The polarized degree of irrationality of $K3$ surfaces

Given a polarized variety $(X,L)$, we construct and study projections of low degree $X\dashrightarrow \mathbb{P}(H^0(L^\vee)) \dashrightarrow \mathbb P ^n$ using the associated kernel bundles. As an application, we can show that the degree of irrationality of a very general $(1,6)$ abelian surface, as well as that of a very general $K3$ surface of genus $6$ is $3$. We also give new upper bounds for $K3$ surfaces of any genus. Moreover, in the case of surfaces, this observation can be used to show that maps of the degree at most $d$ move in families. We study the family of projections of minimal degree of a very general $K3$ surface of genus $4,5,6$. As a different application of our construction, we exhibit new rational maps of low degree for some hyper-K\"ahler varieties, abelian varieties and Gushel--Mukai threefolds.