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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2003, Volume 241, Pages 132–168
(Mi tm393)
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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
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
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
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https://www.mathnet.ru/eng/tm393 https://www.mathnet.ru/eng/tm/v241/p132
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