On a~Classical Correspondence between K3 Surfaces

C. G. Madonna ; V. V. Nikulin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Number theory, algebra, and algebraic geometry, Tome 241 (2003), pp. 132-168

Résumé

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.