Fano visitor problem for K3 surfaces

TL;DR

This paper proves that certain K3 surfaces can be embedded into the derived categories of Fano or weak Fano varieties, expanding understanding of their geometric and categorical relationships.

Contribution

It demonstrates that K3 surfaces with specific genus conditions are Fano visitors, constructing explicit embeddings into derived categories of Fano or weak Fano varieties.

Findings

K3 surfaces with genus g not congruent to 3 mod 4 are Fano visitors.

Constructs explicit embeddings of derived categories for these K3 surfaces.

Uses birational geometry and stability conditions to establish the embeddings.

Abstract

Let $X$ be a K3 surface with Picard number 1 and genus $g$, such that $g\not\equiv 3 \mod 4$. In this paper, we show that $X$ is a Fano visitor, i.e., there is a smooth Fano variety $Y$ and an embedding $D^b(X)\hookrightarrow D^b(Y)$ given by a fully faithful functor. If $g\equiv 3\mod 4$, we construct a smooth weak Fano variety $Y$. Our proof is based on several results concerning a sequence of flips associated with a K3 surface and an ample line bundle. This sequence is constructed by using the work of Bayer and Macr\`i on the description of the birational geometry of a moduli space of sheaves on a K3 surface through Bridgeland stability conditions, and the study of the fixed locus of antisymplectic involutions on hyperk\"ahler manifolds by Sacc\`a, Macr\`i, O'Grady, and Flapan.