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Hilbert squares of genus 16 K3 surfaces

Junyu Meng

TL;DR

The article advances the understanding of genus $16$ K3 surfaces by proving that the Hilbert scheme $S^{[2]}$ is isomorphic to the moduli space $M_{S'}(2,h',7)$ for the FM partner $S'$, and by constructing a natural embedding of a projectivized bundle $X'= extbf{P}_{S'}(F'^*)$ into both hyper-Kähler fourfolds in a way compatible with the Fourier–Mukai correspondence. It provides an explicit geometric realization of this isomorphism via a short exact sequence involving the rank-$8$ bundle $E_8'$, yielding a concrete description of the isomorphism in terms of stable sheaves and Brill–Noether loci, and it situates these objects within Mukai’s projective models of genus $16$ K3s. The paper further connects to the Debarre–Voisin program by explicitly constructing a trivector $t_1$ on $V_{10}$ that should realize $S^{[2]}$ as a Debarre–Voisin fourfold $DV(t_1)$, thereby offering a geometric explanation for the isomorphism and linking the genus $16$ hyper-Kähler geometry to modern projective constructions. Overall, it provides a coherent geometric picture tying together Hilbert schemes, moduli of sheaves, FM partners, and Debarre–Voisin varieties in the genus $16$ setting, with explicit, verifiable structures.

Abstract

We consider the geometry of a general polarized K3 surface $(S,h)$ of genus 16 and its Fourier-Mukai partner $(S',h')$. We prove that $S^{[2]}$ is isomorphic to the moduli space $M_{S'}(2,h',7)$ of stable sheaves with Mukai vector $(2,h',7)$ and describe the embeddings of the projectivization of the stable vector bundle of Mukai vector $(2,-h',8)$ over $S'$ into these two isomorphic hyper-Kähler fourfolds. Following the work of FrédÉric Han in arXiv:2501.16013, we explicitly construct an interesting 3-form $t_1\in \wedge^3 V_{10}^*$ which potentially gives an isomorphism between $S^{[2]}$ and the Debarre-Voisin fourfold in $G(6,V_{10})$ associated to $t_1\in \wedge^3 V_{10}^*$. This would provide a geometric explanation of the existence of such an isomorphism, which was proved in arXiv:2102.11622 by a completely different argument.

Hilbert squares of genus 16 K3 surfaces

TL;DR

The article advances the understanding of genus K3 surfaces by proving that the Hilbert scheme is isomorphic to the moduli space for the FM partner , and by constructing a natural embedding of a projectivized bundle into both hyper-Kähler fourfolds in a way compatible with the Fourier–Mukai correspondence. It provides an explicit geometric realization of this isomorphism via a short exact sequence involving the rank- bundle , yielding a concrete description of the isomorphism in terms of stable sheaves and Brill–Noether loci, and it situates these objects within Mukai’s projective models of genus K3s. The paper further connects to the Debarre–Voisin program by explicitly constructing a trivector on that should realize as a Debarre–Voisin fourfold , thereby offering a geometric explanation for the isomorphism and linking the genus hyper-Kähler geometry to modern projective constructions. Overall, it provides a coherent geometric picture tying together Hilbert schemes, moduli of sheaves, FM partners, and Debarre–Voisin varieties in the genus setting, with explicit, verifiable structures.

Abstract

We consider the geometry of a general polarized K3 surface of genus 16 and its Fourier-Mukai partner . We prove that is isomorphic to the moduli space of stable sheaves with Mukai vector and describe the embeddings of the projectivization of the stable vector bundle of Mukai vector over into these two isomorphic hyper-Kähler fourfolds. Following the work of FrédÉric Han in arXiv:2501.16013, we explicitly construct an interesting 3-form which potentially gives an isomorphism between and the Debarre-Voisin fourfold in associated to . This would provide a geometric explanation of the existence of such an isomorphism, which was proved in arXiv:2102.11622 by a completely different argument.

Paper Structure

This paper contains 10 sections, 40 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Statement of the result
  4. Acknowledgements.
  5. Genus 16 Polarized K3 surfaces
  6. Fourier-Mukai Partner $S'$
  7. The $\mathbb{P}^1$-bundle $X'=\mathbb{P}_{S'}(F'^*)$ as a prime exceptional divisor
  8. Explicit isomorphism $S^{[2]}\cong M_{S'}(2,h',7)$
  9. Back to Mukai's model
  10. More Projective Geometry

Key Result

Theorem 1.2

There exist natural embeddings of $X'$ into both $S^{[2]}$ and $M_{S'}(2,h',7)$. Moreover, the embedding of $X'$ into $S^{[2]}$ induces an identification of the restriction of the rank 6 subbundle of $V_{10}\otimes\mathcal{O}_{S^{[2]}}$ to $X'$ with the rank 6 subbundle of $V_{10}'^*\otimes \mathcal{O}_{X'}$, which are constructed in DHOV and Frederic respectively.

Theorems & Definitions (89)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 79 more