On Correspondences of a K3 Surface with Itself. I

Let $X$ be a K3 surface with a polarization $H$ of degree $H^2=2rs$, $r,s\ge 1$. Assume that $H\cdot N(X)=\mathbb Z$ for the Picard lattice $N(X)$. The moduli space of sheaves over $X$ with the isotropic Mukai vector $(r,H,s)$ is again a K3 surface $Y$. We prove that $Y\cong X$ if there exists $h_1\in N(X)$ with $h_1^2=f(r,s)$, $H\cdot h_1\equiv 0\mathrm {\,mod}\ g(r,s)$, and $h_1$ satisfies some condition of primitivity. These conditions are necessary if $X$ is general with $\mathop {\mathrm{rk}}N(X)=2$. The existence of such kind of a riterion is surprising, and it also gives some geometric interpretation of elements in $N(X)$ with negative square. We describe all irreducible 18-dimensional components of the moduli space of pairs $(X,H)$ with $Y\cong X$. We prove that their number is always infinite. Earlier, similar results have been known only for $r=s$.