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Geometry of genus sixteen K3 surfaces

Frederic Han

TL;DR

We address explicit projective models for genus-16 K3 surfaces via Mukai's rank-2 bundle F, constructing X = Proj(F^∨) in $\mathbb{P}_9$ and proving it is defined by ten quadrics. The paper gives an effective Mukai unirationalization-based method to compute these quadrics and describes the resulting double cover ramified along a degree-10 hypersurface, connecting to hyperkähler geometry and Debarre-Voisin theory. It then develops a canonical framework tying S and X to constructions on S (via Lazarsfeld-Mukai bundles) and to trivectors t2 and t1 that realize Debarre-Voisin varieties and Hilb^2(S), including discriminant and ramification phenomena with concrete computations and an F_8 example. The results suggest a rich interplay between vector-bundle syzygies, DV varieties, Peskine loci, and K3 geometry, with several conjectures guiding future work.

Abstract

Polarized K3 surfaces of genus sixteen have a Mukai vector bundle of rank two. We study the geometry of the projectivization of this bundle. We prove that it has an embedding in $\mathbb{P}_9$ with an ideal generated by quadrics. We give an effective method to compute these quadrics from a general choice in Mukai's unirationalization of the moduli space. This linear system gives a double cover of $\mathbb{P}_9$ ramified on a degree $10$ hypersurface. It gives relative Weddle/Kummer surfaces over a Peskine variety associated to an explicit trivector. This work is also motivated by hyperkähler geometry and Debarre-Voisin varieties. Oberdieck showed that the Hilbert square of a general K3-surface of genus $16$ is a Debarre-Voisin variety for some trivector. We start to investigate the relationship between these two trivectors.

Geometry of genus sixteen K3 surfaces

TL;DR

We address explicit projective models for genus-16 K3 surfaces via Mukai's rank-2 bundle F, constructing X = Proj(F^∨) in and proving it is defined by ten quadrics. The paper gives an effective Mukai unirationalization-based method to compute these quadrics and describes the resulting double cover ramified along a degree-10 hypersurface, connecting to hyperkähler geometry and Debarre-Voisin theory. It then develops a canonical framework tying S and X to constructions on S (via Lazarsfeld-Mukai bundles) and to trivectors t2 and t1 that realize Debarre-Voisin varieties and Hilb^2(S), including discriminant and ramification phenomena with concrete computations and an F_8 example. The results suggest a rich interplay between vector-bundle syzygies, DV varieties, Peskine loci, and K3 geometry, with several conjectures guiding future work.

Abstract

Polarized K3 surfaces of genus sixteen have a Mukai vector bundle of rank two. We study the geometry of the projectivization of this bundle. We prove that it has an embedding in with an ideal generated by quadrics. We give an effective method to compute these quadrics from a general choice in Mukai's unirationalization of the moduli space. This linear system gives a double cover of ramified on a degree hypersurface. It gives relative Weddle/Kummer surfaces over a Peskine variety associated to an explicit trivector. This work is also motivated by hyperkähler geometry and Debarre-Voisin varieties. Oberdieck showed that the Hilbert square of a general K3-surface of genus is a Debarre-Voisin variety for some trivector. We start to investigate the relationship between these two trivectors.

Paper Structure

This paper contains 18 sections, 29 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Equations of $X$ via Mukai's unirationalization
  3. Initial choices in Mukai construction
  4. Explicit construction
  5. Equations of $X$ in $\mathbb{P}_9$
  6. A canonical involution ${i_X}$ on $\mathbb{P}_9$
  7. Canonical constructions from $(S,h)$
  8. Equations of $S$ in $\mathbb{P}_{16}$
  9. Smoothness of $X$
  10. Basic pieces
  11. Syzygies of $X$ and the trivector $t_2$
  12. Geometry of $t_2$ and the double cover of ${\mathbb{P}(V_{10}^{\vee})}$
  13. Further results and observations
  14. The trivector $t_1$ and $\mathop{\mathrm{Hilb}}\nolimits^2(S)$
  15. Orthogonality of trivectors
  16. ...and 3 more sections

Key Result

Theorem 1

A generic polarized $K3$ surface $(S,h)$ of genus $16$ carries a rank $2$ vector bundle $F$ such that

Theorems & Definitions (70)

  • Theorem : mukaigenus16
  • Definition 1.0.1
  • Theorem : Th \ref{['refequationsX']}, Cor \ref{['refXsmooth']}, Cor \ref{['refdoublecover']}
  • Theorem : Th \ref{['refequationsS']}
  • Remark 2.1.1
  • Lemma 2.2.1
  • proof
  • Proposition 2.2.2
  • proof
  • Lemma 2.2.3
  • ...and 60 more