On rational maps between $\mathrm{K3}$ surfaces

I want to discuss the proof of the following theorem: if there is a dominant rational map $W \dasharrow S$ between two complex projective $\mathrm{K3}$ surfaces $W$ and $S$, then Picard number $\rho(S)$ is strictly greater than $1$. I will also bring some examples illustrating this theorem. Based on joint work with Ilya Karzhemanov.