[Paper Review] Two polarized K3 surfaces associated to the same cubic fourfold

Emma Brakkee|arXiv (Cornell University)|Aug 3, 2018

Algebraic Geometry and Number Theory15 references3 citations

TL;DR

This paper establishes that for a generic polarized K3 surface (S,L) of degree d ≡ 0 mod 6, the associated K3 surface S^τ under Hassett's involution τ is isomorphic to the moduli space of stable sheaves on S with Mukai vector (3,L,d/6). It further proves that Hilbert schemes of two points on S and S^τ are birational if and only if the Diophantine equation 3p² − (d/6)q² = −1 has an integral solution, yielding derived equivalent but non-birational Hilbert schemes for certain d.

ABSTRACT

For infinitely many $d$, Hassett showed that special cubic fourfolds of discriminant $d$ are related to polarized K3 surfaces of degree $d$ via their Hodge structures. For half of the $d$, each associated K3 surface $(S,L)$ canonically yields another one, $(S^τ,L^τ)$. We prove that $S^τ$ is isomorphic to the moduli space of stable coherent sheaves on $S$ with Mukai vector $(3,L,d/6)$. We also explain for which $d$ the Hilbert schemes $ ext{Hilb}^n(S)$ and $ ext{Hilb}^n(S^τ)$ are birational.

Motivation & Objective

• To geometrically describe the involution τ on the moduli space of polarized K3 surfaces of degree d ≡ 0 mod 6.
• To determine when the Hilbert schemes of two points on a K3 surface and its τ-partner are birational.
• To construct examples of derived equivalent but non-birational Hilbert schemes of two points on K3 surfaces.
• To clarify the relationship between the Fourier–Mukai partner S^τ and the moduli space of sheaves on S with Mukai vector (3,L,d/6).

Proposed method

• Uses the Mukai lattice and Hodge-theoretic correspondence between cubic fourfolds and K3 surfaces to define the involution τ on the moduli space M_d.
• Applies lattice-theoretic techniques, including discriminant groups and orthogonal extensions, to analyze the action of τ.
• Establishes that S^τ is isomorphic to the moduli space M_S(3,L,d/6) of stable sheaves on S with Mukai vector (3,L,d/6).
• Reduces the birationality of Hilbert schemes Hilb²(S) and Hilb²(S^τ) to the solvability of the Diophantine equation 3p² − (d/6)q² = −1.
• Employs the derived equivalence of S and S^τ via the Kuznetsov category of the associated cubic fourfold to support the geometric construction.
• Analyzes the Néron–Severi group and line bundle structures on Hilb²(S) to derive the birationality criterion.

Experimental results

Research questions

• RQ1 What is the geometric nature of the involution τ on the moduli space of degree d polarized K3 surfaces when d ≡ 0 mod 6?
• RQ2 When are the Hilbert schemes of two points on a K3 surface and its τ-partner birational?
• RQ3 Can derived equivalent Hilbert schemes of two points on K3 surfaces fail to be birational, and if so, under what conditions?
• RQ4 How does the moduli space of stable sheaves with Mukai vector (3,L,d/6) relate to the τ-partner K3 surface S^τ?
• RQ5 What role does the solvability of 3p² − (d/6)q² = −1 play in the birational geometry of Hilbert schemes of two points?

Key findings

• The involution τ maps a polarized K3 surface (S,L) to a surface S^τ that is isomorphic to the moduli space M_S(3,L,d/6) of stable sheaves on S with Mukai vector (3,L,d/6).
• For generic (S,L) with ρ(S) = 1, the Hilbert schemes Hilb²(S) and Hilb²(S^τ) are birational if and only if the Diophantine equation 3p² − (d/6)q² = −1 has an integral solution.
• There exist infinitely many d for which the equation 3p² − (d/6)q² = −1 is solvable, but not all d satisfy this condition.
• For d = 6·73, the equation is not solvable, so Hilb²(S) and Hilb²(S^τ) are not birational, despite being derived equivalent.
• The paper constructs the first known example of derived equivalent but non-birational Hilbert schemes of two points on K3 surfaces.
• For n ≥ 3, the birationality of Hilb^n(S) and Hilb^n(S^τ) depends on the solvability of either 3p²(n−1) − (d/6)q² = −1 or 3p² − (d/6)q²(n−1) = −1, with solvability conditions depending on d/6 modulo 4 or 8.