On a classical correspondence between \(K3\) surfaces.

*(English. Russian original)*Zbl 1076.14046
Proc. Steklov Inst. Math. 241, 120-153 (2003); translation from Tr. Mat. Inst. Im. V. A. Steklova 241, 132-168 (2003).

Let \(X\) be a \(K3\) surface that is the intersection of three quadrics in \({\mathbb P}^5\). The three quadrics define the plane \(\mathbb P^2\) (the net) of the quadrics such that the curve \(C \subset {\mathbb P}^2\) of degenerate quadrics is of degree \(6\). Let \(Y\) be the minimal resolution of the double cover of \({\mathbb P}^2\) branched over \(C\). Then \(Y\) is also a \(K3\) surface. This correspondence between \(X\) and \(Y\) is a very classical one. In the article under review, the authors consider the following problems.

Question 1. When is \(X \cong Y\)?

Question 2. What conditions on the Picard lattice \(N(X)\) and its primitive vector \(H\) with \(H^2 = 8\) are sufficient for \(Y\) to be isomorphic to \(X\) and are necessary if \(X\) is a general \(K3\) surface with the Picard lattice \(N(X)\)?

The second question has been answered successfully (see the article for technical formulation). As a consequence, when the Picard number \(\rho(X) = \operatorname{rank} N(X) \geq 12\), then \(X \cong Y\) holds if and only if there exists an \(x \in N(X)\) such that \(x . H \equiv 1\pmod 2\). It is known that those \(X\) form a moduli space of dimension 19. So one sees that those \(X\) satisfying \(X \cong Y\) form a divisor (modulo codimension 2) in this 19-dimensional space. The authors show that the general member \(X\) in this divisor is of Picard number 2 and discriminant \(-d\) for some \(d > 0\) with \(d \equiv 1\pmod 8\). Further, they show that the set \(D\) of those \(d\) satisfies either \(a^2 - d b^2 = 8\), or \(a^2 - d b^2 = -8\) for some integers \(a, b\). The set \(D\) gives the label to the connected components of the divisor above. E.g. \(d = 17\) gives the classical example of \(X\) containing a line such that \(X \cong Y\). Another pretty corrollary is: \(Y \cong X\) holds if there exists an \(h_1 \in N(X)\) such that the primitive sublattice \([H, h_1]_{\text{pr}}\) in \(N(X)\) generated by \(H\) and \(h_1\) has an odd determinant and satisfies \(h_1^2 = \pm 4\), \(h_1 . H \equiv 0\pmod 2\). These conditions are also necessary if either \(\rho(X) = 1\), or \(\rho(X) = 2\) and \(X\) is a general \(K3\) surface with its Picard lattice. Finally the authors give sufficient conditions for an isomorphism \((T(X) \otimes {\mathbb Q}, H^{2,0}(X)) \cong (T(Y) \otimes {\mathbb Q}, H^{2,0}(Y))\) of transcendental periods to be factorized as a sequence of correspondences defined by isotropic Mukai vectors.

For the entire collection see [Zbl 1059.11002].

Question 1. When is \(X \cong Y\)?

Question 2. What conditions on the Picard lattice \(N(X)\) and its primitive vector \(H\) with \(H^2 = 8\) are sufficient for \(Y\) to be isomorphic to \(X\) and are necessary if \(X\) is a general \(K3\) surface with the Picard lattice \(N(X)\)?

The second question has been answered successfully (see the article for technical formulation). As a consequence, when the Picard number \(\rho(X) = \operatorname{rank} N(X) \geq 12\), then \(X \cong Y\) holds if and only if there exists an \(x \in N(X)\) such that \(x . H \equiv 1\pmod 2\). It is known that those \(X\) form a moduli space of dimension 19. So one sees that those \(X\) satisfying \(X \cong Y\) form a divisor (modulo codimension 2) in this 19-dimensional space. The authors show that the general member \(X\) in this divisor is of Picard number 2 and discriminant \(-d\) for some \(d > 0\) with \(d \equiv 1\pmod 8\). Further, they show that the set \(D\) of those \(d\) satisfies either \(a^2 - d b^2 = 8\), or \(a^2 - d b^2 = -8\) for some integers \(a, b\). The set \(D\) gives the label to the connected components of the divisor above. E.g. \(d = 17\) gives the classical example of \(X\) containing a line such that \(X \cong Y\). Another pretty corrollary is: \(Y \cong X\) holds if there exists an \(h_1 \in N(X)\) such that the primitive sublattice \([H, h_1]_{\text{pr}}\) in \(N(X)\) generated by \(H\) and \(h_1\) has an odd determinant and satisfies \(h_1^2 = \pm 4\), \(h_1 . H \equiv 0\pmod 2\). These conditions are also necessary if either \(\rho(X) = 1\), or \(\rho(X) = 2\) and \(X\) is a general \(K3\) surface with its Picard lattice. Finally the authors give sufficient conditions for an isomorphism \((T(X) \otimes {\mathbb Q}, H^{2,0}(X)) \cong (T(Y) \otimes {\mathbb Q}, H^{2,0}(Y))\) of transcendental periods to be factorized as a sequence of correspondences defined by isotropic Mukai vectors.

For the entire collection see [Zbl 1059.11002].

Reviewer: De-Qi Zhang (Singapore)

##### MSC:

14J28 | \(K3\) surfaces and Enriques surfaces |

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\textit{C. Madonna} and \textit{V. V. Nikulin}, in: Number theory, algebra, and algebraic geometry. Collected papers dedicated to the 80th birthday of Academician Igor' Rostislavovich Shafarevich. Transl. from the Russian. Moskva: Maik Nauka/Interperiodika. 120--153 (2003; Zbl 1076.14046); translation from Tr. Mat. Inst. Im. V. A. Steklova 241, 132--168 (2003)