Fano threefolds and K3 surfaces

TL;DR

This paper characterizes which K3 surfaces can be realized as anticanonical divisors in Fano threefolds, linking their Picard lattices and polarization classes to the Fano threefold's Picard group and intersection form.

Contribution

It establishes a precise criterion connecting K3 surfaces with given Picard lattices to Fano threefolds via their anticanonical divisors.

Findings

A general K3 surface with specified Picard lattice and polarization can be embedded as an anticanonical divisor in a Fano threefold.

The isomorphism between the Picard lattice and that of a Fano threefold determines the embedding.

The intersection product on the Picard lattice corresponds to the Fano threefold's first Chern class and intersection pairing.

Abstract

We discuss in this note which K3 surfaces appear as anticanonical divisors in a Fano threefold. We prove in particular that a general K3 surface with given Picard lattice P and polarization class h in P is an anticanonical divisor in a Fano threefold if and only if (P,h) is isomorphic to (Pic(V), c_1(V)) for some Fano threefold V, where Pic(V) is equipped with the intersection product (L,M) --> (L.M.c_1(V)).