The polarized degree of irrationality of $K3$ surfaces

Abstract

Given a polarized variety $(X,L)$, we construct and study projections of low degree $X⇢\mathbb{P}(H^0(L^{\vee }))⇢{\mathbb{P}}^n$ using the associated kernel bundles. As an application, we can show that the degree of irrationality of a very general $(1,6)-$polarized abelian surface, as well as that of a very general $K3$ surface of genus $6$ is $3$. We also give new upper bounds on the degree of irrationality of $K3$ surfaces of any genus. 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 generically finite rational maps of low degree from some hyper-Kähler varieties and abelian varieties to projective spaces.

Accepted version to appear in Annali SNS. For reference purposes, note that the number of certain sections changed with respect to some older versions (Lemma 2.1 is now Proposition 1.5+Corollary 6.5, Section 2(.0) is now Section 1+ Section 6, Section 2.1 is now Section 4, Section 2.2 is now Section 2, Section 3 is now Section 5, Section 3 is new only on (1,6) abelian surfaces)