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Izv. RAN. Ser. Mat., 2008, Volume 72, Issue 3, Pages 89–102 (Mi izv1130)  

This article is cited in 2 scientific papers (total in 2 papers)

Explicit correspondences of a K3 surface with itself

C. G. Madonnaa, V. V. Nikulinbc

a Spanish National Research Council (Consejo Superior de Investigaciones Científicas)
b Steklov Mathematical Institute, Russian Academy of Sciences
c University of Liverpool

Abstract: Let $X$ be a K3-surface with a polarization $H$ of degree $H^2=2rs$, $r,s\ge1$. We consider the moduli space $Y$ of sheaves over $X$ with a primitive isotropic Mukai vector $(r,H,s)$. This space is again a K3-surface. In earlier papers, we gave necessary and sufficient conditions (in terms of the Picard lattice $N(X)$) for $Y$ and $X$ to be isomorphic. Here we show that these conditions imply the existence of an isomorphism between $Y$ and $X$ which is a composite of certain universal geometric isomorphisms between moduli of sheaves over $X$ and Tyurin's geometric isomorphism between moduli of sheaves over $X$ and $X$ itself. It follows that a general K3-surface $X$ with $\rho(X)=\operatorname{rk}N(X)\le2$ is isomorphic to $Y$ if and only if there is an isomorphism $Y\cong X$ which is a composite of universal isomorphisms and Tyurin's isomorphism.

DOI: https://doi.org/10.4213/im1130

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English version:
Izvestiya: Mathematics, 2008, 72:3, 497–508

Bibliographic databases:

Document Type: Article
UDC: 512.774+512.723
MSC: 14J28, 14J60
Received: 10.07.2006

Citation: C. G. Madonna, V. V. Nikulin, “Explicit correspondences of a K3 surface with itself”, Izv. RAN. Ser. Mat., 72:3 (2008), 89–102; Izv. Math., 72:3 (2008), 497–508

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Viacheslav V. Nikulin, “Self-correspondences of K3 surfaces via moduli of sheaves and arithmetic hyperbolic reflection groups”, Proc. Steklov Inst. Math., 273 (2011), 229–237  mathnet  crossref  mathscinet  zmath  isi  elib
    2. Madonna C.G., “On some moduli spaces of bundles on $K3$ surfaces, II”, Proc. Amer. Math. Soc., 140:10 (2012), 3397–3408  crossref  mathscinet  zmath  isi  elib  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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