Projective models for Hilbert squares of $K3$ surfaces

TL;DR

The paper investigates the projective geometry of the Hilbert square $S^{[2]}$ of a very general polarized $K3$ surface, focusing on the linear system $|L_2-2\delta|$ of quadrics containing $S$. It proves that $|L_2-2\delta|$ is very ample for genus $g\ge7$ and analyzes the genus-7 and genus-8 cases via Mukai models, realizing $S^{[2]}$ as corank-2 or rank-4 degeneracy loci inside ambient homogeneous spaces and deriving explicit presentations of the homogeneous ideal and syzygies. It also establishes Le Potier's strange duality for these polarizations and examines deformations to nearby hyperkähler fourfolds, including projective normality results and detailed minimal free resolutions in low genus. The work reveals striking parallels between genus-7 and genus-8 constructions, connects $S^{[2]}$ to its Fourier--Mukai partner, and sets a foundation for a systematic understanding of projective models and syzygies of hyperkähler fourfolds, with implications for locally complete families and higher-dimensional analogues.

Abstract

For a very general polarized $K3$ surface $S\subset \mathbb{P}^g$ of genus $g\ge 5$, we study the linear system on the Hilbert square $S^{[2]}$ parametrizing quadrics in $\mathbb{P}^g$ that contain $S$. We prove its very ampleness for $g\geq 7$. In the cases of genus 7 or 8, we describe in detail the projective geometry of the corresponding embedding by making use of the Mukai model for $S$. In both cases, it can be realized as a degeneracy locus on an ambient homogeneous space, in a strikingly similar fashion. In consequence, we give explicit descriptions of its ideal and syzygies. Furthermore, we extract new information on the locally complete families, in a first step towards the understanding of their projective geometry.

Theorems & Definitions (74)

• Theorem A
• Corollary A
• Theorem B
• Theorem C
• Theorem D
• Theorem 2.1: Green's duality theorem
• Remark 1
• Lemma 1: ogrady
• Remark 2
• Lemma 2