)
Rok vydání: 2018
Předmět:
Zdroj: Mathematische Annalen
Mathematische Annalen, Springer Verlag, 2019, 373 (1-2), pp.597-623. ⟨10.1007/s00208-018-1717-5⟩
Mathematische Annalen, 2019, 373 (1-2), pp.597-623. ⟨10.1007/s00208-018-1717-5⟩
ISSN: 1432-1807
0025-5831
DOI: 10.1007/s00208-018-1717-5
Popis: A Nikulin configuration is the data of $16$ disjoint smooth rational curves on a K3 surface. According to a well known result of Nikulin, if a K3 surface contains a Nikulin configuration $\mathcal{C}$, then $X$ is a Kummer surface $X=Km(B)$ where $B$ is an Abelian 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 $X=Km(B)$ be the associated Kummer surface. To the natural Nikulin configuration $\mathcal{C}$ on $X=Km(B)$, we associate another Nikulin configuration $\mathcal{C}'$; we denote by $B'$ the Abelian surface associated to $\mathcal{C}'$, so that we have also $X=Km(B')$. For $k\geq2$ we prove that $B$ and $B'$ are not isomorphic. We then construct an infinite order automorphism of the Kummer surface $X$ that occurs naturally from our situation. Associated to the two Nikulin configurations $\mathcal{C},$ $\mathcal{C}'$, there exists a natural bi-double cover $S\to X$, which is a surface of general type. We study this surface which is a Lagrangian surface in the sense of Bogomolov-Tschinkel, and for $k=2$ is a Schoen surface.
22 pages, refereed version
Databáze: OpenAIRE
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