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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].

##### MSC:
 14J28 $$K3$$ surfaces and Enriques surfaces