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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2003, Volume 241, Pages 132–168 (Mi tm393)  

This article is cited in 13 scientific papers (total in 13 papers)

On a Classical Correspondence between K3 Surfaces

C. G. Madonnaa, V. V. Nikulinbc

a Università degli Studi di Roma — Tor Vergata
b Steklov Mathematical Institute, Russian Academy of Sciences
c University of Liverpool
References:
Abstract: Let $X$ be a K3 surface that is the intersection (i.e. a net $\mathbb P^2$) of three quadrics in $\mathbb P^5$. The curve of degenerate quadrics has degree 6 and defines a natural double covering $Y$ of $\mathbb P^2$ ramified in this curve which is again a K3. This is a classical example of a correspondence between K3 surfaces that is related to the moduli of sheaves on K3 studied by Mukai. When are general (for fixed Picard lattices) $X$ and $Y$ isomorphic? We give necessary and sufficient conditions in terms of Picard lattices of $X$ and $Y$. For example, for the Picard number 2, the Picard lattice of $X$ and $Y$ is defined by its determinant $-d$, where $d>0$, $d\equiv 1\mod 8$, and one of the equations $a^2-db^2=8$ or $a^2-db^2=-8$ has an integral solution $(a,b)$. Clearly, the set of these $d$ is infinite: $d\in \{(a^2\mp 8)/b^2\}$, where $a$ and $b$ are odd integers. This gives all possible divisorial conditions on the 19-dimensional moduli of intersections of three quadrics $X$ in $\mathbb P^5$, which imply $Y\cong X$. One of them, when $X$ has a line, is classical and corresponds to $d=17$. Similar considerations can be applied to a realization of an isomorphism $(T(X)\otimes \mathbb Q, H^{2,0}(X)) \cong (T(Y)\otimes \mathbb Q, H^{2,0}(Y))$ of transcendental periods over $\mathbb Q$ of two K3 surfaces $X$ and $Y$ by a fixed sequence of types of Mukai vectors.
Received in November 2002
Bibliographic databases:
Document Type: Article
UDC: 512.7
Language: Russian
Citation: C. G. Madonna, V. V. Nikulin, “On a Classical Correspondence between K3 Surfaces”, Number theory, algebra, and algebraic geometry, Collected papers. Dedicated to the 80th birthday of academician Igor' Rostislavovich Shafarevich, Trudy Mat. Inst. Steklova, 241, Nauka, MAIK «Nauka/Inteperiodika», M., 2003, 132–168; Proc. Steklov Inst. Math., 241 (2003), 120–153
Citation in format AMSBIB
\Bibitem{MadNik03}
\by C.~G.~Madonna, V.~V.~Nikulin
\paper On a~Classical Correspondence between K3 Surfaces
\inbook Number theory, algebra, and algebraic geometry
\bookinfo Collected papers. Dedicated to the 80th birthday of academician Igor' Rostislavovich Shafarevich
\serial Trudy Mat. Inst. Steklova
\yr 2003
\vol 241
\pages 132--168
\publ Nauka, MAIK «Nauka/Inteperiodika»
\publaddr M.
\mathnet{http://mi.mathnet.ru/tm393}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2024049}
\zmath{https://zbmath.org/?q=an:1076.14046}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2003
\vol 241
\pages 120--153
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  • This publication is cited in the following 13 articles:
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