Fano visitor problem for K3 surfaces

TL;DR

This work establishes that a K3 surface $X$ of Picard number $1$ and genus $g$ is a Fano visitor when $g\not\equiv 3\pmod{4}$ and a weak Fano visitor when $g\equiv 3\pmod{4}$, by constructing a chain of birational transformations among smooth hyperkähler-type varieties tied to Bridgeland stability on the derived category $D^b(X)$. Central to the method is a Beaville–Mukai-type moduli space $\mathcal{M}$ with Mukai vector $v=(0,h,1-g)$, whose wall-crossings induce a sequence of Mukai flops and standard flips; restricting these to fixed loci of an antisymplectic involution yields a controllable series of birational steps $M_i\dashrightarrow M_{i+1}$, starting from $M_1=\mathrm{Bl}_X\mathbb{P}^g$. Through twisted Bondal–Orlov-type fully faithful functors and detailed analysis of ample cones, the authors obtain embeddings $D^b(X)\hookrightarrow D^b(M_j)$ into a Fano (or weak Fano) model, with consequences for Hilbert schemes of $X$ as additional visitors. In the genus $g\equiv 3\pmod{4}$ case, a Mukai dual $\hat{X}$ provides a derived-equivalence symmetry, sending the flip sequence to its reverse and showing $\hat{X}$ is typically non-isomorphic to $X$. A divisorial contraction for $4\mid g$ further describes a contraction to a singular Fano with Lagrangian/orthogonal Grassmannian fibers, enriching the landscape of Fano visitors via hyperkähler geometry.

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 Macrì 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 hyperkähler manifolds by Saccà, Macrì, O'Grady, and Flapan.

Theorems & Definitions (65)

• Theorem A
• Theorem B
• Conjecture 1
• Corollary 1.1
• Proposition 1.2
• Proposition 2.1
• Remark 2.2
• Proposition 3.1
• Lemma 4.1
• proof