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Tr. Mat. Inst. Steklova, 2004, Volume 246, Pages 217–239 (Mi tm157)  

This article is cited in 4 scientific papers (total in 4 papers)

On Correspondences of a K3 Surface with Itself. I

V. V. Nikulinab

a Steklov Mathematical Institute, Russian Academy of Sciences
b University of Liverpool

Abstract: Let $X$ be a K3 surface with a polarization $H$ of degree $H^2=2rs$, $r,s\ge 1$. Assume that $H\cdot N(X)=\mathbb Z$ for the Picard lattice $N(X)$. The moduli space of sheaves over $X$ with the isotropic Mukai vector $(r,H,s)$ is again a K3 surface $Y$. We prove that $Y\cong X$ if there exists $h_1\in N(X)$ with $h_1^2=f(r,s)$, $H\cdot h_1\equiv 0\mathrm { mod} g(r,s)$, and $h_1$ satisfies some condition of primitivity. These conditions are necessary if $X$ is general with $\mathop {\mathrm{rk}}N(X)=2$. The existence of such kind of a riterion is surprising, and it also gives some geometric interpretation of elements in $N(X)$ with negative square. We describe all irreducible 18-dimensional components of the moduli space of pairs $(X,H)$ with $Y\cong X$. We prove that their number is always infinite. Earlier, similar results have been known only for $r=s$.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2004, 246, 204–226

Bibliographic databases:
UDC: 512.7
Received in February 2004

Citation: V. V. Nikulin, “On Correspondences of a K3 Surface with Itself. I”, Algebraic geometry: Methods, relations, and applications, Collected papers. Dedicated to the memory of Andrei Nikolaevich Tyurin, corresponding member of the Russian Academy of Sciences, Tr. Mat. Inst. Steklova, 246, Nauka, MAIK Nauka/Inteperiodika, M., 2004, 217–239; Proc. Steklov Inst. Math., 246 (2004), 204–226

Citation in format AMSBIB
\Bibitem{Nik04}
\by V.~V.~Nikulin
\paper On Correspondences of a~K3 Surface with Itself.~I
\inbook Algebraic geometry: Methods, relations, and applications
\bookinfo Collected papers. Dedicated to the memory of Andrei Nikolaevich Tyurin, corresponding member of the Russian Academy of Sciences
\serial Tr. Mat. Inst. Steklova
\yr 2004
\vol 246
\pages 217--239
\publ Nauka, MAIK Nauka/Inteperiodika
\publaddr M.
\mathnet{http://mi.mathnet.ru/tm157}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2101295}
\zmath{https://zbmath.org/?q=an:1130.14030}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2004
\vol 246
\pages 204--226


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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. Nikulin V.V., “On correspondences of a K3 surface with itself. II”, Algebraic Geometry, Contemporary Mathematics Series, 422, 2007, 121–172  crossref  mathscinet  zmath  isi
    2. C. G. Madonna, V. V. Nikulin, “Explicit correspondences of a K3 surface with itself”, Izv. Math., 72:3 (2008), 497–508  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. 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
    4. 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
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